Proof. The existence of a unique short-time solution \(u \in C^{2k+\alpha, k+\alpha/2}(\overline{X_S})\) when \(k=0,1\) for some maximal time \(T_{max} > 0\) and any \(S < T_{max}\) follows from the standard theory of fully non-linear parabolic equations with first-order corner compatibility conditions (cf. [53]). Our primary task is to establish the higher-order corner regularity, specifically \(u \in C^{4+\alpha, 2+\alpha/2}(\overline{X_S})\) under the second-order compatibility condition (\(k=2\)), via a bootstrapping argument.

Let \(v := \partial_t u\). Differentiating the complex Hessian flow equation 5 with respect to \(t\), we observe that \(v\) satisfies the linearized parabolic equation: \[\label{eq:linearized95v} \partial_t v = F^{i\bar{j}}(A[u]) v_{i\bar{j}} - \partial_t \psi \quad \text{in } X_S,\tag{8}\] with boundary condition \(v = \partial_t \varphi\) on \(\partial_s X_S\) and initial condition \(v(\cdot, 0) = v_1\) on \(\overline{X}\).

Since we already know from the base step that \(u \in C^{2+\alpha, 1+\alpha/2}(\overline{X_S})\), the coefficients \(F^{i\bar{j}}(A[u])\) of the linearized operator naturally belong to the Hölder space \(C^{\alpha, \alpha/2}(\overline{X_S})\). The source term \(\partial_t \psi\) is smooth, hence also in \(C^{\alpha, \alpha/2}(\overline{X_S})\). Thus, equation 8 is a linear, uniformly parabolic equation with \(C^{\alpha, \alpha/2}\) coefficients.

To obtain \(v \in C^{2+\alpha, 1+\alpha/2}(\overline{X_S})\) from the linear Schauder theory (cf. [53]), the Cauchy-Dirichlet data for \(v\) must satisfy the first-order compatibility condition at the corner \(\partial X \times \{0\}\). That is to say: \[\partial_t (\partial_t \varphi)(0,z) = F^{i\bar{j}}(A[\varphi_0])(v_1)_{i\bar{j}}(z) - \partial_t \psi(0,z), \quad \forall z \in \partial X.\] This is precisely the second-order compatibility condition 7 assumed in the theorem. It follows that \(v = \partial_t u \in C^{2+\alpha, 1+\alpha/2}(\overline{X_S})\).

Finally, we return to the original nonlinear equation, which can be written as \[F(A[u]) = \partial_t u + \psi.\] Since we have already obtained \(\partial_t u \in C^{2+\alpha, 1+\alpha/2}(\overline{X_S})\) and \(\psi\) is smooth, the right-hand side belongs to \(C^{2+\alpha, 1+\alpha/2}(\overline{X_S})\).

Thus, by the parabolic Evans–Krylov theory for concave uniformly parabolic equations in domains [53]–[56], and equivalently by the complex-geometric perturbative estimates of Tosatti–Wang–Weinkove–Yang and Chu [57], [58], we first obtain \(u \in C^{2+\alpha,1+\alpha/2}(\overline{X_S})\) with quantitative control. Then, differentiating the equation in space and time and applying linear Schauder estimates to the resulting equations for derivatives of \(u\), we can bootstrap the regularity. In particular, under the second-order compatibility condition, this yields \[u \in C^{4+\alpha, 2+\alpha/2}(\overline{X_S}).\] Iterating this argument gives higher-order regularity for arbitrary \(k\). ◻

Proof. Differentiating the complex Hessian flow 9 with respect to \(t\), we observe that \(v := \partial_t u\) satisfies the following linearized parabolic equation: \[\partial_t v = \partial_t F(A[u]) - \partial_t\psi = F^{i\bar{j}} \partial_t(u_{i\bar{j}}) - \partial_t\psi = F^{i\bar{j}} v_{i\bar{j}} - \partial_t\psi.\] Define the linear uniformly parabolic operator \(L := \partial_t - F^{i\bar{j}} \partial_i \partial_{\bar{j}}\). Let \(B := \sup_{X_T} |\partial_t \psi|\). It follows that: \[L(v - Bt) = -\partial_t \psi - B \le 0 \quad \text{and} \quad L(v + Bt) = -\partial_t \psi + B \ge 0.\] By the parabolic weak maximum principle, the maximum of subsolutions and the minimum of supersolutions must be attained on the parabolic boundary \(\partial_0 X_T\). Therefore, we have: \[\sup_{X_T} (v - Bt) \le \sup_{\partial_0 X_T} (v - Bt) \quad \text{and} \quad \inf_{X_T} (v + Bt) \ge \inf_{\partial_0 X_T} (v + Bt).\] Since \(0 \leq t \leq T\), this immediately yields the global bounds in \(X_T\): \[\label{eq:v95bound95intermediate} \inf_{\partial_0 X_T} v - BT \le v(z, t) \le \sup_{\partial_0 X_T} v + BT \quad \text{in } X_T.\tag{10}\] because \(\partial_tu=\partial_t\varphi\) on \(\partial_sX_T\) and \(\partial_tu=F(A[u(0,z)])-\psi(0,z)=F(A[\varphi_0(z)])-\psi(0,z)\) on \(\partial_bX_T\) are both well dominated. This gives the uniform bound of \(v\). ◻

Proof. As in [39], let \(b \in C^\infty(\overline{X_T})\) be the solution of the following linear elliptic Dirichlet problem, solved slice-wise for each \(t \in [0, T]\): \[\label{construction32of32b} \Delta_\omega b + \operatorname{tr}_\omega\chi = 0 \quad \operatorname{in} X, \quad b = \varphi \quad \operatorname{on} \partial X.\tag{11}\] The existence and smoothness of \(b\) are guaranteed by standard linear elliptic theory, since \(\varphi\) is smooth in both space and time. We claim that \(\underline{u} \le u \le b\) on \(\overline{X_T}\).

For the lower bound, since \(\underline{u}\) is an admissible parabolic subsolution, we have \(\partial_t \underline{u} - F(A[\underline{u}]) \le -\psi = \partial_t u - F(A[u])\) in \(X_T\). On the parabolic boundary \(\partial_0 X_T\), the definition requires \(\underline{u} \le u\). The parabolic comparison principle immediately yields \(\underline{u} \le u\) in \(\overline{X_T}\).

For the upper bound, we can check it slice-wise. Fix a time \(t\in[0,T)\), it follows from the definition of \(b\) that the eigenvalues of \(g^{j\overline{k}}(\chi_{j\overline{k}}+b_{j\overline{k}})\) lie outside \(\{\lambda,\sigma_m(\lambda)\geq e^{\partial_tu+\psi}\}\) due to 1. The comparison principle [40] (see also [33]) then yields the desired estimate \[\sup_{X_T}|u|\leq C\] and we have concluded the proof. ◻

Proof of 3. Set \(v := u - \underline{u}\). Since we are dealing with the Cauchy-Dirichlet problem, \(v\) is uniformly bounded globally by the \(C^0\) estimates (2).

Consider the following test function on \(X_T\): \[G := \log\lambda_1 + \phi(|\nabla u|^2) + \varphi(v),\] where \(\lambda_1\) is the largest eigenvalue of the endomorphism \(A[u]\) (suitably perturbed to be distinct, as in [38]), and the functions \(\phi, \varphi\) are defined as: \[\phi(s) := -\frac{1}{2}\log\left(1 - \frac{s}{2K_0}\right), \quad K_0 := \sup_{X_T}(1 + |\nabla u|^2), \quad \varphi(v) := D_1 e^{-D_2 v},\] for some large constants \(D_1, D_2 > 0\) to be chosen.

Suppose that \(G\) attains its maximum at a point \((x_0, t_0)\). If \((x_0, t_0)\) lies on the parabolic boundary \(\partial_0 X_T\), we immediately obtain: \[\sup_{X_T} G \leq \sup_{\partial_0 X_T} G,\] which yields the desired estimate ?? by bounding \(\lambda_1\) in terms of its boundary values, the gradient bound \(K_0\), and the global \(C^0\) bound of \(v\).

If \((x_0, t_0)\) lies in the interior of \(X \times (0, T]\), we evaluate the linearized parabolic operator \(\mathcal{L} := -\partial_t + F^{k\bar{k}}\nabla_k\nabla_{\bar{k}}\) at this maximum point. At \((x_0, t_0)\), we have \(\nabla G = 0\) and \(\mathcal{L} G \leq 0\).

The calculation of \(\mathcal{L} G\) is purely local and follows word-for-word the computation in [41]. The spatial derivatives of the equation generate terms involving the curvature and torsion of \(\omega\), the derivatives of \(\chi\), and the spatial derivatives of \(\psi\). Because \(v\) is globally bounded independently of time, we can select \(D_2 \gg 1\) and \(D_1 \gg 1\) such that \(\varphi'(v) < 0\) and \(\varphi''(v) \gg |\varphi'(v)|\).

Dividing into two cases based on the eigenvalues (either \(\theta \lambda_1 \leq -\lambda_n\) or \(\theta \lambda_1 > -\lambda_n\) for a small \(\theta > 0\)) and invoking 2 in the latter case to control the positive sum of \(F^{i\bar{i}}\), we derive: \[0 \geq \mathcal{L}G \geq c_1 \frac{\lambda_1^2}{K_0} - c_2,\] where \(c_1, c_2\) are uniform positive constants depending on the quantities specified in the proposition statement. This forces: \[\lambda_1(x_0, t_0) \leq C K_0^{\frac{1}{2}} \leq C K_0.\] This implies the global bound ?? across \(X_T\). ◻

Proof. The argument is a generalization of [40] with some simplifications in the parabolic case. We first show the last two statements. The comparison principle yields \(u\geq\underline{u}\). By a Taylor expansion, we deduce that \(d(z)\leq A|z|\) for some constant \(A>0\) in \(\Omega_\delta\) if we take \(\delta\) small enough. This implies that \(v\geq d(c_0-Nd)\geq d(c_0-NA|z|)\geq d(c_0-NA\delta)\geq\frac{c_0}{2}d\geq0\) if we take further \[\label{choice32of32delta} \delta<\frac{c_0}{2NA}.\tag{16}\] Now we deal with the first statement. We work at a point \(p\in\Omega_{\delta}\times\{t_0\}\) with coordinates such that \(g_{j\bar{k}}=\delta_{jk}\) and \(h_{j\bar{k}}=\lambda_j\delta_{j\bar{k}}\). We can calculate \[F^{p\bar{q}}v_{p\bar{q}}= F^{p\bar{q}}(u-\underline{u})_{p\bar{q}}+c_0 F^{p\bar{q}}d_{p\bar{q}}-2NF^{p\bar{q}}d_pd_{\bar{q}}-2NdF^{p\bar{q}}d_{p\bar{q}}.\] For the first term, note that \(F^{1\bar{1}}\leq...\leq F^{n\bar{n}}\) (see e.g. equation (67) in [38]), we have by the Schur-Horn’s Lemma \[F^{p\bar{q}}(u-\underline{u})_{p\bar{q}}=F^{p\bar{p}}(\lambda_p-(\chi_{p\bar{p}}+\underline{u}_{p\bar{p}}))\leq F^{p\bar{p}}\lambda_p-F^{p\bar{p}}\underline{\lambda}_p=-\tau\mathcal{F}+F^{p\bar{p}}(\lambda_p-(\underline{\lambda}_p-\tau)) ,\] for each \(\tau>0\). Here, we use the notation \(\underline{\lambda}_1\geq...\geq\underline{\lambda}_n\) to denote the eigenvalues of \(\chi+\sqrt{-1}\partial\bar{\partial}\underline{u}\) with respect to \(\omega\) at a point. Now we choose \(\tau>0\) such that \(\underline{\lambda}-\tau\boldsymbol{1}\in\Gamma_m(\omega)\). Indeed, by the definition of subsolutions and the uniform bounds on \(\|\partial_t\underline{u}\|_\infty,\|\psi\|_\infty\), we have \[\label{subsolution32l} \chi_{\underline{u}}^m\wedge\omega^{n-m}\geq e^{\partial_t\underline{u}+\psi}\omega^n\geq C^{-1}\omega^n.\tag{17}\] Gårding’s inequality yields furthermore that \[\chi_{\underline{u}}^k\wedge\omega^{n-k}\geq C^{-\frac{k}{m}}\omega^n\] for each \(1\leq k\leq m\). As we have \((\chi_{\underline{u}}-\tau\boldsymbol{1})^k\wedge\omega^{n-k}=\chi_{\underline{u}}^k\wedge\omega^{n-k}+O(\tau)\), the constant \(\tau>0\) can be chosen uniformly. By the concavity of \(f(\lambda)=\log\sigma_m(\lambda)\) on \(\Gamma_m(\omega)\), we can write \[F^{p\bar{p}}(\lambda)[\lambda_p-(\underline{\lambda}_p-\tau)]\leq F(\lambda)-F(\underline{\lambda}-\tau\boldsymbol{1})=\partial_tu+\psi-F(\underline{\lambda}-\tau\boldsymbol{1}).\] This implies that \[\label{eq:estimate1} F^{p\bar{q}}(u-\underline{u})_{p\bar{q}}\leq-\tau\mathcal{F}+C.\tag{18}\] Next, since \(d\) is smooth in \(\Omega_\delta\) for \(\delta>0\) small enough and \(|\nabla d|=\frac{1}{2}\), we can estimate the remaining terms as follows: \[\label{eq:estimate2} c_0 F^{p\bar{q}}d_{p\bar{q}}-2NdF^{p\bar{q}}d_{p\bar{q}}\leq c_0C\mathcal{F}+NdC\mathcal{F},\tag{19}\] and \[\label{eq:estimate323} -2NF^{p\bar{q}}d_pd_{\bar{q}}\leq-2NF^{1\bar{1}}|\partial d|^2=-\frac{N}{2}F^{1\bar{1}}.\tag{20}\] Combining 18 , 19 , 20 we arrive at \[F^{p\bar{q}}v_{p\bar{q}}\leq-\tau\mathcal{F}-\frac{N}{2}F^{1\bar{1}}+C(c_0+Nd)\mathcal{F}+C.\] Since by our \(\partial_tu\)- estimate we have \(|\partial_tv|=|\partial_t(u-\underline{u})|\leq C\), we can infer furthermore that \[\mathcal{L}v=-\partial_tv+F^{p\overline{q}}v_{p\overline{q}}\leq-\tau\mathcal{F}-\frac{N}{2}F^{1\bar{1}}+C(c_0+Nd)\mathcal{F}+C.\] Now, since \(d(z) \leq A|z|\) for some constant \(A>0\) in \(\Omega_\delta\), we can choose \(c_0\) and \(\delta\) sufficiently small such that \[C(c_0 + NA\delta) \leq \frac{\tau}{4}.\] This allows us to absorb the positive \(\mathcal{F}\) error term, yielding: \[\mathcal{L}v \leq -\frac{3\tau}{4}\mathcal{F} - \frac{N}{2}F^{1\bar{1}} + C.\] To control the remaining terms, we write: \[-\frac{\tau}{2}\mathcal{F} - \frac{N}{2}F^{1\bar{1}} = - \sum_{i=1}^n F^{i\bar{i}} q_i = - \frac{1}{\sigma_m(\lambda)} \sum_{i=1}^n \sigma_{m-1}(\lambda|i) q_i,\] where \(q = \frac{1}{2}(\tau+N, \tau, \dots, \tau)\). By Gårding’s inequality [60], we have: \[\sum_{i=1}^n \sigma_{m-1}(\lambda|i) q_i \geq m \sigma_m(\lambda)^{\frac{m-1}{m}} \sigma_m(q)^{\frac{1}{m}}.\] Dividing both sides by \(\sigma_m(\lambda)\), we get: \[-\frac{\tau}{2}\mathcal{F} - \frac{N}{2}F^{1\bar{1}} \leq -m \sigma_m(\lambda)^{-\frac{1}{m}} \sigma_m(q)^{\frac{1}{m}}.\] Recall that \(\sigma_m(\lambda) = e^{\partial_t u + \psi}\). Thanks to our uniform \(\partial_t u\) estimate (1) and the boundedness of \(\psi\), we have an upper bound \(\sigma_m(\lambda) \leq C_1\) for some uniform constant \(C_1 > 0\). Thus, \(\sigma_m(\lambda)^{-\frac{1}{m}} \geq C_1^{-1/m} > 0\).

On the other hand, a direct calculation of \(\sigma_m(q)\) gives: \[\sigma_m(q) = \frac{1}{2^m} \left[ \tau^{m-1}(\tau+N)\binom{n-1}{m-1} + \tau^m\binom{n-1}{m} \right] \geq \frac{N \tau^{m-1}}{2^m}.\] Therefore, we obtain the estimate: \[-\frac{\tau}{2}\mathcal{F} - \frac{N}{2}F^{1\bar{1}} \leq - \frac{m}{2} C_1^{-1/m} \tau^{\frac{m-1}{m}} N^{\frac{1}{m}}.\] By choosing \(N\) sufficiently large (depending on \(m\), \(\tau\), \(C\), and \(C_1\)), we can ensure that: \[-\frac{\tau}{2}\mathcal{F} - \frac{N}{2}F^{1\bar{1}} \leq -C - \frac{\tau}{4}.\] Substituting this back into our estimate for \(\mathcal{L}v\), we arrive at: \[\mathcal{L}v \leq -\frac{\tau}{4}\mathcal{F} - \frac{\tau}{4} = -\frac{\tau}{4}(1+\mathcal{F}).\] Renaming \(\frac{\tau}{4}\) as our new \(\tau\), we conclude that \(\mathcal{L}v \leq -\tau(1+\mathcal{F})\) on \(\Omega_{\delta,p}\). Finally, we remark that all the constants \(\delta, c_0, N, \tau\) can be chosen uniformly and independently of the time \(t\), owing to the short time existence and the fact that our coordinates depend continuously on \(t\). ◻

Proof. By a basic calculation, we can write \[\begin{align} \partial_t(T_\alpha u)=\partial_{s_\alpha}\partial_tu-\frac{\rho_{s_\alpha}}{\rho_{x_n}}\partial_{x_n}\partial_tu=F^{p\bar{q}}(\nabla_{s_\alpha}\chi_{p\bar{q}}+\nabla_{s_\alpha}u_{p\bar{q}})-\frac{\rho_{s_\alpha}}{\rho_{x_n}}F^{p\bar{q}}(\nabla_{x_n}\chi_{p\bar{q}}+\nabla_{x_n}u_{p\bar{q}})+\frac{\rho_{s_\alpha}}{\rho_{x_n}}\psi_{x_n}. \end{align}\] While \[\begin{align} F^{p\bar{q}}\partial_p\partial_{\bar{q}}T_\alpha(u)=F^{p\bar{q}}u_{p\bar{q}s_\alpha}-\frac{\rho_{s_\alpha}}{\rho_{x_n}}F^{p\bar{q}}u_{p\bar{q}x_n}-2\operatorname{Re}\left(F^{p\bar{q}}\left(\frac{\rho_{s_\alpha}}{\rho_{x_n}}\right)_pu_{\bar{q}x_n}\right)-F^{p\bar{q}}\left(\frac{\rho_{s_\alpha}}{\rho_{x_n}}\right)_{p\bar{q}}u_{x_n}. \end{align}\] Here we have made the following calculation: \[\begin{align} \nabla_{s_\alpha}u_{p\bar{q}}:&=(\nabla_{s_\alpha}u)(\partial_p,\partial_{\bar{q}})=u_{p\bar{q}s_\alpha}-\frac{\partial g^{p\bar{j}}}{\partial s_\alpha}u_{j\bar{q}}=u_{p\bar{q}s_\alpha}-\frac{1}{2}\left(\frac{\partial g^{p\bar{j}}}{\partial z_i}+\frac{\partial g^{p\bar{j}}}{\partial \bar{z}_i}\right)u_{j\bar{q}}\\ &=u_{p\bar{q}s_\alpha}-\frac{1}{2}\left(\Gamma_{ip}^j+\Gamma_{\bar{i}\bar{p}}^{\bar{j}}\right)u_{j\bar{q}}, \end{align}\] if \(s_\alpha=x_i=\frac{1}{2}(z_i+\bar{z}_i)\). Similarly we have the corresponding formula for \(\nabla_{x_n}u_{p\bar{q}}\). Adding both equations we can cancel the third-order derivatives of \(u\) and write \[\begin{align} \mathcal{L}({T_\alpha(u)})= \mathcal{E}+\frac{1}{2}F^{p\bar{q}}\left(\Gamma_{ip}^j+\Gamma_{\bar{i}\bar{p}}^{\bar{j}}\right)u_{j\bar{q}}+F^{p\bar{q}}\frac{1}{2}\left(\Gamma_{ip}^j+\Gamma_{\bar{i}\bar{p}}^{\bar{j}}\right)u_{j\bar{q}}-2\operatorname{Re}\left(F^{p\bar{q}}\left(\frac{\rho_{s_\alpha}}{\rho_{x_n}}\right)_pu_{\bar{q}x_n}\right). \end{align}\] As in [40], we may further estimate \[\begin{align} |F^{p\bar{q}}\Gamma_{ip}^ju_{j\bar{q}}|=\left|\frac{1}{\sigma_m}\sum_k\sigma_{m-1}(\lambda|k)\lambda_kS^k_j\Gamma^{j}_{ip}\bar{S^k_p}\right|\leq C\sum_k\sigma_{m-1}(\lambda|k)|\lambda_k|, \end{align}\] where \(S=S^k_p\) is a unitary matrix which simultaneously diagonalizes \(u_{j\bar{q}}\) and \(F^{p\bar{q}}\) and we have used that fact that \(\sigma_m=e^{\partial_tu+\psi}\) is uniformly bounded.

