June 15, 2026
We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extension \(v\) of the boundary data \(g\), we prove maximal regularity estimates in mixed Lebesgue norms \(L^p_tL^q_x\) for any order derivative of \(v\) in terms of mixed Besov and Lizorkin–Triebel type norms of \(g\). We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity of \(g\) from the mixed-norm regularity of \(v\). As a model case, our results show that the natural \[\accentset{\large\bfseries .}{W}^{1,p}\big(\mathbb{R};L^q(\mathbb{R}^d_+)\big)\cap L^p\big(\mathbb{R};\accentset{\large\bfseries .}{W}^{2,q}(\mathbb{R}^d_+)\big)\] regularity norm of \(v\) is controlled by the \[\accentset{\large\bfseries .}{F}^{1-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)\cap L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{2-\frac{1}{q}}(\mathbb{R}^{d-1})\big)\] norm of \(g\). The maximal regularity estimate holds for \(1\leq p,q<\infty\), while the caloric trace estimate holds for \(1<p<\infty\) and \(1\leq q\leq\infty\). In particular, the endpoint cases \(p=1\) or \(q=1\) in the maximal regularity estimate are included and appear to be new. These endpoint estimates may be useful in the analysis of free-boundary Navier–Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier–Stokes flows exhibiting strong boundary singularities.
35K20, 42B25, 46E35
This note is concerned with maximal regularity and trace estimates for the heat equation. Let us first recall some known results. For \(1<p<\infty\) and sufficiently smooth domains \(\Omega\) in \(\mathbb{R}^d\), \(d\ge2\), the classical trace theorem, first proved by Gagliardo [1], states that \[\left\|v|_{\partial\Omega}\right\|_{W^{1-\frac{1}{p},p}(\partial\Omega)} \lesssim \left\|v\right\|_{W^{1,p}(\Omega)}.\] Here \(W^{1-\frac{1}{p},p}(\partial\Omega)\) denotes the Sobolev-Slobodeckij space, which agrees with the Besov space \(B^{1-\frac{1}{p}}_{p,p}(\mathbb{R}^{d-1})\) in this range of \(p\) if \(\partial\Omega=\mathbb{R}^{d-1}\) [2]. Gagliardo also proved the corresponding extension theorem for \(1<p<\infty\), while in the endpoint case \(p=1\) he showed that the natural trace space is \(L^1(\partial\Omega)\). We refer to [3] for an English account of these results.
We next recall the corresponding result for harmonic extensions in the half-space. For \(g\in C_c^\infty(\mathbb{R}^{d-1})\), its harmonic extension is the harmonic function \(v\) in \(\mathbb{R}^d_+\) given by the convolution of \(g\) with the Poisson kernel, \(v(x)=\int_{\mathbb{R}^{d-1}} \frac{2x_d\,g(y') \,{\rm d}y'}{|\partial B_1|\cdot |(y',0)-x|^d}\). For \(1\leq p<\infty\), \[\left\|g\right\|_{\accentset{\large\bfseries .}{B}^{1-\frac{1}{p}}_{p,p}(\mathbb{R}^{d-1})} \lesssim \left\|v\right\|_{\accentset{\large\bfseries .}{W}^{1,p}(\mathbb{R}^d_+)} \overset{\mathrm{def}}{=\joinrel\!\!=} \sum_{i=1}^d \left\|\partial_{x_i}v\right\|_{L^p(\mathbb{R}^d_+)} \lesssim \left\|g\right\|_{\accentset{\large\bfseries .}{B}^{1-\frac{1}{p}}_{p,p}(\mathbb{R}^{d-1})}.\] See, e.g., Mironescu-Russ [4]. Here the dot on top of \(B\) and \(W\) indicates homogeneous spaces. Hence the boundary homogeneous Besov norm of \(g\) is equivalent to the homogeneous Sobolev norm of its harmonic extension. The first inequality should be distinguished from the classical trace theorem. Indeed, due to the additional harmonicity constraint on \(v\), the endpoint case \(p=1\) is also included, whereas the corresponding classical trace estimate does not hold in general at \(p=1\). In particular, there exist functions in \(C_c^\infty(\mathbb{R}^{d-1})\) whose Besov seminorm \(\left\|g\right\|_{\accentset{\large\bfseries .}{B}^0_{1,1}}\) is infinite (see [4]), even though one can easily construct smooth extensions to \(\mathbb{R}^d_+\) with finite \(\accentset{\large\bfseries .}{W}^{1,1}(\mathbb{R}^d_+)\) norm. For this reason, we shall call the first inequality the harmonic trace estimate. We shall refer to the second inequality as the maximal regularity estimate for the harmonic extension, since the harmonic trace estimate shows that it is optimal.