Since \(\underline{u}\in C^\infty(X_T)\), we infer that \[|\mathcal{L}(T_\alpha(u-\underline{u}))|=-2\operatorname{Re}\left(F^{p\bar{q}}\left(\frac{\rho_{s_\alpha}}{\rho_{x_n}}\right)_p(u-\underline{u})_{\bar{q}x_n}\right)+\mathcal{E}.\] Note that \(\frac{\partial}{\partial z_n}=\frac{1}{2}(\frac{\partial}{\partial x_n}-\sqrt{-1}\frac{\partial}{\partial y_n})\), we can write \[F^{p\bar{q}}\left(\frac{\rho_{s_\alpha}}{\rho_{x_n}}\right)_p(u-\underline{u})_{\bar{q}x_n}=2F^{p\bar{q}}\left(\frac{\rho_{s_\alpha}}{\rho_{x_n}}\right)_p(u-\underline{u})_{\bar{q}n}+\sqrt{-1}F^{p\bar{q}}\left(\frac{\rho_{s_\alpha}}{\rho_{x_n}}\right)_p(u-\underline{u})_{\bar{q}y_n}.\] The argument above gives that the first term can be dominated by \(C\sum_k\sigma_{m-1}(\lambda|k)|\lambda_k|\), while the Cauchy-Schwarz inequality yields that \[\left|F^{p\bar{q}}(\frac{\rho_{s_\alpha}}{\rho_{x_n}})_p(u-\underline{u})_{\bar{q}y_n})\right|\leq \frac{1}{K^{\frac{1}{2}}}F^{p\bar{q}}(u-\underline{u})_{py_n}(u-\underline{u})_{\bar{q}y_n}+\mathcal{E}.\] The proof is finally concluded. ◻

Proof. We calculate the action of the linearized parabolic operator \(\mathcal{L} = -\partial_t + F^{p\bar{q}}\partial_p\partial_{\bar{q}}\) on the term \(\frac{1}{K^{1/2}}(\partial_{y_i}(u-\underline{u}))^2\). First, we compute the time derivative: \[\begin{align} \frac{1}{K^{\frac{1}{2}}}\partial_t(\partial_{y_i}(u-\underline{u}))^2 &= \frac{2}{K^{\frac{1}{2}}}(u-\underline{u})_{y_i}\partial_{y_i}(\partial_tu-\partial_t\underline{u}) \\ &= \frac{2}{K^{\frac{1}{2}}}(u-\underline{u})_{y_i}\left(\partial_{y_i}F(A[u]) - \psi_{y_i} - \partial_{y_i}\partial_t\underline{u}\right) \\ &= \frac{2}{K^{\frac{1}{2}}}(u-\underline{u})_{y_i}\left(F^{p\bar{q}}(\nabla_{y_i}\chi_{p\bar{q}} + \nabla_{y_i}u_{p\bar{q}}) - \psi_{y_i} - \partial_{y_i}\partial_t\underline{u}\right) + \mathcal{E}. \end{align}\] Here, passing from partial derivatives to covariant derivatives introduces commutator terms involving the Christoffel symbols of \(\omega\). As demonstrated in the proof of 4, these terms are linear in \(u_{j\bar{k}}\) and can be bounded by \(C\sum_k\sigma_{m-1}(\lambda|k)|\lambda_k|\), thus they are absorbed into the error term \(\mathcal{E}\).

Next, we compute the spatial second-order derivatives: \[\begin{align} \frac{1}{K^{\frac{1}{2}}}F^{p\bar{q}}\partial_p\partial_{\bar{q}}(\partial_{y_i}(u-\underline{u}))^2 &= \frac{2}{K^{\frac{1}{2}}}F^{p\bar{q}}(u-\underline{u})_{y_ip}(u-\underline{u})_{y_i\bar{q}} + \frac{2}{K^{\frac{1}{2}}}F^{p\bar{q}}(u-\underline{u})_{y_i}(u-\underline{u})_{y_ip\bar{q}}. \end{align}\] Adding both terms together, the third-order derivatives of \(u\) cancel exactly, leaving: \[\begin{align} \mathcal{L}\left(\frac{1}{K^{\frac{1}{2}}}(\partial_{y_i}(u-\underline{u}))^2\right) &\geq \frac{2}{K^{\frac{1}{2}}}F^{p\bar{q}}(u-\underline{u})_{y_ip}(u-\underline{u})_{y_i\bar{q}} \\ &\quad - \frac{2}{K^{\frac{1}{2}}}|(u-\underline{u})_{y_i}|\cdot\left| F^{p\bar{q}}\nabla_{y_i}\chi_{p\bar{q}} - \psi_{y_i} - \partial_{y_i}\partial_t\underline{u} \right| - |\mathcal{E}|. \end{align}\] By the definition of \(K\), we have \(|(u-\underline{u})_{y_i}| \leq CK^{1/2}\). Therefore, the remaining first-order term is uniformly bounded by a constant multiple of \(\mathcal{F}\) and can be absorbed into \(-|\mathcal{E}|\). This yields ?? . ◻

Proof. We begin by computing the time derivative: \[\begin{align} \partial_t|\nabla_a(u-\underline{u})|^2 &= \partial_t\left(e_a^i\bar{e_a^j}(u-\underline{u})_i(u-\underline{u})_{\bar{j}}\right)\\ &= e_a^i\bar{e_a^j}\left[F^{p\bar{q}}(\nabla_i u_{p\bar{q}} + \nabla_i\chi_{p\bar{q}}) - \psi_i - \underline{u}_{it}\right](u-\underline{u})_{\bar{j}} \\ &\quad + e_a^i\bar{e_a^j}\left[F^{p\bar{q}}(\nabla_{\bar{j}} u_{p\bar{q}} + \nabla_{\bar{j}}\chi_{p\bar{q}}) - \psi_{\bar{j}} - \underline{u}_{\bar{j}t}\right](u-\underline{u})_i + \mathcal{E}. \end{align}\] Next, differentiating spatially gives: \[\begin{align} F^{p\bar{q}} \partial_p \partial_{\bar{q}} |\nabla_a (u - \underline{u})|^2 &= F^{p\bar{q}} \partial_p \partial_{\bar{q}} (e_i^a\bar{e_j^a})(u - \underline{u})_i (u - \underline{u})_{\bar{j}} + F^{p\bar{q}} \partial_{\bar{q}} (e_i^a \bar{e_j^a})(u - \underline{u})_{pi} (u - \underline{u})_{\bar{j}} \\ &\quad + F^{p\bar{q}} \partial_{\bar{q}} (e_i^a\bar{e_j^a})(u - \underline{u})_i (u - \underline{u})_{\bar{j}p} + F^{p\bar{q}} \partial_p (e_i^a\bar{e_j^a})(u - \underline{u})_{\bar{q}i} (u - \underline{u})_{\bar{j}} \\ &\quad + F^{p\bar{q}} (e_i^a\bar{e_j^a})(u - \underline{u})_{p\bar{q}i} (u - \underline{u})_{\bar{j}} + F^{p\bar{q}} (e_i^a \bar{e_j^a})(u - \underline{u})_{\bar{q}i} (u - \underline{u})_{\bar{j}p} \\ &\quad + F^{p\bar{q}} \partial_p (e_i^a \bar{e_j^a})(u - \underline{u})_i (u - \underline{u})_{\bar{q}\bar{j}} + F^{p\bar{q}} e_i^a \bar{e_j^a} (u - \underline{u})_{pi} (u - \underline{u})_{\bar{q}\bar{j}} \\ &\quad + F^{p\bar{q}} e_i^a \bar{e_j^a} (u - \underline{u})_i (u - \underline{u})_{p\bar{q}\bar{j}}. \end{align}\] Summing the terms inside the operator \(\mathcal{L}\), the third-order derivatives of \(u\) cancel directly: \[\begin{align} \mathcal{L}\left(\frac{1}{K^{\frac{1}{2}}}|\nabla_a (u - \underline{u})|^2\right) &= \frac{1}{K^{\frac{1}{2}}}F^{p\bar{q}} (e_i^a \bar{e_j^a})(u - \underline{u})_{\bar{q}i} (u - \underline{u})_{\bar{j}p} + \frac{1}{K^{\frac{1}{2}}}F^{p\bar{q}} e_i^a \bar{e_j^a} (u - \underline{u})_{pi} (u - \underline{u})_{\bar{q}\bar{j}} \\ &\quad + \frac{2}{K^{\frac{1}{2}}}\operatorname{Re}F^{p\bar{q}} \partial_{\bar{q}} (e_i^a \bar{e_j^a})(u - \underline{u})_{pi} (u - \underline{u})_{\bar{j}} + \mathcal{E}. \end{align}\] This identity is precisely the parabolic counterpart to Equation (4.27) in [40]. To extract the positivity, note that \(F^{p\bar{q}} = \frac{\sigma_m^{p\bar{q}}}{\sigma_m(\lambda)}\). Because the time derivative bound (1) strictly bounds \(\sigma_m(\lambda) = e^{\partial_t u + \psi}\) away from zero and infinity, we can carry over the algebraic manipulations from [40]. ◻

Proof. We argue by contradiction. It is clear that we can assume without loss of generality that \(T<\infty\). Suppose that there are points \(p_i\in X\) and positive times \(t_i\) with \(p_i\to p_\infty\in\overline{X}\) and \(t_i\to t_\infty\in(0,T]\) such that \(|\nabla u|(p_i,t_i)\to\infty\). Up to a new choice and rescaling we may suppose that \[|\nabla u|(p_i,t_i)=\sup_{X_{t_i}}|\nabla u|:=M_i\to\infty.\] Following the philosophy of [40], there are two cases to deal with:

Case 1. We first assume that \(p_\infty\) lies in the interior of \(X\), where the situation is similar to that in [41]. We can then select a coordinate chart \(B\) centered at \(p_\infty\) (that is, we identify \(p_\infty\) and the origin \(0\)) such that \(\omega(0)=Id\). Since \(p_i\to p_\infty\), we may assume without loss of generality that all the \(p_i\) lie in \(B\). Now consider the complex space \(\mathbb{C}^n\), for each \(R>0\), define a function \(u_i:B_R(0)\to\mathbb{R}\) by \[u_i(w):=u(M_i^{-1}w+p_i,t_i).\] It follows that \(|\nabla u_i|(w)=M_i^{-1}|\nabla u|(M_i^{-1}w+p_i,t_i)\leq1\) and \(|\nabla u_i|(0)=1\). By the \(C^0\)-estimates, we have \(\|u_i\|_\infty\leq C\). The interior estimates 3 combined with all the boundary second-order estimates then give that \[\begin{align} \|\sqrt{-1}\partial\bar{\partial}u_i\|_\infty\leq CM_i^{-2}\sup_{X_{t_i}}|\sqrt{-1}\partial\bar{\partial}u|\leq CM_i^{-2}(1+\sup_{X_{t_i}}|\nabla u|^2)=CM_i^{-2}(1+M_i^2)\leq C. \end{align}\] The Laplacian estimate combined with the Sobolev embedding theorem yields a Hölder norm bound of \(u_i\): \[\|u_i\|_{C^{1,\alpha}\left(B_{\frac{R}{2}}(0)\right)}\leq C,\] for some \(0<\alpha<1\). Therefore, one can extract a subsequence of \(u_i\) that converges in \(C^{1,\frac{\alpha}{2}}\) norm to a function \(u_\infty\) defined on \(B_{\frac{R}{2}}(0)\). Letting \(R\to\infty\) and using a diagonal argument we get a function \(u_\infty:\mathbb{C}^n\to\mathbb{R}\) with \(|\nabla u_\infty|(0)=1\).

At the time \(t_i\), we have the following equation on \(X\): \[\label{eq:Hessian32slice} (\chi(z)+\sqrt{-1}\partial\bar{\partial}u(z,t_i))^m\wedge\omega(z)^{n-m}=e^{\partial_tu(z,t_i)+\psi(z,t_i)}\omega(z)^n.\tag{25}\] Define a biholomorphic map \(\Phi_i:\mathbb{C}^n\to\mathbb{C}^n\) by \[\Phi_i(w):=M_i^{-1}w+p_i.\] Pulling back equation 25 under \(\Phi_i\), the form \(\chi(z)=\sqrt{-1}\chi_{j\bar{k}}dz^j\wedge d\bar{z}^k\) is changed to \(M_i^{-2}\sqrt{-1}\tilde{\chi}_{j\bar{k}}^i(w)dw^j\wedge d\bar{w}^k\), where \(\tilde{\chi}_{j\bar{k}}^i(w)=\chi_{j\bar{k}}(M_i^{-1}w+p_i)\). We obtain that \[\begin{align} &\left(\frac{\sqrt{-1}}{M_i^2}\tilde{\chi}_{j\bar{k}}^i(w)dw^j\wedge d\bar{w}^k+\sqrt{-1}\partial\bar{\partial}u_i(w)\right)^m\wedge\left(\frac{\sqrt{-1}}{M_i^2}\tilde{\omega}_{j\bar{k}}^i(w)dw^j\wedge d\bar{w}^k\right)^{n-m}\\ =&e^{\partial_tu(M_i^{-1}w+p_i,t_i)+\psi(M_i^{-1}w+p_i,t_i)}\left(\frac{\sqrt{-1}}{M_i^2}\tilde{\omega}_{j\bar{k}}^i(w)dw^j\wedge d\bar{w}^k\right)^{n}. \end{align}\] Multiplying both sides of the equation by \(M_i^{2(n-m)}\), the left-hand side stabilizes, while the right-hand side acquires a factor of \(M_i^{-2m}\). Since \(m \geq 1\) and the term \(e^{\partial_t u + \psi}\) is uniformly bounded globally thanks to our a priori \(\partial_t u\)-estimates, the right-hand side strictly converges to \(0\) as \(M_i \to \infty\).

It is clear by definition that \(M_i^2 \left(\frac{\sqrt{-1}}{M_i^2}\tilde{\omega}_{j\bar{k}}^i(w)dw^j\wedge d\bar{w}^k\right) \to \beta = \sqrt{-1}\partial\bar{\partial}|w|^2\) uniformly on compact sets of \(\mathbb{C}^n\). Therefore, letting \(i\to\infty\) we get by standard Bedford-Taylor convergence that the following holds on the whole \(\mathbb{C}^n\): \[(\sqrt{-1}\partial\bar{\partial}u_\infty)^m\wedge(\sqrt{-1}\partial\bar{\partial}|w|^2)^{n-m}=0.\] The Liouville-type theorem for \(m\)-subharmonic functions established in [42] then gives that \(u_\infty\) is constant, contradicting the fact that \(|\nabla u_\infty|(0)=1\).

Case 2. We now deal with the case where \(p_\infty\in\partial X\). Let \(\Omega=B_{2s}\cap\{\rho\leq0\}\) be a boundary coordinate chart centered at \(p_\infty\) such that \(\omega(p_\infty)=Id\). Recall that by our choice of coordinates, the inner normal direction is given by the real part \(x_n\). We may assume without loss of generality that all the \(p_i\) lie in \(B_{s}\cap\{\rho\leq0\}\). Choose a point \(y_i\in\partial X\cap\Omega\) minimizing the Euclidean distance between \(p_i\) and \(\partial X\) with \(r_i:=|y_i-p_i|\).

Case 2a. Assume that \(\liminf_{i}M_ir_i=\infty\). Set \[u_i(z):=u(M_i^{-1}z+p_i, t_i).\] As in the first case, we have \(|\nabla u_i(0)|=1\), \(|\nabla u_i|\leq 1\) and \[\label{eq:case2a} \|u_i\|_{L^\infty(\Omega_i)}+\|\Delta u_i\|_{L^{\infty}(\Omega_i)}\leq C,\tag{26}\] where \(\Omega_i:=\{z:M_i^{-1}z+p_i\in B_{2s}\}\cap\{\rho\leq0\}\). Set \(\hat{\rho}_i(z):=\rho(M_i^{-1}z+p_i)\). It is easy to see that we have an inclusion of sets: \[B_{\frac{M_i r_i}{2}}(0) \subset B_{s M_i}(0)\cap\{\hat{\rho}_i\leq0\}\subset\Omega_i.\] Since \(\liminf_{i}M_ir_i=\infty\), the sequence of domains \(B_{sM_i}\cap\{\hat{\rho}_i\leq0\}\) exhausts the whole space \(\mathbb{C}^n\). Invoking 26 , standard elliptic theory and the Sobolev embedding theorem gives a constant \(0<\alpha<1\) such that \(\|u_i\|_{C^{1,\alpha}}\leq C\) uniformly on compact subsets and we can extract a \(C^{1,\frac{\alpha}{2}}\) limit \(u_\infty\) of \(u_i\) on every compact subset of \(\mathbb{C}^n\). The contradiction follows exactly as in Case 1 from the Liouville-type theorem.

Case 2b. Suppose now that \(\liminf_{i \to \infty} M_i r_i = l \in [0, \infty)\). Up to extracting a subsequence, we may assume \(M_i r_i \to l\). Choose a ball of radius \(r^*\) centered at a point \(y^*\) such that \(B_{r^*}(y^*)\subset B_{s}\cap\{\rho\leq0\}\) and that \(B_{r^*}(y^*)\) is tangent to \(\partial X\) at \(p_\infty\). Shrinking \(r^*\) if necessary we may also assume that there are points \(y_i^*\) such that \(B_{r^*}(y_i^*)\) is tangent to \(\partial X\) at \(y_i\). It is then clear that \(y_i\to p_\infty\) and that \(y_i^*\to y^*\). For each \(i\), we can find a unitary transformation (an isometry of \(\mathbb{C}^n\)) \(\mathcal{R}_i : \mathbb{C}^n \to \mathbb{C}^n\) such that \(\mathcal{R}_i(0) = y_i\), and the inward unit normal vector to \(\partial X\) at \(y_i\) is pulled back precisely to the positive real direction of the \(w_n\)-axis, i.e., \(\frac{\partial}{\partial \operatorname{Re}(w_n)}\). Since \(p_i\) and \(y_i^*\) lie exactly on the inward normal line originating from \(y_i\) at distance \(r_i\) and \(r^*\), their preimage under this transformation are strictly: \[\mathcal{R}_i^{-1}(p_i) = (0, \dots, 0, r_i),\quad \mathcal{R}_i^{-1}(y_i^*) = (0, \dots, 0, r^*).\] Now, we define the rescaled sequence \(u_i\) centered at \(y_i\) with the spatial scaling factor \(M_i\): \[u_i(w) := u\left(\mathcal{R}_i(M_i^{-1}w), t_i\right).\] The condition \(\mathcal{R}_i(M_i^{-1}w) \in B_{r^*}(y_i^*)\) is equivalent to \(M_i^{-1}w \in B_{r^*}((0, \dots, 0, r^*))\), which in turn means \(w \in M_i B_{r^*}((0, \dots, 0, r^*)) = B_{M_i r^*}((0, \dots, 0, M_i r^*))\). As \(M_i \to \infty\), the radius \(M_i r^* \to \infty\) and the centers move to infinity along the positive real \(w_n\)-axis. Consequently, this sequence of scaled balls exhaust the right half-space \(H := \{ w \in \mathbb{C}^n \mid \operatorname{Re}(w_n) > 0 \}\). Therefore, looking at the restriction of \(u\) to \(B_{r^*}(y_i^*)\), the domain of our rescaled functions \(u_i\) exhausts \(H\).

Observe that the original maximal gradient point \(p_i\) corresponds to the point \(w_i^*\) in our new coordinates, where: \[w_i^* = M_i \mathcal{R}_i^{-1}(p_i) = (0, \dots, 0, M_i r_i).\] Notice that \(w_i^* \to (0, \dots, 0, l)\) as \(i \to \infty\). By definition, we have \(|\nabla u_i|(w_i^*) = 1\) and \(|\nabla u_i| \leq 1\).

By standard elliptic theory and the Sobolev embedding theorem, we have a uniform \(C^{1, \alpha}\) bound on compact subsets of \({H}\). Thus, we can extract a subsequence of \(u_i\) converging uniformly in \(C^{1, \frac{\alpha}{2}}\) to a limit function \(u_\infty\) on \(H\cup\{0\}\). Due to the convergence of the points \(w_i^*\), the limit function must satisfy: \[\label{eq:case2b} |\nabla u_\infty|(0, \dots, 0, l) = 1.\tag{27}\]

The final contradiction is derived from the global boundary barriers. Recall that we have the global bound \(\underline{u} \leq u \leq b\) on \(X_T\). We define the corresponding rescaled functions for the subsolution and the upper barrier exactly as we did for \(u\): \[\underline{u}_i(w) := \underline{u}(\mathcal{R}_i(M_i^{-1}w), t_i), \quad \text{and} \quad b_i(w) := b(\mathcal{R}_i(M_i^{-1}w), t_i).\] This immediately implies that \(\underline{u}_i(w) \leq u_i(w) \leq b_i(w)\) on their domains of definition.

Fix an arbitrary compact subset \(K \subset H\cup\{0\}\). As \(M_i \to \infty\), for any \(w \in K\), the points \(\mathcal{R}_i(M_i^{-1}w)\) uniformly converge to \(p_\infty \in \partial X\). Recall that both \(\underline{u}\) and \(b\) are fixed smooth functions that smoothly coincide with the boundary data \(\varphi\) on the lateral boundary \(\partial_s X_T\). Because \(y_i \to p_\infty\) and \(t_i \to t_\infty\), both sequences of functions \(\underline{u}_i(w)\) and \(b_i(w)\) converge uniformly on \(K\) to the exact same constant value, namely \(\varphi(p_\infty, t_\infty)\).

Since \(\underline{u}_i(w) \leq u_i(w) \leq b_i(w)\), the squeeze theorem forces the limit function \(u_\infty(w)\) to be identically equal to the constant \(\varphi(p_\infty, t_\infty)\) on \(H\cup\{0\}\). This implies \(\nabla u_\infty \equiv 0\) everywhere, which contradicts 27 . The proof is thus complete. ◻

End of proof of 1. Suppose \(T_{\max}\) is the maximal existence time of the flow 5 . Combining 1, 2, and 7, we have established the following uniform a priori estimates for the smooth solution \(u\) on each finite time interval: \[\|u\|_{L^\infty(X_T)} +\|\partial_t u\|_{L^\infty(X_T)} +\|\nabla_\omega u\|_{L^\infty(X_T)} +\|\sqrt{-1}\partial\bar{\partial}u\|_{L^\infty(X_T)} \le C,\] where \(C\) depends only on the background data and on \(T\).

Since \[\sigma_m(\lambda[\chi_u]) = e^{\partial_tu+\psi}\] is bounded from above and below, and since \(\chi_u=\chi+dd^cu\) is uniformly bounded, the eigenvalues \(\lambda[\chi_u]\) remain in a compact subset of the admissible cone \(\Gamma_m\). Hence the linearized operator is uniformly elliptic, and the equation is uniformly parabolic.

We now apply the standard Evans–Krylov–Schauder regularity theory. This is a local issue. In holomorphic coordinate charts, and in boundary charts after flattening \(\partial X\), the complex Hessian flow can be viewed as a concave fully nonlinear parabolic equation in the real Hessian variables. More precisely, since the eigenvalues of \(\chi_u\) stay in a compact subset of \(\Gamma_m\), one may extend the operator off a small neighborhood of the compact set traced by the solution to obtain a concave uniformly parabolic operator in the real Hessian. This is the standard reduction used in the complex Evans–Krylov theory of Wang [62], Tosatti–Wang–Weinkove– Yang [57], and Chu [58]; near the boundary we use the same localization and flattening philosophy as in Collins–Picard [40].

The boundary \(C^{2+\alpha,1+\alpha/2}\) estimate then follows from the parabolic Evans–Krylov theory for concave uniformly parabolic Dirichlet problems, together with standard boundary Schauder theory; see Krylov [56] and Lieberman [53]. Consequently, for some \(\alpha\in(0,1)\), \[\|u\|_{C^{2+\alpha,1+\alpha/2}(\overline{X_T})}\le C.\] Here the estimate is first obtained away from the initial corner; the regularity up to \(t=0\) follows from the imposed compatibility conditions and the standard short-time theory.

Once this estimate is obtained, the coefficients of the linearized operator belong to \(C^{\alpha,\alpha/2}\) up to the parabolic boundary. Standard parabolic Schauder estimates then yield, for every integer \(k\ge0\), \[u\in C^\infty(X\times(0,T]) \cap C^{2k+\alpha,k+\alpha/2}(\overline{X_T}).\] The uniform estimates allow the solution to be continued past \(T_{\max}\) unless \(T_{\max}=T\). Hence the maximal existence time is the prescribed time, completing the proof. ◻

Proof of 1. Since \(M\) is a smooth \(m\)-pseudoconvex Hermitian manifold, we may choose a smooth defining function \(\rho\in C^\infty(\overline{M})\) such that \[\rho<0 \quad \text{in } M,\qquad \rho=0 \quad \text{on }\partial M,\] and \(\rho\) is strictly \((\omega,m)\)-subharmonic on \(\overline{ M}\). Set \[\chi:=dd^c\rho .\] Then \(\chi\) is a smooth strictly \((\omega,m)\)-positive \((1,1)\)-form on \(\overline{ M}\).

We now reduce ?? to the form treated in 1. Let \[v:=u-\rho.\] Since \(\rho\) is independent of \(t\), the equation ?? becomes \[\label{eq:transformed-flow} \begin{cases} \partial_t v =\log\displaystyle\frac{(\chi+dd^cv)^m\wedge\omega^{n-m}}{\omega^n}-\psi \quad \text{in } M_T,\\[1.2ex] v=\widehat{\varphi}\quad \text{on }\partial_0 M_T, \end{cases}\tag{28}\] where \[\widehat{\varphi}:=\varphi-\rho .\] Since \(\rho|_{\partial M}=0\), the lateral boundary datum of \(v\) is still \(\varphi\). Moreover, \[\widehat{\varphi}_0:=\widehat{\varphi}|_{\overline{ M}\times\{0\}} =\varphi_0-\rho,\] and \[\chi+dd^c\widehat{\varphi}_0 =dd^c\rho+dd^c(\varphi_0-\rho)=dd^c\varphi_0.\] Hence \(\widehat{\varphi}_0\) is strictly \((\chi,\omega,m)\)-subharmonic because \(\varphi_0\) is strictly \((\omega,m)\)-subharmonic by assumption. Also, since \(\rho\) is time-independent and vanishes on \(\partial M\), the compatibility conditions of order \(k\) for \(\varphi\) are inherited by \(\widehat{\varphi}\).

It therefore remains only to construct an admissible parabolic subsolution for 28 .

Step 1: Extension of the parabolic boundary data. We claim that there exists a function \[\widetilde{\varphi}\in C^{2,1}(\overline{ M_T})\cap C^\infty(\overline{ M}\times(0,T])\] such that \[\widetilde{\varphi}=\varphi \quad \text{on } \partial_s M_T, \qquad \widetilde{\varphi}(\cdot,0)=\varphi_0 \quad \text{on } \overline{ M},\] and such that both \(\partial_t\widetilde{\varphi}\) and \(dd^c\widetilde{\varphi}\) are bounded on \(\overline{ M_T}\).

Indeed, since \(\partial M\) is smooth and compact, we may choose finitely many boundary charts \(\{U_\alpha\}_{\alpha=1}^N\) covering \(\partial M\) such that for each \(\alpha\) there is a smooth diffeomorphism \[\kappa_\alpha:U_\alpha\longrightarrow B_2\subset\mathbb{R}^{2n}\] satisfying \[\kappa_\alpha(U_\alpha\cap M)=B_2^+:=B_2\cap\{x_{2n}>0\}, \qquad \kappa_\alpha(U_\alpha\cap\partial M)=B_2':=B_2\cap\{x_{2n}=0\}.\] Let \(V\Subset \bigcup_{\alpha=1}^N U_\alpha\) be an open neighborhood of \(\partial M\), and choose a partition of unity \[\theta_0,\theta_1,\dots,\theta_N\in C^\infty(\overline{ M}\cup V), \qquad \sum_{\alpha=0}^N\theta_\alpha\equiv 1 \;\text{on }\overline{ M},\] such that \(\operatorname{Supp} \theta_0\Subset M\) and \(\operatorname{Supp}\theta_\alpha\Subset U_\alpha\) for \(\alpha=1,\dots,N\).

Fix a cutoff function \(\eta\in C^\infty([0,\infty))\) such that \[\eta(s)\equiv 1 \quad \text{for } 0\le s\le \tfrac14, \qquad \eta(s)\equiv 0 \quad \text{for } s\ge \tfrac12.\] For each \(\alpha=1,\dots,N\), write \[\kappa_\alpha(x)=(y',y_{2n}),\qquad y'\in\mathbb{R}^{2n-1},\;y_{2n}\ge 0,\] and define on \((U_\alpha\cap\overline{ M})\times[0,T]\) \[H_\alpha(x,t) := \eta(y_{2n}) \Big( \varphi(\kappa_\alpha^{-1}(y',0),t) - \varphi(\kappa_\alpha^{-1}(y',0),0) \Big).\] Since \(\varphi\) is smooth on \(\partial M\times[0,T)\), each \(H_\alpha\) belongs to \[C^{2,1}\big((U_\alpha\cap\overline{ M})\times[0,T]\big) \cap C^\infty\big((U_\alpha\cap\overline{ M})\times(0,T]\big).\] We then set \[\widetilde{\varphi}(x,t) := \varphi_0(x)+\sum_{\alpha=1}^N \theta_\alpha(x)\,H_\alpha(x,t), \qquad (x,t)\in\overline{ M_T}.\]

By construction, \(\widetilde{\varphi}\in C^{2,1}(\overline{ M_T})\cap C^\infty(\overline{ M}\times(0,T])\). Moreover, if \(x\in\partial M\), then \(\theta_0(x)=0\), \(y_{2n}=0\), and hence \(\eta(y_{2n})=1\). Therefore, \[H_\alpha(x,t)=\varphi(x,t)-\varphi(x,0)\] for every \(\alpha\) such that \(x\in U_\alpha\), and thus \[\widetilde{\varphi}(x,t) = \varphi_0(x)+\sum_{\alpha=1}^N\theta_\alpha(x)\bigl(\varphi(x,t)-\varphi(x,0)\bigr) = \varphi(x,t),\] because \(\varphi_0|_{\partial M}=\varphi(\cdot,0)\) and \(\sum_{\alpha=1}^N\theta_\alpha(x)=1\) on \(\partial M\). On the other hand, at \(t=0\) we have \(H_\alpha(x,0)=0\) for every \(\alpha\), so \[\widetilde{\varphi}(x,0)=\varphi_0(x) \qquad\text{for all }x\in\overline{ M}.\] Finally, since \(\widetilde{\varphi}\in C^{2,1}(\overline{ M_T})\) and \(\overline{ M_T}\) is compact, both \(\partial_t\widetilde{\varphi}\) and \(dd^c\widetilde{\varphi}\) are bounded on \(\overline{ M_T}\).

Step 2: Construction of a subsolution. For a constant \(A>1\) to be chosen, define \[\underline{v}:=\widetilde{\varphi}+(A-1)\rho .\] Then \[\partial_t\underline{v}=\partial_t\widetilde{\varphi}, \qquad \chi+dd^c\underline{v} =dd^c\rho+dd^c\widetilde{\varphi}+(A-1)dd^c\rho =dd^c\widetilde{\varphi}+A\,dd^c\rho .\] Since \(\rho\) is strictly \((\omega,m)\)-subharmonic on \(\overline{ M}\) and \(dd^c\widetilde{\varphi}\) is bounded, by compactness there exist constants \(A_0>1\) and \(c_0>0\) such that for all \(A\ge A_0\), \[dd^c\widetilde{\varphi}+A\,dd^c\rho \in \Gamma_m(\omega) \quad\text{on }\overline{ M_T},\] and \[\bigl(dd^c\widetilde{\varphi}+A\,dd^c\rho\bigr)^m\wedge\omega^{n-m} \ge c_0 A^m\,\omega^n \quad\text{on }\overline{ M_T}.\] Therefore, for \(A\ge A_0\), \[\log\frac{(\chi+dd^c\underline{v})^m\wedge\omega^{n-m}}{\omega^n} \ge \log(c_0A^m).\] Choosing \(A\) even larger if necessary so that \[\log(c_0A^m)\ge \sup_{\overline{ M_T}}\bigl(\partial_t\widetilde{\varphi}+\psi\bigr),\] we obtain \[\partial_t\underline{v} \le \log\frac{(\chi+dd^c\underline{v})^m\wedge\omega^{n-m}}{\omega^n}-\psi \quad\text{in } M_T.\] On the lateral boundary \(\partial_s M_T\), since \(\rho=0\) on \(\partial M\) and \(\widetilde{\varphi}=\varphi\), we have \[\underline{v}=\varphi=\widehat{\varphi} \quad\text{on }\partial_s M_T.\] On the bottom boundary \(\partial_b M_T=\overline{ M}\times\{0\}\), we have \[\underline{v}(x,0)=\varphi_0(x)+(A-1)\rho(x)\le \varphi_0(x)-\rho(x) =\widehat{\varphi}(x,0),\] because \(\rho\le 0\) on \(\overline{ M}\) and \(A>0\). Hence \(\underline{v}\) is an admissible parabolic subsolution of 28 in the sense of 1.

Step 3: Application of 1. All the assumptions of 1 are now satisfied for the transformed problem 28 . Therefore there exists a unique solution \[v\in C^\infty( M\times(0,T])\cap C^{2k+\alpha,k+\frac{\alpha}{2}}(\overline{ M_T}).\] Setting \[u:=v+\rho,\] we conclude that \(u\) is the unique solution in \[C^\infty( M\times(0,T])\cap C^{2k+\alpha,k+\frac{\alpha}{2}}(\overline{ M_T})\] of ?? . This proves the corollary. ◻

Proof. The first statement is immediate from the analog of \((\omega,m)\)-subharmonic functions. The second statement can be proved in the same way from [17] (here we have to use [28]). The last statement is a direct consequence of [17]. ◻

Proof. Using 7, the proof is an easy adaptation of [17]. By [29], we have that \(u_0\) is \(\omega\)-subharmonic on \(\Omega\) and hence \((\omega,m)\)-subharmonic by checking the current inequality. By 8, it remains to show that \(\tilde{u}\) is upper semi-continuous at some point \((0,z_0)\), which is an easy consequence of the sub-mean-value inequality 7. Since \(u\) is bounded above, we may assume without loss of generality that \(u\leq0\). Fix small positive numbers \(\delta,r\) such that \(\delta<r<T\) and \(B(z_0,2r)\subset\Omega\). For each \(t\in[0,\delta)\) and \(z\in B(z_0,\delta)\), we apply 7 to the slice \(u_t\) on the ball \(B(z,r+\delta)\) we get \[u(t,z)\leq\frac{1}{r+\delta}\int_0^{r+\delta}ds\int_{\partial B(z,s)}u(t,\zeta)P_{z,s}(z,\zeta)d\sigma(\zeta).\] Fatou’s lemma then gives that \[\limsup_{(t,z)\to(0,z_0)}u(t,z)\leq\frac{1}{r+\delta}\int_0^{r+\delta}ds\int_{\partial B(z,s)}u_0(\zeta)P_{z_0,s}(z_0,\zeta)d\sigma(\zeta).\] Letting \(\delta\to0\) and then \(r\to0\) we conclude by 7 again that \[\limsup_{(t,z)\to(0,z_0)}u(t,z)\leq u_0(z_0).\] The proof is thus finished. ◻

Proof. The proof is an easy adaptation of [17]. Applying 7 to the \(\omega\)-subharmonic function \(u(t_0,\cdot)\) (recall that \(\Omega\) has real dimension \(2n\)), we get \[\begin{align} u(t_0,z_0) &\leq \frac{1}{r}\int_0^r dw \int_{\partial B_w(z_0)} u(t_0,\zeta)P_{z_0,w}(z_0,\zeta)d\sigma_w(\zeta)\\ &= \frac{1}{r}\int_0^r w^{2n-1}dw \int_{\partial B_1(0)} u(t_0,z_0+w\xi) P_{z_0,w}(z_0,z_0+w\xi)d\sigma_1(\xi). \end{align}\] By the definition of \(\kappa_0\), for every \(-1\leq s\leq1\), \[|u(t_0,z_0+w\xi)-u(t_0+\varepsilon s,z_0+w\xi)| \leq \kappa_0\varepsilon |s|.\] Since \[\int_{\partial B_1(0)} w^{2n-1}P_{z_0,w}(z_0,z_0+w\xi)d\sigma_1(\xi)=1,\] we obtain, for every fixed \(s\in[-1,1]\), \[u(t_0,z_0)\leq \frac{1}{r}\int_0^r w^{2n-1}dw \int_{\partial B_1(0)} u(t_0+\varepsilon s,z_0+w\xi) P_{z_0,w}(z_0,z_0+w\xi)d\sigma_1(\xi) +\kappa_0\varepsilon |s|.\] Averaging this inequality over \(s\in[-1,1]\) gives the desired estimate, \(\frac{1}{2}\int_{-1}^1 |s|\,ds=\frac{1}{2}\leq1\). ◻

Proof. One direction is immediate from 9. The other direction is an easy consequence of 7 and the uniform Lipschitz assumption of \(u\). ◻

Proof. Assume that the sequence \(\{u_j\}\) does not converge locally uniformly to \(-\infty\) in \(\Omega_T\). We will first demonstrate that \(\{u_j\}\) is bounded in \(L_{loc}^1(\Omega_T)\).