For time-dependent functions, it is natural to use mixed \(L^p_tL^q_x\) type norms in the trace and extension estimates. This direction seems to have been initiated by Weidemaier [5]–[7]. In particular, he proved the trace estimate [7] \[\begin{align} \label{eq:Weidemaier-05} W^{1,p}\big(\mathbb{R};L^q(\mathbb{R}^d_+)\big) \cap L^p\big(\mathbb{R}; W^{2,q}(\mathbb{R}^d_+)\big) \to F^{1-\frac{1}{2q}}_{p,q} \big(\mathbb{R};\,L^q(\mathbb{R}^{d-1})\big) \cap L^p\big(\mathbb{R};\,B^{2-\frac{1}{q}}_{q,q}(\mathbb{R}^{d-1})\big), \end{align}\tag{1}\] for \(\frac{3}{2}<q\leq p<\infty\). Here \(F^{s}_{p,q}(\mathbb{R};X)\) denotes the Bochner-Lizorkin-Triebel space; see Section 2 for the definition. Weidemaier also studied the corresponding extension estimate in [6]. The exponents have been relaxed to \(1<p,q<\infty\) in Denk-Hieber-Prüss [8]. Further developments in this direction include the work of Johnsen-Sickel [9] on mixed-norm Lizorkin-Triebel spaces and that of Denk-Roodenburg [10] on weighted Sobolev spaces.
Related mixed-norm estimates for parabolic initial-boundary value problems were obtained by Denk–Hieber–Prüss [8]. In the model case of the heat equation in the half-space, \[\label{eq:HBV} \left\{ \begin{align} \,&\partial_t v-\Delta v=0, \quad x\in\mathbb{R}^d_+,\; t\in\mathbb{R},\\[3pt] & v(x',x_d,t)|_{x_d=0}=g(x',t), \quad x'\in\mathbb{R}^{d-1},\;t\in\mathbb{R}, \end{align} \right.\tag{2}\] a solution \(v\) is given by the convolution of \(g\) with the Poisson kernel for heat equation, \[\label{eq:def-v} v(x,t)=-\int_\mathbb{R}\int_{\mathbb{R}^{d-1}}2\partial_{x_d} \Gamma(x'-y',x_d,t-s)\,g(y',s)\,{\rm d}y'\,{\rm d}s,\tag{3}\] where \(\Gamma\) is the heat kernel \[\Gamma(x,t)= (4\pi t)^{-\frac{d}{2}}e^{\frac{-|x|^2}{4t}} \text{ for } t>0; \qquad \Gamma(x,t)=0 \text{ for } t\leq 0.\] When \(d=1\), 3 reduces to \[\label{eq:def-v-1D} v(x,t)=-\int_\mathbb{R}2\partial_{x} \Gamma(x,t-s)\,g(s)\,{\rm d}s.\tag{4}\] We shall refer to this solution \(v\) as the caloric extension of \(g\). The results of [8] imply that, for the caloric extension and \(1<p,q<\infty\), \[\begin{align} \label{eq:DHP} \left\|v\right\|_{W^{1,p}\big(\mathbb{R};L^q(\mathbb{R}^d_+)\big)\cap L^p\big(\mathbb{R}; W^{2,q}(\mathbb{R}^d_+)\big)} \lesssim \left\|g\right\|_{{F}^{1-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)\cap L^{p}\big(\mathbb{R};\,{B}_{q,q}^{2-\frac{1}{q}}(\mathbb{R}^{d-1})\big)}. \end{align}\tag{5}\] See [8] and also [11]. We call 5 the caloric extension estimate, or equivalently the maximal regularity estimate for the caloric extension. The corresponding reverse inequality, which estimates the boundary norm of \(g\) by the interior mixed norm of its caloric extension, will be called the caloric trace estimate.
The borderline case \(p=1\), corresponding to \(L^1\) regularity in time, is of particular interest. Such estimates do not follow from the standard maximal regularity theory in Banach spaces that satisfy the unconditional martingale differences condition (see [8] and the references therein). Moreover, recent work shows that \(L^1\)-in-time regularity provides a useful framework for obtaining global-in-time control in fluid equations [12]. In this direction, Ogawa–Shimizu [11] studied \(L^1\)-in-time estimates for parabolic equations. These estimates were later used in their study of the global well-posedness of the Navier–Stokes equations with free boundary conditions [13], [14]. Their results imply that, for caloric extension \(v\) of \(g\), \[\label{eq:Ogawa-Shimizu} \left\|v\right\|_{\accentset{\large\bfseries .}{W}^{1,1}\big(\mathbb{R};L^q(\mathbb{R}^d_+)\big)\cap L^1\big(\mathbb{R}; \accentset{\large\bfseries .}{W}^{2,q}(\mathbb{R}^d_+)\big)} \lesssim \left\|g\right\|_{{\accentset{\large\bfseries .}{F}}^{1-\frac{1}{2q}}_{1,1}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)\cap L^{1}\big(\mathbb{R};\,{B}_{q,1}^{2-\frac{1}{q}}(\mathbb{R}^{d-1})\big)},\tag{6}\] for \(1\leq q<\infty\); see [11]. Compared with the boundary spaces appearing in 5 , the second lower indices in both the Lizorkin–Triebel and Besov norms are \(1\) rather than \(q\). Furthermore, the Besov norm appearing in the last term of 6 is inhomogeneous, whereas the regularity norm on the left-hand side is homogeneous. This suggests that the boundary regularity in the above estimate may not be optimal. These observations motivate the estimates proved in the present paper.