Fix a relatively compact subinterval \(J \subset (0, T)\) and a compact subset \(K \subset \Omega\). By the locally uniformly Lipschitz property of the sequence \(u_j\), for any fixed \(t \in J\), the sequence of slices \(\{u_j(t, \cdot)\}\) does not converge locally uniformly to \(-\infty\) on \(K\). Applying the compactness properties for \(\omega\)-subharmonic functions (see [29]), we deduce that \(\{u_j(t, \cdot)\}\) is bounded in \(L_{loc}^1(\Omega)\). Because \(\{u_j\}\) is locally uniformly Lipschitz in the time variable, it readily follows that the sequence is uniformly bounded in \(L^1(J \times K)\), establishing that \(\{u_j\}\) is bounded in \(L_{loc}^1(\Omega_T)\).

Next, let us consider the dense countable subset \(\mathbb{Q} \cap (0, T)\). For each rational \(r \in \mathbb{Q} \cap (0, T)\), the slices \(\{u_j(r, \cdot)\}\) are locally uniformly bounded above, allowing us to extract a subsequence that converges in \(L_{loc}^1(\Omega)\) to an \((\omega,m)\)-subharmonic function \(u(r, \cdot)\). By utilizing a standard Cantor diagonal process, we can extract a single subsequence, which we still denote as \(\{u_j\}\), such that for every rational \(r \in \mathbb{Q} \cap (0, T)\), \(u_j(r, \cdot)\) converges in \(L_{loc}^1(\Omega)\) to \(u(r, \cdot)\).

Since the functions \(\{u_j\}\) share a local uniform Lipschitz bound in \(t\), the limit function \((r, z) \mapsto u(r, z)\) inherits this identical locally uniform Lipschitz continuity with respect to \(r\). Consequently, we can uniquely extend \(u\) to the entire domain \((0, T) \times \Omega\) by defining \(u(t, z) := \lim_{\mathbb{Q} \ni r \to t} u(r, z)\).

The uniform Lipschitz property in time ensures that for all \(t \in (0, T)\), the slice sequence \(\{u_j(t, \cdot)\}\) converges in \(L_{loc}^1(\Omega)\) to \(u(t, \cdot)\), and that the extended function \(u\) remains locally uniformly Lipschitz in \(t \in (0, T)\). Because limits of \((\omega,m)\)-subharmonic functions in \(L_{loc}^1\) remain \((\omega,m)\)-subharmonic, each slice \(u(t, \cdot)\) belongs to this class, implying \(u \in P_m(\Omega_T)\). Finally, an application of Fubini’s theorem combined with the dominated convergence theorem guarantees that the subsequence \(\{u_j\}\) converges to \(u\) in \(L_{loc}^1(\Omega_T)\). ◻

Proof. Let us fix a compact interval \(J\subset(0,T)\) and a compact subset \(K\subset\Omega\). We are going to show that \(u\in L^p(K\times J)\) for \(1\leq p<\frac{n}{n-m}\). For each fixed \(t_0\in J\), \(u(t_0,\cdot)\) is \((\omega,m)\)-subharmonic and hence lies in \(L^p(K\times J)\) by [44] (note that Fang proved similar integrability on compact Hermitian manifolds, the local case follows easily by an extension argument). We can then apply Fubini’s theorem to write \[\begin{align} \left(\int_{J\times K}|u(t,z)|^pdVdt\right)^{\frac{1}{p}}&\leq\left(\int_{J\times K}|u(t,z)-u(t_0,z)|^pdVdt\right)^{\frac{1}{p}}+\left(\int_{J\times K}|u(t_0,z)|^pdVdt\right)^{\frac{1}{p}}\\ &\leq\kappa_J\left(\int_J|t-t_0|^pdV\right)^{\frac{1}{p}}+\left(\int_{J\times K}|u(t_0,z)|^pdVdt\right)^{\frac{1}{p}}, \end{align}\] where \(\kappa_J\) is the Lipschitz constant of \(u\) on \(J\). This proves the first statement. The proof of the second statement is identical to that in [17]. ◻

Proof. Fix a compact subset \(J \times K \Subset ]0,T[ \times \Omega\). Since \(u\) is locally bounded, there exists a constant \(M > 0\) such that \(|u(t,z)| \le M\) for all \((t,z) \in J \times K\). Moreover, by the locally uniform Lipschitz condition in time, there is a constant \(L_J > 0\) such that \(|\partial_t u(t,z)| \le L_J\) for almost every \((t,z) \in J \times \Omega\).

For each fixed \(t \in J\), the slice \(u_t\) is a locally bounded \((\omega,m)\)-subharmonic function on \(\Omega\) with \(\|u_t\|_{L^\infty(K)} \le M\). By [28], \(u_t\) belongs to the Sobolev space \(W^{1,2}_{\mathrm{loc}}(\Omega)\); in fact, the proof of [28] provides a uniform estimate \[\int_K |\nabla_z u(t,z)|^2 \, dV(z) \le C,\] by using a smooth decreasing approximation and their Chern-Levine-Nirenberg inequalities [28] along with the Mazur lemma in functional analysis, where the constant \(C\) depends only on \(M\), \(K\), \(\Omega\) and the Hermitian metric \(\omega\), but not on the particular \(t \in J\). Consequently, \(\nabla_z u \in L^2(J \times K) \subset L^1(J \times K)\) by the Cauchy–Schwarz inequality.

Combining the spatial gradient bound with the boundedness of \(\partial_t u\), we obtain \[\int_J \int_K \bigl( |\partial_t u(t,z)| + |\nabla_z u(t,z)| \bigr) \, dV(z) \, dt \le |J| \cdot |K| \cdot L_J + |J|^{1/2} \, |K|^{1/2} \, C^{1/2} < \infty.\] Thus the full gradient \(\nabla_{t,z} u\) belongs to \(L^1_{\mathrm{loc}}(\Omega_T)\), which means exactly that \(u \in W^{1,1}_{\mathrm{loc}}(\Omega_T)\). ◻

Proof. Let \(\chi \in C_c^0(\Omega_T)\) be a continuous test function. By definition, evaluating the measure against the test function yields: \[\int_{\Omega_T} \chi dt \wedge (dd^c u^j)^p \wedge \omega^{n-p} = \int_0^T dt \left( \int_\Omega \chi(t, \cdot) (dd^c u^j_t)^p \wedge \omega^{n-p} \right) := \int_0^T F_j(t) dt.\] Since the sequence \((u^j)\) is monotone and converges to \(u\) almost everywhere, for almost every \(t \in (0,T)\), the slice sequence \((u^j_t)\) is a monotone sequence of \(m\)-subharmonic functions converging almost everywhere to \(u_t\). It follows from the fundamental convergence theorem for complex Hessian measures (cf. [28]) that \(F_j(t)\) converges pointwise to \(F(t) := \int_\Omega \chi(t, \cdot) (dd^c u_t)^p \wedge \omega^{n-p}\) for almost every \(t \in (0,T)\).

Furthermore, since the sequence \((u^j)\) converges monotonically to \(u\), it is uniformly bounded on any compact subset \(E_2 \Subset \Omega_T\) containing the support of \(\chi\). The parabolic Chern-Levine-Nirenberg inequality established in 11 guarantees that the sequence of functions \(F_j(t)\) is uniformly bounded by a constant independent of \(j\) and \(t\).

We can therefore apply the Lebesgue Dominated Convergence Theorem to conclude that \[\lim_{j \to \infty} \int_0^T F_j(t) dt = \int_0^T F(t) dt,\] which completes the proof. ◻

Proof. As in [17], the uniform Lipschitz assumption \(t|\partial_t h(t,z)| \le \kappa_h\) implies that for any \(s > 0\) such that \(st < T\), we have the estimate: \[|h(t,z) - h(st,z)| \le \kappa_h \frac{|s-1|}{\min(s,1)}.\] Fix an arbitrary \(0 < S < T\) and let \(s \ge 1/2\) be close enough to \(1\) such that \(sS < T\). We define a scaled and shifted comparison function on the sub-cylinder \(\Omega_S\):\[v^s(t,z) := s^{-1}u(st,z) - C|s-1|(t+1),\]where \(C > 0\) is a large constant to be determined. We claim that \(v^s\) is a subsolution to the parabolic Hessian equation on \(\Omega_S\). Since \(u\) is an exact smooth solution on \(\Omega_T\), evaluating the Hessian operator on \(v^s\) yields: \[(dd^c v^s(t,\cdot))^m \wedge \omega^{n-m} = s^{-m}(dd^c u(st,\cdot))^m \wedge \omega^{n-m}= s^{-m} e^{\partial_\tau u(st,z) + \psi(st,z)} g(z) dV.\] To ensure that \(v^s\) is a pluripotential subsolution, we require: \[s^{-m} e^{\partial_\tau u(st,z) + \psi(st,z)} g(z) dV \ge e^{\partial_t v^s(t,z) + \psi(t,z)} g(z) dV.\] Observing that \(\partial_t v^s(t,z) = \partial_\tau u(st,z) - C|s-1|\), we can take the logarithm of both sides to see that the subsolution condition is equivalent to: \[-m \log s + \psi(st,z) \ge -C|s-1| + \psi(t,z).\] By the uniform Lipschitz property of \(\psi\), we have \(|\psi(t,z) - \psi(st,z)| \le \kappa_\psi t |s-1| \le \kappa_\psi T |s-1|\). Furthermore, for \(s \in [1/2, 3/2]\), elementary calculus gives \(-m \log s \ge -2m|s-1|\). Consequently, the inequality holds as long as we choose \(C\) such that:\[C \ge 2m + \kappa_\psi T.\]Next, we verify the boundary conditions on the parabolic boundary \(\partial_0 \Omega_S\). For any \(t \in [0,S]\) and \(z \in \partial \Omega\), we have: \[\begin{align} v^s(t,z) &= s^{-1}h(st,z) - C|s-1|(t+1)\le h(st,z) + |s^{-1}-1| M - C|s-1|\\ &\le h(t,z) + 2\kappa_h |s-1| + 2M |s-1| - C|s-1| \le h(t,z), \end{align}\] where we used the inequality \(|s^{-1}-1| \le 2|s-1|\) for \(s \ge 1/2\) and we need \(C\geq 2\kappa_h+2M\). Similarly, on the bottom boundary \(\{0\} \times \Omega\), we trivially have \(v^s(0,z) \le h(0,z)\) provided \(C \ge 2M\). Thus, setting \(C := 2M + 2\kappa_h + 2m + \kappa_\psi T\) guarantees that \(v^s\) is a smooth subsolution on \(\Omega_S\) with \(v^s \le u\) on \(\partial_0 \Omega_S\). Since \(u\) is a smooth solution, the smooth parabolic comparison principle (see also 21 below) implies \(v^s \le u\) everywhere in \(\Omega_S\). This yields the point-wise inequality: \[s^{-1}u(st,z) - C|s-1|(t+1) \le u(t,z), \quad \forall (t,z) \in \Omega_S.\] Letting \(s \to 1\) from above (\(s > 1\)) and below (\(s < 1\)), we deduce the differential inequality: \[|\partial_t u(t,z) \cdot t - u(t,z)| \le C(t+1).\] Rearranging this, we obtain: \[t|\partial_t u(t,z)| \le M + C(T+1) =: \kappa_u.\] Since \(M, \kappa_h\), and \(\kappa_\psi\) are uniform bounds independent of \(S\), the constant \(\kappa_u\) is uniform. Letting \(S \to T\) concludes the proof. ◻

Proof. Fix an arbitrary \(0 < S < T\) and a scaling parameter \(s > 1/2\) sufficiently close to \(1\) such that \(sS < T\). We construct a barrier by taking the arithmetic mean of symmetrically scaled solutions. For \((t,z) \in \Omega_S\), we define:\[v^s(t,z) := \frac{s^{-1}u(st,z) + s u(s^{-1}t,z)}{2} - C(t+1)(s-1)^2,\]where \(C > 0\) is a large uniform constant to be chosen later. We claim that \(v^s\) acts as a subsolution to the parabolic complex Hessian equation on \(\Omega_S\).First, we estimate the complex Hessian measure of \(v^s\). Setting \(w_1 = s^{-1}u(st,z)\) and \(w_2 = s u(s^{-1}t,z)\), we evaluate the operators: \[\begin{align} (dd^c w_1)^m \wedge \omega^{n-m} &= s^{-m}(dd^c u(st,\cdot))^m \wedge \omega^{n-m}= s^{-m} e^{\partial_\tau u(st,z) + \psi(st,z)} g(z) dV \end{align}\] and \[(dd^c w_2)^m \wedge \omega^{n-m} = s^m(dd^c u(s^{-1}t,\cdot))^m \wedge \omega^{n-m} = s^m e^{\partial_\tau u(s^{-1}t,z) + \psi(s^{-1}t,z)} g(z) dV.\] The mixed Hessian inequality (cf, [46]) then yields that \[(dd^c v^s)^m \wedge \omega^{n-m} \ge \exp\left( \frac{\partial_\tau u(st,z) + \partial_\tau u(s^{-1}t,z)}{2} + \frac{\psi(st,z) + \psi(s^{-1}t,z)}{2} \right) g(z) dV.\] Since \(\partial_t v^s(t,z) = \frac{1}{2}\big[\partial_\tau u(st,z) + \partial_\tau u(s^{-1}t,z)\big] - C(s-1)^2\), comparing the exponents reduces the subsolution condition to: \[\frac{\psi(st,z) + \psi(s^{-1}t,z)}{2} \ge \psi(t,z) - C(s-1)^2.\] The semi-convex bound \(|\partial_t^2 \psi| \le C_\psi\) on the compact domain \(\overline{\Omega_T}\) ensures that \(\big|\frac{\psi(st) + \psi(s^{-1}t)}{2} - \psi(t)\big| \le 2C_\psi T^2(s-1)^2\).

Next, we verify the boundary condition \(v^s \le h\) on the parabolic boundary \(\partial_0 \Omega_S\). On the side boundary \(\partial_s \Omega_S\), the assumption \(t^2 |\partial_t^2 h| \le C_h\) implies \(\frac{1}{2}[h(st) + h(s^{-1}t)] \le h(t) + \tilde{C}_h(s-1)^2\). Writing \(s = 1+\sigma\) and \(s^{-1} \le 1 - \sigma + 2\sigma^2\) (for \(s\) close to 1), straightforward algebraic expansion combined with the Lipschitz bound \(t|\partial_t h| \le \kappa_h\) and the uniform bound \(\sup|u| \le M\) yields: \[\begin{align} v^s(t,z) &\le \frac{1}{2}\big[s^{-1}h(st,z) + s h(s^{-1}t,z)\big] - C(s-1)^2 \\ &\le \frac{h(st,z) + h(s^{-1}t,z)}{2} + (M + 2\kappa_h)(s-1)^2 - C(s-1)^2 \\ &\le h(t,z) + (\tilde{C}_h + M + 2\kappa_h - C)(s-1)^2. \end{align}\] By selecting \(C \ge \tilde{C}_h + M + 2\kappa_h\), we guarantee that \(v^s(t,z) \le h(t,z)\) on \(\partial_0 \Omega_S\). The comparison principle then forces \(v^s \le u\) everywhere in \(\Omega_S\). This implies that for all \((t,z) \in \Omega_S\): \[\frac{s^{-1}u(st,z) + s u(s^{-1}t,z)}{2} - u(t,z) \le C(t+1)(s-1)^2.\] To extract the bound for the second time derivative, we apply a second-order Taylor expansion to the left-hand side of the above inequality. Let \(s = 1 + \sigma\), then \(s^{-1} = 1 - \sigma + \sigma^2 + O(\sigma^3)\). Expanding \(u(st)\) and \(u(s^{-1}t)\) around \(t\), we have: \[\begin{align} s^{-1}u(st) &= (1 - \sigma + \sigma^2)\left[ u + \sigma t \partial_t u + \frac{1}{2}\sigma^2 t^2 \partial_t^2 u + O(\sigma^3) \right] \\ &= u + \sigma(t\partial_t u - u) + \sigma^2\left( u - t\partial_t u + \frac{1}{2}t^2\partial_t^2 u \right) + O(\sigma^3), \end{align}\] and similarly for the second term: \[\begin{align} s u(s^{-1}t) &= (1 + \sigma)\left[ u + (-\sigma + \sigma^2)t \partial_t u + \frac{1}{2}\sigma^2 t^2 \partial_t^2 u + O(\sigma^3) \right] \\ &= u + \sigma(u - t\partial_t u) + \sigma^2\left( \frac{1}{2}t^2\partial_t^2 u \right) + O(\sigma^3). \end{align}\] Summing these two expansions and dividing by \(2\), we see that the \(O(\sigma)\) terms exactly cancel each other out: \[\frac{s^{-1}u(st) + s u(s^{-1}t)}{2} = u + \frac{\sigma^2}{2} \left( t^2\partial_t^2 u - t\partial_t u + u \right) + O(\sigma^3).\] Substituting this back into our comparison principle estimate and subtracting \(u(t,z)\), we obtain: \[\frac{\sigma^2}{2} \left( t^2\partial_t^2 u - t\partial_t u + u \right) + O(\sigma^3) \le C(t+1)\sigma^2.\] Dividing by \(\sigma^2\) and taking the limit as \(\sigma \to 0\) (i.e., \(s \to 1\)), we arrive at: \[\frac{1}{2} t^2 \partial_t^2 u(t,z) - \frac{1}{2} t \partial_t u(t,z) + \frac{1}{2} u(t,z) \le C(t+1).\] Multiplying by 2 and rearranging the terms yields: \[t^2 \partial_t^2 u(t,z) \le 2C(T+1) + t \partial_t u(t,z) - u(t,z).\] Finally, we invoke our assumed a priori bounds \(\sup |u| \le M\) and \(t|\partial_t u| \le \kappa\). This provides the desired uniform upper bound: \[t^2 \partial_t^2 u(t,z) \le 2C(T+1) + \kappa + M := K.\] This completes the proof. ◻

Proof. By standard convolutions, we can find smooth approximations \(\psi_j,g_j\) of \(\psi\) and \(g\) respectively.

Step 1. We are going to construct smooth boundary data \(h_j\) that approximates \(h\) on \(\partial_0\mathbb{B}_T\) uniformly. Firstly, we use radial extension \(\tilde{h}(z,t):=h\left(\frac{z}{|z|},t\right)\) to extend \(h\) to \(\{\frac{1}{2}\leq |z|\leq\frac{3}{2}\}\times[0,T]\). By constant extension, we can assume that \(\tilde{h}\) is a continuous function defined on \(\{\frac{1}{2}\leq |z|\leq\frac{3}{2}\}\times[-1,T+1]\). Then, we use convolutions to obtain smooth functions \(h_j\) defined in a neighborhood of \(\partial_0\mathbb{B}_T\) such that \(h_j\to h\) uniformly on \(\partial_0\mathbb{B}_T\). It is then clear that \(h_j\) satisfies ?? with a (probably larger) uniform constant \(\kappa\) in the pointwise sense. At this stage, \(h_j|_{\mathbb{B}\times\{0\}}\) may not lie in \(\operatorname{SH}_m(\mathbb{B},\omega)\), we modify \(h_j\) on \(\partial_0\mathbb{B}_T\) as follows.

Since \(h_0\) is a continuous \((\omega,m)\)-subharmonic function near \(\overline{\mathbb{B}}\), we can apply [28] to find a sequence of smooth \((\omega,m)\)-subharmonic functions \(\varphi_j\) decreasing (hence converging uniformly) to \(h_0\) on \(\overline{\mathbb{B}}\), we then consider the following Dirichlet problem of Hessian equations: \[\label{dirichlet32problem} \left\{ \begin{align} &(dd^cw_j)^m\wedge\omega^{n-m}=\frac{1}{j}\omega^n, \quad\operatorname{on}\mathbb{B},\\ &w_j=h_j-\varphi_j\quad\partial{\mathbb{B}}. \end{align} \right.\tag{29}\] By standard stability result (see [28]) we deduce that \(w_j\to0\) uniformly on \(\mathbb{B}\). Set \[\left\{ \begin{align} &\hat{h_j}=h_j,\operatorname{on}\partial\mathbb{B}\times(0,T],\\ &\hat{h}_j=\varphi_j+w_j\quad\partial{\mathbb{B}}\times\{0\}. \end{align} \right.\] It is clear that \(\hat{h}_j\) is smooth and \(0th\)-order compatible at the corner \(\partial\mathbb{B}\times\{0\}\), which converges uniformly to \(h\) on \(\partial_0\mathbb{B}_T\). We next modify \(\hat{h}_j\) to ensure that these approximations are at least second-order compatible at \(\partial\mathbb{B}\times\{0\}\), in order to use 1 (this modification is just for technical convenience and is not essential). Set \[v_{j,1}(z) := \log \frac{(dd^c \hat{h}_{j,0})^m \wedge \omega^{n-m}}{g_j(z)\omega^n} - \psi_j(0,z)\] and \[v_{j,2}(z) := F^{p\bar{q}}(dd^c \hat{h}_{j,0}) \cdot (v_{j,1})_{p\bar{q}} - \partial_t \psi_j(0,z).\] We then define the difference terms to be \[D_{j,1}(z) := v_{j,1}(z) - \partial_t \hat{h}_j(z,0),\quad D_{j,2}(z) := v_{j,2}(z) - \partial_{tt} \hat{h}_j(z,0),\] and, moreover, \[P_j(z,t) := t \cdot D_{j,1}(z) + \frac{t^2}{2} \cdot D_{j,2}(z).\] Let \(\eta:[0,\infty)\to[0,1]\) be a standard cutoff function such that \(\eta|_{[0,1/2]}\equiv1\) and \(\eta\) vanishes on \([1,\infty)\). The final modification of \(\hat{h}_j\) will be defined as \[H_j(z,t) := \hat{h}_j(z,t) + \eta\left(\frac{t}{\delta_j}\right) P_j(z,t),\] where \(\delta_j\to0\) are small constants to be chosen later. It is easy to check that \(H_j\) are smooth boundary data that are second-order compatible at the corner \(\partial\mathbb{B}\times\{0\}\). Finally, we must choose \(\delta_j > 0\) sufficiently small to ensure that this modification does not destroy the uniform bounds on the scaled time derivatives \(t|\partial_t H_j|\) and \(t^2|\partial_{tt} H_j|\). Since \(P_j(z,t) = t D_{j,1}(z) + \frac{t^2}{2} D_{j,2}(z)\) and the cutoff function \(\eta(t/\delta_j)\) is supported on \(t \in [0, \delta_j]\), we can estimate the contribution of the modification to the first scaled time derivative: \[\begin{align} t \partial_t \left( \eta\left(\frac{t}{\delta_j}\right) P_j(z,t) \right) &= \frac{t}{\delta_j} \eta'\left(\frac{t}{\delta_j}\right) P_j(z,t) + t \eta\left(\frac{t}{\delta_j}\right) \partial_t P_j(z,t) \\ &\le C \frac{t}{\delta_j} \left( t |D_{j,1}| + t^2 |D_{j,2}| \right) + C t \left(|D_{j,1}| + t |D_{j,2}|\right) \\ &\le C \left( \delta_j |D_{j,1}| + \delta_j^2 |D_{j,2}| \right), \end{align}\] where we crucially used the fact that \(t \le \delta_j\) on the support of \(\eta\). A similar calculation for the second scaled derivative yields: \[\begin{align} t^2 \partial_{tt} \left( \eta\left(\frac{t}{\delta_j}\right) P_j(z,t) \right) &= \left(\frac{t}{\delta_j}\right)^2 \eta'' P_j + 2\frac{t}{\delta_j} t \eta' \partial_t P_j + t^2 \eta \partial_{tt} P_j \\ &\le C \left( \delta_j |D_{j,1}| + \delta_j^2 |D_{j,2}| \right). \end{align}\] Although the compatibility correction terms \(D_{j,1}\) and \(D_{j,2}\) may be very large and depend on \(j\) (as they involve the linearized operator acting on the approximation \(\varphi_j\)), they are bounded by constants for each fixed \(j\). Therefore, by choosing \(\delta_j>0\) sufficiently small such that \(C \left( \delta_j |D_{j,1}| + \delta_j^2 |D_{j,2}| \right) \le \frac{1}{j}\), the uniform conditions \(t|\partial_t H_j| \le \kappa_h + \frac{1}{j}\) and \(t^2|\partial_{tt} H_j| \le C_h + \frac{1}{j}\) are perfectly preserved. Moreover, the modification itself satisfies \(|\eta(t/\delta_j) P_j| \le C(\delta_j |D_{j,1}| + \delta_j^2 |D_{j,2}|) \le \frac{1}{j}\), ensuring \(H_j \to h\) uniformly on \(\partial_0\mathbb{B}_T\). Note also that \(H_j(\cdot, 0) = \varphi_j + w_j\) is strictly \((\omega,m)\)-subharmonic since \((dd^c w_j)^m \wedge \omega^{n-m} = \frac{1}{j}\omega^n > 0\). Thus, \(H_j\) fully satisfies all the assumptions required for the smooth flow in 1.

Step 2. Using 1, we can find smooth solutions \(u_j\in C^\infty\left(\overline{\mathbb{B}}\times(0,T]\right)\cap C^{4,2}(\overline{\mathbb{B}_T})\) such that \[\label{eq:hessian32appro32sequence} \left\{ \begin{align} dt\wedge(dd^cu_j)^m\wedge\omega^{n-m}=e^{\partial_t u_j+\psi_j(t,z)}g_j(z)dt\wedge dV_{\mathbb{B}} \quad\operatorname{on}\mathbb{B}_T,\\ u_j=H_j\quad\operatorname{on}\partial_0\mathbb{B}_T. \end{align} \right.\tag{30}\] By 13 and 14, we have that the sequence \(\{u_j\}\) is locally uniformly Lipschitz and locally uniformly semi-concave on \((0,T]\). Here, we shall note that the explicit constants calculated in 13 and 14 depends only on the uniform bounds of the boundary values \(H_j\) of \(u_j\), so a priori we do not need a uniform bound of \(u_j\). We then claim that \(u_j\) is uniformly bounded on \(\overline{\mathbb{B}}_T\). Indeed, on each slice \(\overline{\mathbb{B}}\times\{t\}\), \(u_j(t,\cdot)\) is \((\omega,m)\)-subharmonic and hence \(u_j(t,\cdot)\leq H_j(t,\cdot)\). Consequently, \(u_j\le H_j\leq C\). For the lower bound, set \[v(t,z):=A\rho(z)-\sup_{\partial{\mathbb{B}}_T}|h|,\] where \(\rho \le 0\) is a bounded continuous \((\omega, m)\)-subharmonic function on \(\Omega\) solving \((dd^c \rho)^m \wedge \omega^{n-m} = g dV\) with zero boundary values. It is clear that we have \[dt\wedge(dd^cv)^m\wedge\omega^{n-m}\geq e^{\partial_t v+\psi_j(t,z)}g_j(z)dt\wedge dV_{\mathbb{B}}=e^{\psi_j(t,z)}g_j(z)dt\wedge dV_{\mathbb{B}},\] for \(A\) large enough. The parabolic comparison principle 21 (whose proof is independent from here) below then yields that \(v\leq u_j\), whence the uniform lower bound of \(u_j\).

Now, invoking 9 we can find \(u\in P_{m,\omega}(\mathbb{B}_T)\) such that \(u_j\to u\) in \(L^1(\mathbb{B}_T)\). From the discussions above, all the conditions in 13 are fulfilled, this implies that \[\lim_{j\to\infty}\dot{u}_j(z,t)=\dot{u}(z,t)\] for almost all \((z,t)\in\mathbb{B}_T\), and, moreover, \[e^{\partial_t u_j+\psi_j(t,z)}g_j(z)dt\wedge dV_{\mathbb{B}} \to e^{\partial_t u+\psi(t,z)}g(z)dt\wedge dV_{\mathbb{B}}\] in the weak sense of Radon measures on \(\mathbb{B}_T\).

Step 3. It remains to show the convergence of parabolic Hessian measures on the left-hand side of 30 . By Fubini’s theorem, it is easy to see that for almost all \(t\in(0,T]\), we have \(\lim_{j\to\infty}\dot{u}_j(z,t)=\dot{u}(z,t)\) for almost all \(z\in\mathbb{B}\). For all such \(t\), we use the stability result [28] again to conclude that \(u_j(t,\cdot)\to u_t\in\operatorname{SH}_m(\mathbb{B},\omega)\cap C^0(\mathbb{B})\), which necessarily coincides with \(u(t,\cdot)\) by the proof of 9 (see also [17]). Due to the locally uniformly Lipschitz property of \(u_j\) and \(u\), we conclude that \(u_j(t,\cdot)\to u(t,\cdot)\) uniformly for all \(t\in(0,T]\) and hence \(u\) is continuous on \((0,T]\times\overline{\mathbb{B}}\) and \(u\) has the correct boundary value on \((0,T]\times\partial\mathbb{B}\).

Indeed, fix an arbitrary \(t\in(0,T]\), choose a sequence \(t_i\to t\) such that \(u_j(t_i,\cdot)\to u(t_i,\cdot)\) uniformly as \(j\to\infty\). Note that for this fixed \(t\), we have \(|u_j(t,\cdot)-u_j(t_k,\cdot)|\leq\kappa_t|t-t_k|\) for all \(j\) and \(k\) large enough. For each \(\varepsilon>0\), we first choose \(k\) large such that \[|u_j(t,\cdot)-u_j(t_k,\cdot)|\leq\frac{\varepsilon}{3},\quad |u(t,\cdot)-u(t_k,\cdot)|\leq\frac{\varepsilon}{3},\quad\forall j\in \mathbb{Z}^+,\] and then choose \(j\) large enough for this fixed \(t_k\) such that \(|u(t_k,\cdot)-u_j(t_k,\cdot)|\leq\frac{\varepsilon}{3}\). It follows from the triangle inequality that \(u_j(t,\cdot)\to u(t,\cdot)\) uniformly. As a consequence, each slice \(u(t,\cdot)\) is a continuous \((\omega,m)\)-subharmonic function, it follows easily from the locally uniformly Lipschitz condition of \(u\) that \(u\) is continuous globally on \(\mathbb{B}_T\). Furthermore, we have the weak convergence \[(dd^cu_j(t,\cdot))^m\wedge\omega^{n-m}\to (dd^cu(t,\cdot))^m\wedge\omega^{n-m}\] by [28].

Now, fix a test function \(\chi\in C_c^0(\mathbb{B}_T)\), we can write \[\begin{align} & \int_{\mathbb{B}_T}\chi dt \wedge(dd^cu_j)^m\wedge\omega^{n-m}:=\int_0^Tdt\int_{\mathbb{B}}\chi(t,\cdot)(dd^cu_j(t,\cdot))^m\wedge\omega^{n-m}\\ \to&\int_0^Tdt\int_{\mathbb{B}}\chi(t,\cdot)(dd^cu(t,\cdot))^m\wedge\omega^{n-m}= \int_{\mathbb{B}_T}\chi dt \wedge(dd^cu)^m\wedge\omega^{n-m}, \end{align}\] where thanks to the CLN inequality established in 11, we could apply the dominated convergence theorem to obtain the convergence above. The proof is finally concluded.

Step 4. We show that \(u(t,\cdot)\to h_0\) in \(L^1(\mathbb{B})\) as \(t\searrow0\). From below, we construct a continuous subbarrier adapting the idea of [17]. Define \[\underline{v}(t,z) := h_0(z) + t(\rho(z) - C) + m[t \log(t/T) - t],\] where \(\rho \le 0\) is a continuous \((\omega,m)\)-subharmonic function in \(\mathbb{B}\) satisfying \((dd^c \rho)^m \wedge \omega^{n-m} = g dV\) with \(\rho = 0\) on \(\partial \mathbb{B}\). By choosing \(C := \max \{ \sup \rho + M_\psi - m \log T, \kappa_h - m \}\), where \(M_\psi=\sup_{\mathbb{B}_T}|\psi|\), it is not difficult to see that \(\underline{v}\) is a pluripotential subsolution with boundary value \(\leq h\).

Since \(H_j \to h\) uniformly on \(\partial_0 \mathbb{B}_T\), for any \(\varepsilon > 0\), we have \(H_j \ge \underline{v} - \varepsilon\) on \(\partial_0 \mathbb{B}_T\) for all sufficiently large \(j\). The parabolic comparison principle (see 21 again) then yields \(u_j \ge \underline{v} - \varepsilon\) throughout \(\mathbb{B}_T\). Taking \(j \to \infty\) and then \(\varepsilon \to 0\), we obtain \(u \ge \underline{v}\) in \(\mathbb{B}_T\). This implies that \[\liminf_{t \searrow 0} u(t,z) \ge \lim_{t \searrow 0} \underline{v}(t,z) = h_0(z).\]

From above, we use an integral monotonicity argument inspired by [17] (see also 7 below). Since \(u_j\) are smooth solutions, the Hessian Chern-Levine-Nirenberg inequality [28] guarantees that the measure mass \(\int_{\mathbb{B}} \chi (dd^c u_j)^m \wedge \omega^{n-m}\) is uniformly bounded for any continuous test function \(\chi \ge 0\) compactly supported in \(\mathbb{B}\). We claim that there exists a uniform constant \(C > 0\) such that \[\int_{\mathbb{B}} \chi u_j(t,z) g_j dV \le \int_{\mathbb{B}} \chi H_j(0,z) g_j dV + C t, \quad \forall t \in (0,T).\] Indeed, Since \(u_j\) is a smooth solution to the complex Hessian flow, it satisfies the equation: \[(dd^c u_j(t,\cdot))^m \wedge \omega^{n-m} = e^{\partial_t u_j(t,\cdot) + \psi_j(t,\cdot)} g_j dV \ge e^{\partial_t u_j(t,\cdot) - K} g_j dV,\] where \(K > 0\) is a uniform constant such that \(\psi_j \ge -K\) on \(\overline{\mathbb{B}_T}\). Multiplying both sides by the non-negative test function \(\chi\) and integrating over \(\mathbb{B}\), we obtain: \[\int_{\mathbb{B}} \chi e^{\partial_t u_j(t,\cdot) - K} g_j dV \le \int_{\mathbb{B}} \chi (dd^c u_j(t,\cdot))^m \wedge \omega^{n-m}\le C.\] Let \(M_j := \int_{\mathbb{B}} \chi g_j dV\). Since \(g_j \to g\) uniformly on \(\overline{\mathbb{B}}\) and \(g > 0\), the mass \(M_j\) is uniformly bounded away from zero and infinity. Applying Jensen’s inequality to the convex function \(x \mapsto e^x\) with respect to the probability measure \(d\mu_j := M_j^{-1} \chi g_j dV\), we get: \[\exp \left( \int_{\mathbb{B}} \partial_t u_j(t,\cdot) d\mu_j \right) \le \int_{\mathbb{B}} e^{\partial_t u_j(t,\cdot)} d\mu_j \le \frac{C}{M_j}.\] Taking the logarithm yields a uniform upper bound \(C > 0\) for the integral of the time derivative: \[\int_{\mathbb{B}} \chi \partial_t u_j(t,z) g_j(z) dV \le C, \quad \forall t \in (0,T).\] Integrating this inequality with respect to time from \(0\) to \(t\) gives: \[\int_{\mathbb{B}} \chi u_j(t,z) g_j(z) dV - \int_{\mathbb{B}} \chi u_j(0,z) g_j(z) dV \le C t.\] Noting that \(u_j(0,z) = H_j(0,z)\) by the initial condition, the proof of the claim is therefore finished.

We have established that \(u_j(t,\cdot) \to u(t,\cdot)\) uniformly on slices \(t>0\), and by construction, \(H_j(0,\cdot) \to h_0\) uniformly on \(\overline{\mathbb{B}}\). Passing to the limit \(j \to \infty\) yields \[\int_{\mathbb{B}} \chi u(t,z) g dV \le \int_{\mathbb{B}} \chi h_0(z) g dV + C t.\] Let \(w_0 \in \operatorname{SH}_m(\mathbb{B}, \omega)\) be any weak cluster point of \(u(t,\cdot)\) as \(t \searrow 0\). Taking the limit \(t \to 0^+\) gives \(\int \chi w_0 g dV \le \int \chi h_0 g dV\). Since the test function \(\chi\) was arbitrary, this forces \(w_0 \le h_0\) almost everywhere, and hence everywhere. This implies that any cluster point of \(u(t,\cdot)\) as \(t \searrow 0\) must be \(h_0\) and hence \(u(t,\cdot)\to h_0\) in \(L^1\) as \(t\searrow0\). It thus remains to show that the convergence is actually uniform.

Step 5. We finally upgrade the \(L^1\) convergence to uniform convergence on \(\overline{\mathbb{B}}\), showing that \(u(t,\cdot) \to h_0\) uniformly as \(t \searrow 0\). This will complete the proof that \(u \in C^0(\overline{\mathbb{B}_T})\).

From the lower bound established in Step 4, we have \(u(t,z) \ge \underline{v}(t,z)\). Since \(\underline{v}\) is continuous on \(\overline{\mathbb{B}_T}\) and \(\underline{v}(0,z) = h_0(z)\), we immediately obtain the uniform lower bound: \[\liminf_{t \searrow 0} \inf_{z \in \overline{\mathbb{B}}} (u(t,z) - h_0(z)) \ge 0.\]

To establish the uniform upper bound, we adapt the boundary strip argument from [17]. For each \(t \in [0,T)\), let \(H_t\) be the unique continuous function on \(\overline{\mathbb{B}}\) such that \((dd^c H_t)^m \wedge \omega^{n-m} = 0\) in \(\mathbb{B}\) and \(H_t = h(t,\cdot)\) on \(\partial \mathbb{B}\). Since the boundary data \(h(t, \zeta)\) is uniformly Lipschitz in time, the stability result [28] ensures that \(H_t(z) \le H_0(z) + \kappa_h t\) for all \(z \in \overline{\mathbb{B}}\). Note also that for \(t>0\), \(u(t,\cdot)\) is an \((\omega,m)\)-subharmonic function with boundary values \(h(t,\cdot)\). The comparison principle yields \(u(t,z) \le H_t(z)\) in \(\mathbb{B}\).

Since \(u(t,\cdot) \to h_0\) in \(L^1(\mathbb{B})\) as \(t \searrow 0\), Hartogs’ lemma for \(\omega\)-subharmonic functions (cf. [29]) ensures that this convergence holds uniformly bounded from above on any compact subset. Thus, for any compact subset \(K \Subset \mathbb{B}\), we have \[\label{eq:hartogs95interior} \limsup_{t \searrow 0} \sup_{z \in K} (u(t,z) - h_0(z)) \le 0.\tag{31}\]

Now, fix an arbitrary \(\varepsilon > 0\). Both \(H_0\) and \(h_0\) are continuous on \(\overline{\mathbb{B}}\) and they coincide exactly on the boundary \(\partial \mathbb{B}\). Therefore, we can find a narrow boundary strip \(\mathbb{B} \setminus K\) (where \(K\) is a compact subset of \(\mathbb{B}\)) such that \[\sup_{z \in \mathbb{B} \setminus K} (H_0(z) - h_0(z)) \le \varepsilon.\] Evaluating the upper bound in this boundary strip, we obtain: \[u(t,z) - h_0(z) \le H_t(z) - h_0(z) \le H_0(z) - h_0(z) + \kappa_h t \le \varepsilon + \kappa_h t.\] Taking the limit supremum as \(t \searrow 0\) yields \(\limsup_{t \searrow 0} \sup_{\mathbb{B} \setminus K} (u(t,z) - h_0(z)) \le \varepsilon\). Combining this with the interior bound 31 on \(K\), we deduce that globally: \[\limsup_{t \searrow 0} \sup_{z \in \overline{\mathbb{B}}} (u(t,z) - h_0(z)) \le \varepsilon.\] Since \(\varepsilon > 0\) is arbitrary, we conclude that \(\limsup_{t \searrow 0} \sup_{\overline{\mathbb{B}}} (u(t,z) - h_0(z)) \le 0\). Together with the uniform lower bound, this proves that \(u(t,\cdot) \to h_0\) uniformly on \(\overline{\mathbb{B}}\) as \(t \searrow 0\), hence \(u\) is globally continuous on \(\overline{\mathbb{B}_T}\). The proof is now complete. ◻

Proof. See [17]. ◻

Proof. As in [17], this is an immediate consequence of 11 and the plurifine locality [28]. ◻

Proof. We first establish the uniform upper bound. Given any candidate \(v \in \mathcal{S}_{h,g,\psi}(\Omega_T)\), the time slice \(v_t := v(t, \cdot)\) is an \((\omega, m)\)-subharmonic function on \(\Omega\) for each fixed \(t \in (0, T)\). The boundary condition ensures that \(\limsup_{z \to \zeta} v_t(z) \le h(t,\zeta) \le M_h\) for any boundary point \(\zeta \in \partial \Omega\). Invoking the standard maximum principle for \((\omega,m)\)-subharmonic functions, we immediately obtain \(v_t(z) \le M_h\) on \(\Omega\). Taking the supremum over all such admissible \(v\) yields \(U(t,z) \le M_h\), and consequently \(U^* \le M_h\).

Next, we verify that \(\mathcal{S}_{h,g,\psi}(\Omega_T)\) is non-empty by constructing a concrete global sub-barrier. Consider the following time-independent test function: \[\underline{v}(t,z) := A\rho(z) - M_h.\] It is straightforward to see that \(\underline{v}\) respects the Cauchy-Dirichlet boundary data, as \(\underline{v}^* \le -M_h \le h\) on the parabolic boundary \(\partial_0 \Omega_T\). Since \(\partial_t \underline{v} = 0\), evaluating its parabolic Hessian measure yields: \[dt \wedge (dd^c \underline{v})^m \wedge \omega^{n-m} = A^m (dd^c \rho)^m \wedge \omega^{n-m} \wedge dt = A^m g dV \wedge dt.\] To guarantee that \(\underline{v}\) is a pluripotential subsolution, we require the inequality \(A^m g \ge e^{\partial_t \underline{v} + \psi} g = e^\psi g\) to hold. Our choice of \(A = e^{M_\psi / m}\) naturally satisfies this condition, confirming that \(\underline{v} \in \mathcal{S}_{h,g,\psi}(\Omega_T)\). This proves the non-emptiness of the admissible class and establishes the lower bound for the envelope.

Finally, the fact that \(\mathcal{S}_{h,g,\psi}(\Omega_T)\) is closed under finite maxima follows directly from 15. ◻

Proof. Under the assumption that \(U\) is a pluripotential subsolution, the proof follows exactly the same as the a priori semi-concavity estimate established for smooth solutions in 14. For a scaling parameter \(s\) close to \(1\), one considers the barrier \[v^s(t,z) := \frac{s^{-1}U(st,z) + s U(s^{-1}t,z)}{2} - C_U(t+1)(s-1)^2.\] Relying on the mixed Hessian inequality for bounded \((\omega,m)\)-subharmonic functions (cf. [46]), a completely analogous verification confirms that \(v^s \in \mathcal{S}_{h,g,\psi}(\Omega_T)\) when the constant \(C_U\) is chosen as above. The definition of the envelope then enforces \(v^s \le U\) everywhere in \(\Omega_T\). We omit the repetitive calculations here. ◻

Proof. We construct an explicit global pluripotential supersolution to dominate the envelope \(U\). For each fixed \(t \in [0,T]\), let \(H_t(z) := H(t,z)\) be the unique continuous \((\omega,m)\)-subharmonic function on \(\overline{\Omega}\) solving the homogeneous Dirichlet problem for the complex Hessian equation: \[\left\{ \begin{align} &(dd^c H_t)^m \wedge \omega^{n-m} = 0 \quad \text{in } \Omega, \\ &H_t(z) = h(t, z) \quad \text{on } \partial \Omega. \end{align} \right.\] The existence and continuity of such \(H_t\) follow from [28]. Since the boundary data \(h(t,z)\) is locally uniformly Lipschitz in \(t\), the stability result [28] ensures that the solution \(H(t,z)\) is also locally uniformly Lipschitz in \(t\) and hence continuous on \(\Omega_T\).

We claim that \(H(t,z)\) is a pluripotential supersolution to the parabolic complex Hessian equation in \(\Omega_T\). Indeed, it is clear that \[(dd^c H_t)^m \wedge \omega^{n-m} = 0 \le e^{\partial_t H(t,z) + \psi(t,z)} g(z) dV.\] On the other hand, the comparison principle easily yields that \(H\geq h\) on \(\partial_0\Omega_T\).

For any admissible subsolution \(v \in \mathcal{S}_{h,g,\psi}(\Omega_T)\), the comparison principle gives that \(v_t \le H_t\) in \(\Omega\). Taking the supremum over all such subsolutions yields: \[U(t,z) \le H(t,z) \quad \text{in } \Omega_T.\] Finally, fixing \(t_0 \in (0,T]\) and \(\zeta \in \partial \Omega\), the continuity of \(H\) up to the boundary implies: \[\limsup_{\Omega_T \ni (t,z) \to (t_0, \zeta)} U(t,z) \le \lim_{(t,z) \to (t_0, \zeta)} H(t,z) = H(t_0, \zeta) = h(t_0, \zeta).\] This establishes the desired upper bound on the lateral boundary. ◻

Proof. The construction of these sub-barriers relies on the resolution of the homogeneous Dirichlet problem for the elliptic complex Hessian equation to match the boundary values, the arguments are completely analogous to those in the complex Monge-Ampère case. We omit the details and refer the reader to Step 4 of 14 and [17] for the exact barrier formulas. ◻

Proof. For boundary points located on the side \((\tau,\zeta) \in (0,T) \times \partial\Omega\), the required upper bound is a direct consequence of the global supersolution barrier constructed in 17.

We now turn our attention to the initial bottom boundary where \(\tau = 0\). The argument is similar to that in Step 4 of 14. Fix an arbitrary relatively compact domain \(D \Subset \Omega\) and choose a non-negative test function \(\chi \in C_c^0(D)\). According to a Chern-Levine-Nirenberg type estimate (cf. [28]), the total mass \(\int_D (dd^c v_{j,t})^m \wedge \omega^{n-m}\) is uniformly bounded from above for all \(j\) and all \(t \in (0,T)\), since the sequence is uniformly bounded in \(L^\infty(\Omega_T)\).

By virtue of 19, we can extract a uniform constant \(A > 0\) such that for all \(j\) and all \(t \in (0,T)\), \[\int_D \chi v_{j,t} g dV \le \int_D \chi v_{j,0} g dV + A t \le \int_D \chi h_0 g dV + A t.\] Thanks to 10 and the convergence \(v_j \to v\) in \(L_{loc}^1(\Omega_T)\) (hence in \(L^1(\Omega_T)\) by the uniform boundedness of functions), we can pass to the limit as \(j \to \infty\) on slices to yield \[\int_D \chi v_t g dV \le \int_D \chi h_0 g dV + A t, \quad \text{for a.e. } t \in (0,T).\] Let \(v^*\) be any weak cluster point of \(v_t\) as \(t \searrow 0\). Taking the limit supremum as \(t \to 0^+\) in the above inequality gives \[\int_D \chi v^* g dV \le \int_D \chi h_0 g dV.\] Because the choice of the test function \(\chi \ge 0\) and the subset \(D\) was arbitrary, it follows that \(v^* \le h_0\) pointwise in \(\Omega\). Combining this pointwise inequality with the upper semi-continuous extension property of \(m\)-parabolic potentials at \(t=0\) (8), we finally conclude that \(\limsup_{(t,z) \to (0,\zeta)} v(t,z) \le h_0(\zeta)\). ◻

Proof. Taking into account the corresponding properties established in [28], the proof of [17] can be easily adapted here. The first statement follows immediately from 17) and 18.

The lower bound \(\liminf_{t \to 0} U(t,z_0) \ge h_0(z_0)\) is guaranteed by the initial sub-barrier in 18. Thus, it remains to prove the upper bound: \[\limsup_{\Omega_T \ni (t,z) \to (0,z_0)} U^*(t,z) \le h_0(z_0).\] By 5, we have that \(U\) is locally uniformly Lipschitz in \(t \in (0,T)\), hence \(U^*(t,\cdot) = (U_t)^*\) in \(\Omega\) for all \(t \in (0,T)\) thanks to 8. Consequently, we can use 8 to reduce the problem to showing that \[\lim_{t \searrow 0} (U_t)^*(z_0) \le h_0(z_0), \quad \forall z_0 \in \Omega.\]

Let \(\chi \in C_c^0(\Omega)\) be a non-negative continuous test function compactly supported in a subdomain \(D \Subset \Omega\). Fix an arbitrary slice \(t_0 \in (0,T)\). By Choquet’s lemma, there exists an increasing sequence \(\{v^j\} \subset \mathcal{S}_{h,g,\psi}(\Omega_T)\) bounded uniformly by \(M_U\), such that \((U_{t_0})^* = (\lim_{j \to \infty} v^j_{t_0})^*\) in \(\Omega\).

For the compact subset \(D\), the Chern-Levine-Nirenberg inequality for the complex Hessian operator (see [28]) implies that the total mass of the Hessian measures is uniformly bounded: \[\int_D \chi (dd^c v^j_t)^m \wedge \omega^{n-m} \le C_1, \quad \forall t \in (0,T),\] where \(C_1\) depends on \(D\), \(\chi\), \(\omega\), and \(M_U\), but is independent of \(t\) and \(j\). Applying 19, we obtain a uniform constant \(C_2 > 0\) such that \[\int_D \chi v^j_{t_0} g dV \le \int_D \chi h_0 g dV + C_2 t_0.\] By [28], the \(m\)-negligible set \(\{z \in \Omega \mid \lim_{j \to \infty} v^j_{t_0}(z) < (U_{t_0})^*(z) \}\) is \(m\)-polar and hence has Lebesgue measure zero. Thus, applying the monotone convergence theorem [28] to pass the limit \(j \to \infty\) inside the integral, we obtain \[\int_D \chi (U_{t_0})^* g dV \le \int_D \chi h_0 g dV + C_2 t_0.\] Let \(w_0 \in \operatorname{SH}_m(\Omega, \omega)\) be any cluster point of \((U_t)^*\) as \(t \searrow 0\). By standard pluripotential compactness, \((U_t)^*\) converges to \(w_0\) in \(L^q(\Omega)\) for any \(1<q<\frac{n}{n-m}\). While \(g\in L^p(\Omega)\) with \(p>\frac{n}{m}\), we can taking the limit \(t_0 \to 0^+\) in the integral inequality above to write \[\int_D \chi w_0 g dV \le \int_D \chi h_0 g dV.\] Since the non-negative test function \(\chi\) and the subset \(D \Subset \Omega\) were arbitrary, and the density \(g\) is strictly positive almost everywhere, we deduce that \(w_0 \le h_0\) almost everywhere. As both \(w_0\) and \(h_0\) are \((\omega,m)\)-subharmonic functions, this inequality extends everywhere in \(\Omega\). This confirms the desired upper bound and concludes the proof. ◻

Proof. When \(h\) is uniformly Lipschitz on \([0,T]\), the proof is the same as that in Step 5 of 14. For the general case, fix a small \(\varepsilon > 0\). We define the time-shifted boundary data on the lateral side of the sub-cylinder \(\Omega \times [0, T-\varepsilon]\) by: \[h^\varepsilon(t, \zeta) := h(t+\varepsilon, \zeta) \quad \text{for } (t,\zeta) \in (0, T-\varepsilon] \times \partial\Omega.\] To ensure continuity at the corner \(\{0\} \times \partial\Omega\), we must match the initial data. We define the initial slice at \(t=0\) by \[h^\varepsilon(0,z) := h_0(z) + \phi_\varepsilon(z), \quad \text{for } z \in \overline{\Omega},\] where \(h_0(\cdot) := h(0, \cdot)\), and \(\phi_\varepsilon \in \operatorname{SH}_m(\Omega, \omega) \cap C^0(\overline{\Omega})\) is defined as the unique continuous solution to the homogeneous Dirichlet problem for the complex Hessian equation: \[\left\{ \begin{align} &(dd^c \phi_\varepsilon)^m \wedge \omega^{n-m} = 0 \quad \text{in } \Omega, \\ &\phi_\varepsilon(z) = h(\varepsilon, z) - h_0(z) \quad \text{on } \partial\Omega. \end{align} \right.\] By construction, it is clear that \(h^\varepsilon\) is globally continuous on the parabolic boundary \(\partial_0 \Omega_{T-\varepsilon}\). By the stability theorem (cf. [28]), we see that \(h^\varepsilon\) converges uniformly to \(h\) on the whole parabolic boundary. Consequently, it follows from the definition that \(U^\varepsilon:=U_{h^\varepsilon,\psi,g,T-\varepsilon}\) converges to \(U\) uniformly on \([0,T-\varepsilon)\) as \(\varepsilon\to 0\) (we have used the identity principle 16 below).

It is clear that \(h^\varepsilon\) is uniformly Lipschitz on \([0,T-\varepsilon]\), hence we have that \(U^\varepsilon(t,\cdot)\) converges uniformly to \(h_0\) as \(t\to0\) for each fixed \(\varepsilon\). The proof is finally concluded. ◻

Sketch of proof. We prove the result on finite cylinders; the general case follows by restriction to \(\Omega_S\), \(S<T\), and the identity principle. The boundary assertions follow from 7 and 20; it remains to prove \(U\in\mathcal{S}_{h,g,\psi}(\Omega_T)\).

We first assume that \(h\) is uniformly Lipschitz on \([0,T]\) and that \(h_0\in C^0(\overline{\Omega})\). Set \[H_m(u_t):=(dd^cu_t)^m\wedge\omega^{n-m},\qquad b:=\min(1,T/2).\] For \(\kappa>0\), let \(\mathcal{S}^\kappa_{h,g,\psi}\) be the subclass of \(\mathcal{S}_{h,g,\psi}\) consisting of all \(u\) such that \[\operatorname*{ess\,sup}_{\Omega}|\partial_tu(t,\cdot)| \le \frac{\kappa}{\min(t,b)} \quad\text{for a.e. }t,\] and set \(U^\kappa:=\sup\{u:u\in\mathcal{S}^\kappa_{h,g,\psi}\}\). The truncated class is stable under finite maxima. By Choquet’s lemma, there is an increasing sequence \(u_j\in\mathcal{S}^\kappa_{h,g,\psi}\) such that \[(U^\kappa)^*=\left(\sup_j u_j\right)^* .\] Since the \(u_j\)’s have a common local Lipschitz bound in time, they increase to \((U^\kappa)^*\) a.e. and \((U^\kappa)^*\in P_m(\Omega_T)\). By the monotone convergence theorem for Hermitian Hessian measures, \[dt\wedge H_m(u_{j,t})\longrightarrow dt\wedge H_m(((U^\kappa)^*)_t)\] weakly. Moreover \(\partial_tu_j+\psi\to \partial_t(U^\kappa)^*+\psi\) in the sense of distributions and is locally uniformly bounded. The exponential lower semicontinuity lemma therefore gives \[e^{\partial_t (U^\kappa)^*+\psi}g\,dt\wedge dV \le \liminf_{j\to\infty} e^{\partial_tu_j+\psi}g\,dt\wedge dV .\] Thus \((U^\kappa)^*\) is a pluripotential subsolution. Since \((U^\kappa)^*\le U^*\le h\) on \(\partial_0\Omega_T\), and the same time-Lipschitz bound is inherited by the limit, we get \[(U^\kappa)^*=U^\kappa\in\mathcal{S}^\kappa_{h,g,\psi}(\Omega_T).\]

We next show that \(U^\kappa\) is independent of \(\kappa\) for large \(\kappa\). Choose \(\rho\le0\), bounded and \((\omega,m)\)-subharmonic, such that \(H_m(\rho)\ge e^{-B}g\,dV\). For \(u\in\mathcal{S}^\kappa_{h,g,\psi}\), \(s\) close to \(1\), and \(a:=1-2|s-1|\), define on \(\Omega_S\), \(sS<T\), \[w_s(t,z):=a s^{-1}u(st,z)+(1-a)\rho(z)-C(1-a)(t+1).\] By the mixed Hessian inequality and homogeneity of \(H_m\), \(w_s\) is a subsolution once \(C\) absorbs the errors coming from \(-m\log s\), the Lipschitz variation of \(\psi\), the boundedness of \(\psi\), and the lower bound for \(H_m(\rho)\). Enlarging \(C\), the uniform Lipschitz bound of \(h\) also gives \(w_s^*\le h\) on \(\partial_0\Omega_S\). The factor \(a\) ensures that, for \(\kappa\ge\kappa_0\), \[\operatorname*{ess\,sup}_{\Omega}|\partial_tw_s(t,\cdot)| \le \frac{\kappa}{\min(t,b)} .\] Using the truncated identity principle, \(w_s\le U^\kappa\) in \(\Omega_S\). Taking the supremum over \(u\in\mathcal{S}^\kappa\) and letting \(s\to1^\pm\), we obtain \(t|\partial_tU^\kappa|\le \kappa_0\). Hence \(U^\kappa\in\mathcal{S}^{\kappa_0}\), and so \[U^\kappa=U^{\kappa_0}\qquad(\kappa\ge\kappa_0).\]

Now let \(v\in\mathcal{S}_{h,g,\psi}(\Omega_T)\). For \(S<T\) and \(\varepsilon>0\), set \[v_\varepsilon(t,z):= v(t+\varepsilon,z)-C\varepsilon(t+1)-\theta(\varepsilon), \qquad (t,z)\in\Omega_S,\] where \(C\) is large and \(\theta(\varepsilon)=\sup_{\overline{\Omega}}|U^*(\varepsilon,\cdot)-h_0|\). Then \(v_\varepsilon\in\mathcal{S}^\kappa_{h,g,\psi}(\Omega_S)\) for some \(\kappa\), hence \(v_\varepsilon\le U^{\kappa_0}\). Letting \(\varepsilon\to0\), then \(S\to T\), gives \(v\le U^{\kappa_0}\). Taking the supremum over \(v\) yields \(U\le U^{\kappa_0}\), while the reverse inequality is immediate. Thus \[U=U^{\kappa_0}\in\mathcal{S}_{h,g,\psi}(\Omega_T).\]

If \(h_0\) is not continuous, take \(h_0^j\in SH_m(\Omega,\omega)\cap C^0(\overline{\Omega})\) with \(h_0^j=h_0\) on \(\partial\Omega\) and \(h_0^j\searrow h_0\). Let \(U^j:=U_{h^j,g,\psi,\Omega_T}\). The previous case gives \(U^j\in\mathcal{S}_{h^j,g,\psi}\), and \(U^j\searrow V\ge U\). The monotone convergence of Hessian measures and the same exponential lower semicontinuity show that \(V\) is a subsolution with \(V^*\le h\); hence \(V\le U\), so \(V=U\).

Finally, for boundary data satisfying only the weighted Lipschitz condition, fix \(S<T\) and use shifted data \[h^\varepsilon(t,\zeta)=h(t+\varepsilon,\zeta),\qquad h^\varepsilon(0,\cdot)=h_0+\phi_\varepsilon,\] where \(H_m(\phi_\varepsilon)=0\) and \(\phi_\varepsilon|_{\partial\Omega}=h(\varepsilon,\cdot)-h_0\). Then \(h^\varepsilon\) is uniformly Lipschitz on \([0,S]\) and \(h^\varepsilon\to h\) uniformly on \(\partial_0\Omega_S\). The corresponding envelopes converge uniformly to \(U\) on \(\Omega_S\), and the preceding limit argument gives that \(U\) is a subsolution on \(\Omega_S\). Since \(S<T\) is arbitrary, \(U\in\mathcal{S}_{h,g,\psi}(\Omega_T)\). ◻

Proof. Taking \(\Phi\) as the subsolution and \(\Psi\) as the supersolution, and noting that they share the same boundary data \(h_\Phi = h_\Psi = h\) which satisfies condition ?? , the comparison principle directly yields: \[\Phi \le \Psi \quad \text{in } \Omega_T.\]

Symmetrically, by reversing their roles, taking \(\Psi\) as the subsolution and \(\Phi\) as the supersolution, the comparison principle again applies and gives: \[\Psi \le \Phi \quad \text{in } \Omega_T.\]

Combining these two inequalities, we conclude that \(\Phi = \Psi\) everywhere in \(\Omega_T\). ◻

Proof. Fix \(S \in (0, T)\) and a small constant \(\varepsilon > 0\). We aim to show that \[\Phi \le \Psi + 2\varepsilon t \quad \text{in } \Omega_S.\] Consider the upper semi-continuous function on \([0, S] \times \overline{\Omega}\) defined by \[W(t,z) := \Phi(t,z) - \Psi(t,z) - 2\varepsilon t.\] Since \(h_\Phi \le h_\Psi\), we have \(W \le 0\) on the parabolic boundary \(\partial_0 \Omega_S\). We argue by contradiction and assume that the maximum of \(W\) is strictly positive and is attained at some interior point \((t_0, z_0) \in (0, S] \times \Omega\).

Define the compact set where the maximum is achieved at time \(t_0\): \[K := \{ z \in \Omega \mid W(t_0, z) = W(t_0, z_0) \}.\] By the maximum principle in the time variable, we must have \[\partial_t \Phi(t_0, z) \ge \partial_t^- \Psi(t_0, z) + 2\varepsilon, \quad \forall z \in K.\] Since \(\Psi\) is locally uniformly semi-concave in \(t\) and continuous on \([0,T) \times \Omega\), its left time-derivative \(\partial_t^- \Psi(t, z)\) is upper semi-continuous in \([0,T) \times \Omega\). Thus, there exists a sufficiently small radius \(r > 0\) such that the following inequality holds on the open neighborhood \(B := \{ z \in \Omega \mid \operatorname{dist}(z, K) < r \}\): \[\label{eq:time95deriv95ineq} \partial_t \Phi(t_0, z) \ge \partial_t^- \Psi(t_0, z) + \varepsilon, \quad \forall z \in B.\tag{33}\]

To simplify notations, let \(\varphi := \Phi(t_0, \cdot)\) and \(v := \Psi(t_0, \cdot)\) be the spatial slices at \(t_0\). Since \(\Phi\) is a pluripotential subsolution and \(\Psi\) is a supersolution, 15 implies that in the sense of measures on \(B\): \[\begin{align} (dd^c \varphi)^m \wedge \omega^{n-m} &\ge e^{\partial_t \Phi(t_0, z) + \psi(t_0, z)} g(z) dV, \\ (dd^c v)^m \wedge \omega^{n-m} &\le e^{\partial_t^- \Psi(t_0, z) + \psi(t_0, z)} g(z) dV. \end{align}\] Combining these with 33 , we obtain the strict measure inequality on \(B\): \[\label{eq:measure95strict95ineq} (dd^c \varphi)^m \wedge \omega^{n-m} \ge e^\varepsilon (dd^c v)^m \wedge \omega^{n-m} \quad \text{in } B.\tag{34}\]

Now, we define the lifted function \(\varphi_r := \varphi + m_r\), where \(m_r := \min_{\partial B} (v - \varphi)\). By definition, \(v \ge \varphi_r\) on \(\partial B\). The comparison principle [28] then yields that \(\varphi_r \le v\) everywhere in \(B\).

Evaluating this inequality at the maximum point \(z_0 \in K \subset B\), we get: \[\varphi_r(z_0) - v(z_0) = \varphi(z_0) - v(z_0) + \min_{\partial B} (v - \varphi) \le 0,\] which rearranges to: \[\varphi(z_0) - v(z_0) \le \max_{\partial B} (\varphi - v).\] Since \(K \cap \partial B = \emptyset\) by construction, the maximum of \(\varphi - v\) on \(\overline{B}\) is strictly attained in the interior, so \(\varphi(z) - v(z) < \varphi(z_0) - v(z_0)\) for all \(z \in \partial B\). This directly contradicts the inequality above. ◻

Proof. It is clear that \(\Psi\) is continuous on \((0,T) \times \Omega\). Indeed, the boundary data \(h_\Psi\) is continuous on \([0,T) \times \partial\Omega\). Thus, each slice is a supersolution of the elliptic complex Hessian equation and hence is continuous by [28]. However, unlike 21, \(\Phi\) is only assumed to be continuous in the interior, not necessarily on \([0,T) \times \overline{\Omega}\).

Fix an arbitrary \(S \in (0, T)\). We will prove that \(\Phi \le \Psi\) on the sub-cylinder \(\Omega_S\). For a sufficiently small \(s \in (0, (T-S)/2)\), we define a lifted and time-shifted supersolution \(v\) on \([0,S] \times \overline{\Omega}\) as follows: \[v(t,z) := \Psi(t+s, z) + c(s) + \delta(s) + Ast,\] where \(c(s) > 0\) is the initial error term from 22 satisfying \(\lim_{s \to 0^+} c(s) = 0\), \(A > 0\) is a large constant to be chosen, and \(\delta(s)\) is the uniform modulus of continuity of the boundary data \(h_\Psi\) on the \(\partial_s\Omega_S\): \[\delta(s) := \sup \left\{ |h_\Psi(\tau, z) - h_\Psi(t, z)| \;\middle|\; z \in \partial\Omega, \, t,\tau \in [0,S], \, |t-\tau| \le s \right\}.\] Clearly, we have \(\lim_{s \to 0^+} \delta(s) = 0\).

Let us verify that \(v\) is indeed a supersolution on \(\Omega_S\). Since \(\Psi\) is a supersolution on \(\Omega_T\), evaluating the Hessian measure of \(v\) at time \(t\) yields: \[\begin{align} (dd^c v(t,\cdot))^m \wedge \omega^{n-m} &= (dd^c \Psi(t+s, \cdot))^m \wedge \omega^{n-m} \\ &\le e^{\partial_t \Psi(t+s,z) + \psi(t+s,z)} g(z) dV. \end{align}\] To ensure \(v\) is a supersolution, we require this measure to be bounded from above by \(e^{\partial_t v(t,z) + \psi(t,z)} g(z) dV\). Since \(\partial_t v(t,z) = \partial_t \Psi(t+s,z) + As\), this condition reduces to: \[\psi(t+s, z) \le \psi(t,z) + As.,\] and hence it suffices to choose \(A\geq\kappa_\psi\).

Next, we check the boundary conditions of \(v\) on the parabolic boundary \(\partial_0 \Omega_S\). On the lateral boundary \((0,S] \times \partial\Omega\), the definition of \(\delta(s)\) and the fact that \(\Psi = h_\Psi\) on the boundary gives: \[v(t,z) \ge \Psi(t+s, z) + \delta(s) = h_\Psi(t+s, z) + \delta(s) \ge h_\Psi(t,z).\] On the initial bottom slice \(\{0\} \times \Omega\), 22 guarantees that \(\Psi(s,z) \ge h_\Psi(0,z) - c(s)\), which implies: \[v(0,z) = \Psi(s,z) + c(s) + \delta(s) \ge h_\Psi(0,z) + \delta(s) \ge h_\Psi(0,z).\] Thus, \(v\) is a continuous supersolution on \([0,S] \times \overline{\Omega}\) that dominates \(h_\Psi\) on \(\partial_0 \Omega_S\).

It then follows from 21 that \[\Phi(t,z) \le v(t,z) = \Psi(t+s, z) + c(s) + \delta(s) + Ast \quad \text{in } \Omega_S.\] Finally, letting \(s \to 0^+\), the continuity of \(\Psi\) in the interior of the domain, along with the facts that \(c(s) \to 0\) and \(\delta(s) \to 0\), yields: \[\Phi(t,z) \le \Psi(t,z) \quad \text{in } \Omega_S.\] Since \(S < T\) was chosen arbitrarily, the proof is complete. ◻

Proof of 10. Without loss of generality, we may focus on a slightly smaller parabolic cylinder \(\Omega_S\) for any fixed \(S \in (0, T)\), and aim to prove \(\Phi \le \Psi\) on \(\Omega_S\). Letting \(S \to T\) will then yield the global result.

Step 1: Time-shift. For \(s > 0\) sufficiently close to \(1\) (say, \(s \in [1/2, 3/2]\)), we introduce the time-shifted function: \[W^s(t,z) := s^{-1} \Phi(st, z) - C|s-1|(t+1),\] where \(C > 0\) is chosen large enough such that \(W^s\) is a pluripotential subsolution, as in 13. Moreover, on the parabolic boundary \(\partial_0 \Omega_S\), the Lipschitz property of \(h_\Phi\) ensures that \(W^s \le h_\Phi + O(|s-1|)\).

Step 2: Time Convolution via Riemann Sums. Let \(\eta_\varepsilon \ge 0\) be a standard smooth mollifier supported in \((1-\varepsilon, 1+\varepsilon)\) such that \(\int_{\mathbb{R}} \eta_\varepsilon(s) ds = 1\). We define the time-mollified function: \[\Phi^\varepsilon(t,z) := \int_{\mathbb{R}} W^s(t,z) \eta_\varepsilon(s) ds.\] By construction, \(\Phi^\varepsilon\) is of class \(C^1\) in \(t\). We claim that \(\Phi^\varepsilon\) is also a pluripotential subsolution. Instead of using the subsolution theorem in [51], as was done in [17], we prove this directly by approximating the integral with finite Riemann sums, which relies crucially on the Lipschitz and semi-concavity properties of \(\Phi\).

For any integer \(N\), we approximate \(\Phi^\varepsilon\) by the convex combination (Riemann sum): \[\Sigma_N(t,z) := \sum_{i=1}^N c_i W^{s_i}(t,z),\] where \(s_i \in (1-\varepsilon, 1+\varepsilon)\), \(c_i = \eta_\varepsilon(s_i) \Delta s_i \ge 0\), and \(\sum_{i=1}^N c_i = 1\).

Since each \(W^{s_i}\) is locally uniformly semi-concave in \(t\), using 15, we have for each \(i\) and every fixed \(t\), \[(dd^c W^{s_i}(t,\cdot))^m\wedge\omega^{n-m} \ge f_i(t,z)\,g(z)dV, \qquad f_i(t,z):=\exp\bigl(\partial_t^+W^{s_i}(t,z)+\psi(t,z)\bigr).\] By the mixed Hessian inequality (cf. [46]) for bounded \((\omega,m)\)-subharmonic functions, applied to the finite family \(W^{s_1},\ldots,W^{s_N}\), we obtain \[\begin{align} (dd^c \Sigma_N(t,\cdot))^m\wedge\omega^{n-m} &= \left(dd^c\sum_{i=1}^N c_i W^{s_i}(t,\cdot)\right)^m \wedge\omega^{n-m} \\ &\ge \left(\sum_{i=1}^N c_i f_i(t,z)^{1/m}\right)^m g(z)dV \\ &\ge \prod_{i=1}^N f_i(t,z)^{c_i}\,g(z)dV \\ &= \exp\left(\sum_{i=1}^N c_i \bigl(\partial_t^+W^{s_i}(t,z)+\psi(t,z)\bigr)\right)g(z)dV \\ &= \exp\bigl(\partial_t^+\Sigma_N(t,z)+\psi(t,z)\bigr)g(z)dV . \end{align}\] Here the second inequality is the weighted arithmetic-geometric mean inequality, and in the last line we used \[\partial_t^+\Sigma_N = \sum_{i=1}^N c_i\,\partial_t^+W^{s_i},\] which follows from the local uniform semi-concavity in time.

We now pass to the limit as \(N\to\infty\). Fix a compact interval \(J\Subset(0,S)\), and choose \(\varepsilon>0\) so small that \(sJ\Subset(0,S)\) for all \(s\in\operatorname{Supp}\eta_\varepsilon\). Set \[J_\varepsilon:=\bigcup_{s\in\operatorname{Supp}\eta_\varepsilon}sJ .\] Since \(\Phi\) is bounded and locally uniformly Lipschitz in \(t\), there is a constant \(C_J>0\) such that, for all \(s,s'\in\operatorname{Supp}\eta_\varepsilon\), \(t\in J\), and \(z\in\Omega\), \[|W^s(t,z)-W^{s'}(t,z)| \le C_J |s-s'|.\] Indeed, this follows directly from \[W^s(t,z)=s^{-1}\Phi(st,z)-C|s-1|(t+1),\] the uniform boundedness of \(\Phi\) on \(J_\varepsilon\times\Omega\), and the uniform Lipschitz bound of \(\Phi\) in the time variable on \(J_\varepsilon\).

It follows that the Riemann sums \(\Sigma_N\) converge uniformly on \(J\times\Omega\) to \[\Phi^\varepsilon(t,z) := \int_{\mathbb{R}}W^s(t,z)\eta_\varepsilon(s)\,ds .\] In particular, for every fixed \(t\in J\), \(\Sigma_N(t,\cdot)\longrightarrow \Phi^\varepsilon(t,\cdot)\) uniformly on \(\Omega\). Therefore, by the continuity of the Hermitian Hessian operator under uniform convergence of bounded \((\omega,m)\)-subharmonic functions, we obtain \[(dd^c\Sigma_N(t,\cdot))^m\wedge\omega^{n-m} \longrightarrow (dd^c\Phi^\varepsilon(t,\cdot))^m\wedge\omega^{n-m}\] weakly as Radon measures on \(\Omega\), for every \(t\in J\).

Secondly, we analyze the right-hand side. Because \(\Phi\) is locally uniformly semi-concave in time, its time derivative \(\partial_\tau \Phi(\tau, z)\) exists and is continuous outside a countable set of times for each fixed \(z\). Therefore, the map \(s \mapsto \partial_t W^s(t,z)\) is continuous almost everywhere with respect to the Lebesgue measure on \((1-\varepsilon, 1+\varepsilon)\) for each fixed \(z\), making it Riemann integrable. This implies that the sequence of Riemann sums \(\partial_t \Sigma_N(t,z)= \sum_{i=1}^N c_i \partial_tW^{s_i}(t,z)\) converges pointwise to the integral \(\partial_t \Phi^\varepsilon(t,z)= \int_{\mathbb{R}} \partial_tW^s(t,z) \eta_\varepsilon(s) ds\) for almost all \((t,z)\). Furthermore, the locally uniform Lipschitz condition ensures that \(\partial_t \Sigma_N\) is uniformly bounded independently of \(N\). By the Dominated Convergence Theorem, the right-hand side converges in the weak sense of measures: \[\exp( \partial_t \Sigma_N(t,z) + \psi(t,z) ) g(z) dV \to e^{\partial_t \Phi^\varepsilon(t,z) + \psi(t,z)} g(z) dV.\] Combining both limits, the inequality is preserved, yielding: \[(dd^c \Phi^\varepsilon(t,\cdot))^m \wedge \omega^{n-m} \ge e^{\partial_t \Phi^\varepsilon(t,z) + \psi(t,z)} g(z) dV\] in the sense of measures. Thus, \(\Phi^\varepsilon\) is indeed a pluripotential subsolution on \(\Omega_S\).

Step 3: Conclusion. Since \(\Phi^\varepsilon\) is an average of \(W^s\), and \(W^s \le h_\Phi + O(|s-1|)\) on the parabolic boundary, we have: \[\Phi^\varepsilon \le h_\Phi + A \varepsilon \le h_\Psi + A \varepsilon \quad \text{on } \partial_0 \Omega_S\] for some uniform constant \(A > 0\). We can thus define a slightly shifted function: \[\tilde{\Phi}^\varepsilon(t,z) := \Phi^\varepsilon(t,z) - A \varepsilon (t+1),\] which remains a \(C^1\)-in-time subsolution and satisfies \(\tilde{\Phi}^\varepsilon \le h_\Psi\) on \(\partial_0 \Omega_S\).

We are now exactly in the setting to apply the interior comparison principle. Applying 23 to the pair \((\tilde{\Phi}^\varepsilon, \Psi)\), we deduce that: \[\tilde{\Phi}^\varepsilon(t,z) \le \Psi(t,z) \quad \text{in } \Omega_S.\] Finally, letting \(\varepsilon \to 0\), the standard properties of the mollifier ensure that \(\tilde{\Phi}^\varepsilon\) converges pointwise to \(\Phi(t,z)\). This yields \(\Phi \le \Psi\) in \(\Omega_S\), concluding the proof. ◻

Proof. We refer to [48]–[50] for concepts and terminology employed in the proof. Choose an open neighborhood \(X\Subset\mathbb{C}^n\) of \(\overline{\Omega}\) on which \(\omega\) is defined. By Tietze’s extension theorem, after shrinking \(X\) if necessary, we may extend \(f\) to a continuous function, still denoted by \(f\), on \(X\).

Let \(F_\omega\subset J^2(X)\) be the reduced subequation defined by \[J^2_z q\in (F_\omega)_z \quad\Longleftrightarrow\quad dd^cq(z)\in \overline{\Gamma}_m(\omega(z)).\] Equivalently, if \(\lambda_\omega(dd^cq(z))\) denotes the eigenvalues of \(dd^cq(z)\) with respect to \(\omega(z)\), then \[J^2_z q\in (F_\omega)_z \quad\Longleftrightarrow\quad \lambda_\omega(dd^cq(z))\in\overline{\Gamma}_m .\] The value and gradient components of the jet play no role, so this is of the form considered in the obstacle theorem for reduced subequations.

We verify the hypotheses of the Harvey–Lawson–Pliś obstacle theorem [48]. First, since \(\overline{\Gamma}_m\) is a closed convex cone and is stable under addition of non-negative Hermitian forms, \(F_\omega\) is a closed convex degenerate elliptic subequation. Moreover, \[F_\omega+F_\omega\subset F_\omega,\] so \(F_\omega\) itself is a monotonicity cone for \(F_\omega\).

Next, \(F_\omega\) is locally affinely jet-equivalent to the constant coefficient complex Hessian subequation. Indeed, in a local holomorphic coordinate chart write \[\omega(z)=\sqrt{-1}\,g_{j\bar k}(z)\,dz^j\wedge d\bar z^k\] and put \(S(z):=g(z)^{-1/2}\). If \(H\) is the Hermitian matrix representing the \((1,1)\)-part of a real Hessian, define \[T_z(H):=S(z)H S(z)^* .\] Then \[H\in \overline{\Gamma}_m(\omega(z)) \quad\Longleftrightarrow\quad T_z(H)\in \overline{\Gamma}_m(\omega_{\mathrm{eucl}}).\] Extending this transformation by the identity on the gradient part and on the real Hessian directions which do not enter \(dd^c\), we obtain a smooth fiber automorphism of \(J^2_{\mathrm{red}}(X)\). This gives the required local affine jet-equivalence to the constant coefficient complex Hessian subequation.

The strict monotonicity hypothesis is also satisfied. Since \(dd^c|z|^2\) is positive definite, its eigenvalues with respect to the Hermitian metric \(\omega(z)\) lie in \(\Gamma_n\subset\Gamma_m\). Hence \(|z|^2\) is strictly \(F_\omega\)-subharmonic on a neighborhood of \(\overline{\Omega}\).

It remains to check the boundary convexity. Since \(\Omega\) is smoothly strictly \(m\)-pseudoconvex, there exists a smooth defining function \(\rho\) on a neighborhood of \(\overline{\Omega}\) such that \[\Omega=\{\rho<0\},\qquad d\rho\ne0 \text{ on }\partial\Omega, \qquad dd^c\rho\in\Gamma_m(\omega)\] near \(\partial\Omega\). Thus \(\rho\) is strictly \(F_\omega\)-subharmonic near \(\partial\Omega\), and \(\partial\Omega\) is strictly \(F_\omega\)-convex.

Furthermore, since \(F_\omega+F_\omega\subset F_\omega\), we have \[\operatorname{Int}F_\omega\subset \operatorname{Int}\widetilde{F}_\omega .\] Indeed, if \(J\in\operatorname{Int}F_\omega\) and \(-J\in F_\omega\), then by convexity one would get \(0\in\operatorname{Int}F_\omega\), which is impossible for the Hessian cone, since \(0\in\partial F_\omega\). Hence \(-J\notin F_\omega\), which is equivalent to \(J\in\operatorname{Int}\widetilde{F}_\omega\). Therefore the same defining function \(\rho\) is also strictly \(\widetilde{F}_\omega\)-subharmonic near \(\partial\Omega\), and \(\partial\Omega\) is strictly \(\widetilde{F}_\omega\)-convex.

We may now apply the obstacle theorem for locally affinely jet-equivalent convex subequations [48]. It gives that the largest \(F_\omega\)-subharmonic minorant of the continuous obstacle \(f\), namely \[H_f(z):= \sup\{w(z): w\in F_\omega(\Omega),\;w\le f \text{ in }\Omega\},\] is continuous on \(\overline{\Omega}\), belongs to \(F_\omega(\Omega)\), satisfies \[H_f\le f \quad\text{in }\Omega,\] and assumes the boundary value \[H_f=f \quad\text{on }\partial\Omega .\]

We next identify \(F_\omega(\Omega)\) with \(\operatorname{SH}_m(\Omega,\omega)\). For smooth functions this is exactly the definition of the complex Hessian cone. For upper semicontinuous functions, the viscosity and distributional notions agree for convex subequations by the strong Bellman principle [50]. Thus \[F_\omega(\Omega)=\operatorname{SH}_m(\Omega,\omega)\] after taking the usual upper semi-continuous representative.

It remains to compare \(H_f\) with the envelope \(P_\omega(f)\). Since \(H_f\in C^0(\overline{\Omega})\cap\operatorname{SH}_m(\Omega,\omega)\), \(H_f\le f\) in \(\Omega\), and \(H_f=f\) on \(\partial\Omega\), the function \(H_f\) is admissible in the definition of \(P_\omega(f)\). Hence \[P_\omega(f)\ge H_f .\] Conversely, if \(v\in\operatorname{SH}_m(\Omega,\omega)\cap L^\infty(\Omega)\) satisfies \(v^*\le f\) on \(\overline{\Omega}\), then in particular \(v\le f\) in \(\Omega\). Hence \(v\) is a competitor for the obstacle envelope \(H_f\), and therefore \(v\le H_f\). Taking the supremum over all such \(v\), and then upper semi-continuous regularization, gives \[P_\omega(f)\le H_f,\] because \(H_f\) is continuous. Thus \[P_\omega(f)=H_f .\] Consequently \[P_\omega(f)\in \operatorname{SH}_m(\Omega,\omega)\cap C^0(\overline{\Omega}).\] Finally, since \(H_f=f\) on \(\partial\Omega\), we obtain \[\lim_{\Omega\ni z\to\zeta}P_\omega(f)(z) = \lim_{\Omega\ni z\to\zeta}H_f(z) = f(\zeta), \qquad \zeta\in\partial\Omega .\] This proves the lemma. ◻

Proof. By the resolution of the homogeneous Dirichlet problem for the complex Hessian equation (cf. [28]), there exists a unique continuous \((\omega, m)\)-subharmonic function \(v \in \operatorname{SH}_m(\Omega, \omega) \cap C^0(\overline{\Omega})\) such that \[(dd^c v)^m \wedge \omega^{n-m} = 0 \quad \text{in } \Omega,\] with boundary values \(v = \phi\) on \(\partial \Omega\). Since \(u\) shares the same boundary values \(\phi\) and satisfies \((dd^c u)^m \wedge \omega^{n-m} \ge 0\), the standard comparison principle yields \(u \le v\) in \(\Omega\).

We now take a sequence of continuous functions \(f_j \in C^0(\overline{\Omega})\) such that \(f_j \searrow u\) pointwise in \(\overline{\Omega}\). By replacing \(f_j\) with \(\min(f_j, v)\), we may assume without loss of generality that \(f_j = \phi\) on \(\partial \Omega\). For each \(j\), we define the \((\omega, m)\)-subharmonic envelope: \[u_j := \sup \left\{ w \in \operatorname{SH}_m(\Omega, \omega) \mid w^* \le f_j \text{ in } \overline{\Omega} \right\}.\] Since \(u \le f_j\), we clearly have \(u \le u_j \le f_j\). In particular, this forces \(\lim_{z \to \zeta} u_j(z) = \phi(\zeta)\) for all boundary points \(\zeta \in \partial \Omega\). By 24, the envelope \(u_j\) is continuous on \(\overline{\Omega}\). Since the sequence \(\{f_j\}\) is decreasing, \(\{u_j\}\) is also a decreasing sequence of continuous \((\omega,m)\)-subharmonic functions converging pointwise to \(u\). ◻

Proof. Fix a small \(\varepsilon > 0\) and \(S<T\). We define the boundary data \(h^\varepsilon\) on the lateral side of the sub-cylinder \(\mathbb{B} \times [0, S]\) exactly as in 20.

It is clear that \(h^\varepsilon\) is uniformly Lipschitz and semi-concave on \([0,S]\), hence we have a solution \(u^\varepsilon\) of the Hessian flow with boundary data \(h^\varepsilon\). The parabolic comparison principle 10 yields that \(u^\varepsilon=U^\varepsilon:=U_{h^\varepsilon,\psi,g,S}\). By the identity principle 16, we get \(U^\varepsilon\to U:=U_{h,\psi,g,S}\) uniformly on \(\mathbb{B}_S\) as \(\varepsilon\to0\). Since \(h_0\) is continuous, we also have that \(U\) is continuous on \(\overline{\mathbb{B}_S}\) by 7 and 20.

Moreover, for any \(t \in (0, T-\varepsilon]\), since \(\frac{t}{t+\varepsilon} < 1\), we compute: \[t |\partial_t h^\varepsilon(t, \zeta)| = \frac{t}{t+\varepsilon} \cdot (t+\varepsilon) |\partial_t h(t+\varepsilon, \zeta)| \le 1 \cdot \kappa_h = \kappa_h.\] Similarly, the semi-concavity bound is uniformly preserved: \[t^2 \partial_t^2 h^\varepsilon(t, \zeta) = \frac{t^2}{(t+\varepsilon)^2} \cdot (t+\varepsilon)^2 \partial_t^2 h(t+\varepsilon, \zeta) \le 1 \cdot C_h = C_h.\] Since the constants \(\kappa_h\) and \(C_h\) are preserved independently of \(\varepsilon\), the sequence of continuous solutions \(\{u^\varepsilon\}\) satisfies uniform local a priori estimates on any compact time subinterval. By 13, we can argue as in Step 2 of 14 to get the convergence of measures on the right-hand side. The proof is thus finished. ◻

Proof. By our previous a priori estimates and 9, \(U\) is locally uniformly semi-concave in \(t \in (0,T)\), \(U \in \mathcal{S}_{h,g,\psi}(\Omega_T)\), and it assumes the correct boundary values on \(\partial_0 \Omega_T\). It remains to verify that \(U\) actually solves the parabolic complex Hessian equation. We proceed in three steps.

Step 1: Continuous data. We first assume that \(h_0\) and the density \(g\) are continuous on \(\overline{\Omega}\). In this case, the envelope \(U\) is continuous on \([0,T) \times \overline{\Omega}\).

Let \(B \Subset \Omega\) be a small Euclidean ball. The restriction of \(U\) to the parabolic boundary \(\partial_0 B_T\) provides a continuous boundary data \(h_B\). Clearly, the restriction of \(U\) on \(B_T\) is a continuous subsolution to the Cauchy-Dirichlet problem in \(B_T\) with data \(h_B\). By 14, there exists a continuous pluripotential solution \(U_B\) in \(B_T\) such that \(U_B = h_B\) on \(\partial_0 B_T\). Moreover, we have \(U_B \ge U\) in \(B_T\) by the comparison principle 10.

We define the glued function \(V\) by setting \(V = U_B\) in \(B_T\) and \(V = U\) in \(\Omega_T \setminus B_T\). By the local nature of pluripotential subsolutions, it is easy to see that \(V \in \mathcal{S}_{h,g,\psi}(\Omega_T)\). Therefore, by the definition of the envelope, \(V \le U\). Since \(V \ge U\) by construction, we conclude \(V = U\). This implies that \(U = U_B\) in \(B_T\), meaning \(U\) solves the equation in \(B_T\). Since the ball \(B \Subset \Omega\) is arbitrary, \(U\) solves the equation in the whole \(\Omega_T\).

Step 2: \(L^p\) density approximation. We next assume \(h_0\) is continuous, but \(g \in L^p(\Omega)\) with \(p>\frac{n}{m}\). Let \((g_j)\) be a sequence of strictly positive continuous functions on \(\overline{\Omega}\) converging to \(g\) in \(L^p(\Omega)\). Let \(U^j := U_{h,g_j,\psi,\Omega_T}\) be the corresponding envelopes, which are uniformly bounded thanks to 16. Note that the uniform boundedness of \(\rho_j\) corresponding to \(g_j\) can be derived easily by the comparison principle and the stability estimate in [28]. Moreover, 5 and 9 guarantee that the sequence \(\{U^j\}\) is locally uniformly semi-concave and Lipschitz in time with uniform constants.

By 9, up to extracting a subsequence, \(U^j\) converges almost everywhere to a limit \(V \in P_m(\Omega_T)\cap L^\infty(\Omega_T)\). By 13, \(\partial_t U^j \to \partial_t V\) pointwise almost everywhere and we have the weak convergence \[e^{\partial_tU^j+\psi}g_jdV\to e^{\partial_tV+\psi}gdV\] Observe also that all the \(U^j,V\) shares the same boundary data, the stability of complex Hessian equations (see [28]) ensures that \((dd^c U^j_t)^m \wedge \omega^{n-m}\) converges weakly to \((dd^c V_t)^m \wedge \omega^{n-m}\). Hence, passing to the limit, \(V\) solves the equation with density \(g\) in \(\Omega_T\). Also, 17 and 3 yield that \(V^*|_{\partial_0 \Omega_T} \le h\), which means \(V \in \mathcal{S}_{h,g,\psi}(\Omega_T)\) and therefore \(V \le U\).

To prove the reverse inequality \(U \le V\), we use a perturbation argument, following [4], [17], [18]. For each \(j\), let \(\theta_j \in \operatorname{SH}_m(\Omega, \omega) \cap C^0(\overline{\Omega})\) be the unique solution to \((dd^c \theta_j)^m \wedge \omega^{n-m} = |g_j - g|dV\) with \(\theta_j = 0\) on \(\partial \Omega\). The stability theorem again gives \(\lim_{j \to \infty} \sup_{\overline{\Omega}} |\theta_j| = 0\). Fix \(0 < S < T\) and a small \(\varepsilon > 0\), and define on \(\Omega_S\): \[W^j(t,z) := U(t+\varepsilon, z) - \kappa_\psi \varepsilon t + C(\varepsilon) \theta_j(z),\] where \(\kappa_\psi\) is the Lipschitz constant of \(\psi\) in time, and \(C(\varepsilon) > 0\) will be chosen shortly. We verify that \(W^j \in \mathcal{S}_{h,g_j,\psi}(\Omega_S)\). Evaluating its Hessian measure, we have: \[\begin{align} (dd^c W^j)^m \wedge \omega^{n-m} &\ge (dd^c U(t+\varepsilon, \cdot))^m \wedge \omega^{n-m} + C(\varepsilon)^m (dd^c \theta_j)^m \wedge \omega^{n-m} \\ &\ge e^{\partial_t U(t+\varepsilon, z) + \psi(t+\varepsilon, z)} g dV + C(\varepsilon)^m |g - g_j| dV. \end{align}\] By the Lipschitz property of \(\psi\), we have \(\psi(t+\varepsilon, z) \ge \psi(t, z) - \kappa_\psi \varepsilon\). Since \(\partial_t W^j(t,z) = \partial_t U(t+\varepsilon, z) - \kappa_\psi \varepsilon\), it follows that: \[\partial_t U(t+\varepsilon, z) + \psi(t+\varepsilon, z) \ge \partial_t W^j(t,z) + \psi(t,z).\] Thus, the measure is bounded below by: \[e^{\partial_t W^j + \psi} g dV + C(\varepsilon)^m |g - g_j| dV = e^{\partial_t W^j + \psi} g_j dV + \left( C(\varepsilon)^m - e^{\partial_t W^j + \psi} \right) |g - g_j| dV.\] We set \(C(\varepsilon) := (\sup_{\Omega_S} \exp(\partial_t U(t+\varepsilon, z) + \psi(t,z)))^{1/m} < +\infty\), which ensures the second term is non-negative. Consequently, \((dd^c W^j)^m \wedge \omega^{n-m} \ge e^{\partial_t W^j + \psi} g_j dV\). Since \(\theta_j \le 0\) by the comparison principle, modifying \(W^j\) slightly to account for the boundary error (which goes to zero as \(\varepsilon \to 0\)) shows \(W^j \leq U^j+b(\varepsilon)\) for some \(b(\varepsilon)\) such that \(b(\varepsilon)\to0\) as \(\varepsilon\to0\). Letting \(j \to \infty\) and then \(\varepsilon \to 0\), we recover \(U \le V\). Hence \(U = V\) solves the equation.

Step 3: Removing the continuity assumption on \(h_0\). We finally treat the general case where the initial data \(h_0\) is merely bounded and \((\omega,m)\)-subharmonic with continuous boundary values. By 25, we can find a decreasing sequence of continuous \((\omega,m)\)-subharmonic functions \(h_0^j \in C^0(\overline{\Omega})\) converging pointwise to \(h_0\) on \(\Omega\), such that \(h_0^j = h(0, \cdot)\) on \(\partial \Omega\). We then define a sequence of continuous Cauchy-Dirichlet data \(h^j\) by setting \(h^j = h\) on the lateral boundary \([0,T) \times \partial\Omega\) and \(h^j(0,\cdot) = h_0^j\) on \(\Omega\).

By the result of Step 2, the parabolic envelopes \(U^j := U_{h^j, g, \psi, \Omega_T}\) are continuous pluripotential solutions to the complex Hessian flow equation with boundary data \(h^j\). Since the boundary data \(h^j\) is decreasing, the sequence of solutions \(U^j\) is also decreasing. Moreover, since \(U \le U^j\) for all \(j\), the sequence \(U^j\) decreases pointwise to a limit \(V \in P_m(\Omega_T)\) with \(V\geq U\).

Now, 9 gives that \(U^j\) is locally uniformly semi-concave in time. By 13, we have \(\partial_t U^j \to \partial_t V\) almost everywhere, and the Hessian measures \((dd^c U^j_t)^m \wedge \omega^{n-m}\) converge weakly to \((dd^c V_t)^m \wedge \omega^{n-m}\). This guarantees that the limit \(V\) is a pluripotential solution to the complex Hessian flow. It is then easy to see that \(V\) belongs to the admissible class \(\mathcal{S}_{h,g,\psi}(\Omega_T)\), and therefore \(V \le U\) by the definition of the Perron envelope, thus \(U = V\), confirming that the envelope \(U\) is indeed the unique pluripotential solution to the Cauchy-Dirichlet problem. ◻

Proof. For any \(S \in (0,T)\), we define \(U^S := U_{h,g,\psi,\Omega_S}\). Since \(S < +\infty\), our previous result ensures that \(U^S\) solves the complex Hessian flow equation with \(U^S = h\) on \(\partial_0 \Omega_S\).

It follows from the identity principle 16 that, for \(0 < S_1 < S_2 < T\), we have \(U^{S_1} = U^{S_2}\) in \(\Omega_{S_1}\). Letting \(S \to T\), we obtain a function \(V \in P_m(\Omega_T)\) which solves the parabolic complex Hessian equation in \(\Omega_T\) and satisfies \(V = h\) on \(\partial_0 \Omega_T\).

Obviously, \(U \le U^S\) for all \(S \in (0,T)\), which implies \(U \le V\). On the other hand, since \(V\) is a pluripotential solution taking the boundary data \(h\), \(V\) is a valid candidate defining the envelope \(U\) (i.e., \(V \in \mathcal{S}_{h,g,\psi}(\Omega_T)\)), hence \(V \le U\). Therefore, \(V = U\) solves the equation in \(\Omega_T\).

Moreover, by our a priori estimates 5 and 9, \(U^S\) is locally uniformly Lipschitz and semi-concave in \(t \in (0,S)\), hence so is \(U\) in \((0,T)\).

It then follows from 15 that \[(dd^c U_t)^m \wedge \omega^{n-m} = e^{\partial_t U_t + \psi(t,\cdot)} g dV\] for almost every \(t \in (0,T)\). Since \(\partial_t U\) is locally bounded and \(\psi(t,\cdot)\) is continuous, the right-hand side belongs to \(L^p(\Omega)\) for \(p>\frac{n}{m}\). The regularity theorem [28] then ensures that \(U_t\) is continuous on \(\overline{\Omega}\) for almost all \(t \in (0,T)\). Since \(U\) is locally uniformly Lipschitz in \(t\), we infer that \(U\) is continuous in \((0,T) \times \overline{\Omega}\).

Finally, if the initial data \(h_0\) is continuous on \(\overline{\Omega}\), the boundary behavior of the envelope ensures that \(U(t,\cdot)\) continuously approaches \(h_0\) as \(t \to 0^+\). Combined with the interior continuity, this shows that \(U\) is globally continuous on \([0,T) \times \overline{\Omega}\). ◻