For caloric trace estimates, one example is [11]: for \(1\le q\le\infty\), \[\label{eq:Ogawa-Shimizu-trace} \left\|g\right\|_{{\accentset{\large\bfseries .}{F}}^{1-\frac{1}{2q}}_{1,1}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}^0_{q,1}(\mathbb{R}^{d-1})\big)\cap L^{1}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}^{2-1/q}_{q,1}(\mathbb{R}^{d-1})\big)}\lesssim \left\||\partial_tv| + |\nabla^2 v|\right\|_{L^{1}\big(\mathbb{R}; \accentset{\large\bfseries .}{B}^0_{q,1}(\mathbb{R}^d_+)\big)}.\tag{7}\] It shows the optimality of the maximal regularity estimate in [11]. There is no corresponding inequality with \(\accentset{\large\bfseries .}{B}^0_{q,1}\) on the right-hand side replaced by \(L^q\).
This paper is a continuation of our earlier work [15], where we considered the equal exponent case \(p=q\) for both maximal regularity estimates and caloric trace estimates. That work, in turn, extends the one-dimensional first-order caloric trace estimates and maximal regularity estimates established in [16], which were motivated by the blow-up examples of Chang-Kang [17]. The case \(p=q\) is more direct: possibly reflecting the identification \(F^s_{p,p}=B^s_{p,p}\), the natural trace space is the anisotropic Besov space, rather than the intersection space appearing on the right-hand side of 5 . In the setting of anisotropic Besov spaces, see [15] for the corresponding maximal regularity and caloric trace estimates, and Amann [18] for related trace estimates. As we will point out in a comment after Theorem 2, the intersection space appearing on the right-hand side of ?? (homogeneous version of 5 ) for general \(p,q\) reduces to the anisotropic Besov space when \(p=q\). We also note that caloric trace estimates are related to the construction of singular solutions to the Stokes and Navier–Stokes equations near the boundary; see, for instance, [17] and [16].
To overcome the more difficult case when \(p,q\) are not necessarily equal, the new ingredients used in this paper are maximal functions and duality arguments. In particular, we will use the Fefferman-Stein vector-valued maximal inequality and a vector-version of the Peetre’s maximal inequality for frequency localized functions, see Lemmas 5 and 9. They are very suitable for the study of Littlewood-Paley frequency decomposition used by the Lizorkin-Triebel and Besov norms.
In this paper, we study mixed-norm maximal regularity estimates, modeled on 5 , and the corresponding caloric trace estimates for pairs \((v,g)\) satisfying 2 . Our results show that the homogeneous version of 5 holds for \(1\leq p,q<\infty\), thereby improving 5 and 6 . We also prove the corresponding caloric trace estimates for \(1<p<\infty\) and \(1\leq q\leq\infty\), which quantify the sharpness of the maximal regularity estimates.
Theorem 1 (One-dimensional case). Let \(d=1\) and \(n=\alpha+2\beta\) with nonnegative integers \(\alpha,\beta\). Suppose the function \(g(t)\in C_c^{\infty}(\mathbb{R})\), and \(v(x,t)\) is given by 4 . For \(1\leq p,q<\infty\), we have \[\label{eq:max-reg-1D} \left\|\partial_x^{\alpha}\partial_t^{\beta}v\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+)\big)} \lesssim \left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}(\mathbb{R})}.\qquad{(1)}\] Moreover, for \(1<p<\infty\) and \(1\leq q\leq \infty\), as well as \((p,q)=(1,1)\) and \((p,q)=(\infty,\infty)\), we have \[\label{eq:trace-1D} \left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}(\mathbb{R})}\lesssim\left\|\partial_x^{n}v\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+)\big)}.\qquad{(2)}\]
Comments on Theorem 1:
(i) The exponent \(\frac{n}{2}-\frac{1}{2q}\) can be non-positive. It happens when \(n=0\) or when \(n=q=1\).
(ii) When \(p=q\), Theorem 1 recovers \(d=1\) case of [15].
Theorem 2 (Maximal regularity estimates in higher dimensions). Let \(d\geq 2\), \(1\leq p,q<\infty\) and \(n=|\alpha|+2 \beta=\sum_{i=1}^d\alpha_i+2\beta\) with nonnegative integers \(\alpha=(\alpha_1,\ldots,\alpha_d)\) and \(\beta\). Suppose that \(g(x',t)\in C_c^\infty(\mathbb{R}^{d-1}\times \mathbb{R})\), and \(v(x,t)\) is given by 3 . Then we have that for \(n=0\), \[\label{eq:regu-est1} \left\|v\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}\lesssim \left\|g\right\|_{\accentset{\large\bfseries .}{F}^{-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)},\qquad{(3)}\] and for \(n\geq1\), \[\label{eq:regu-est2} \left\|\partial_x^{\alpha}\partial_t^{\beta}v\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}\lesssim \left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)} +\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{n-\frac{1}{q}}(\mathbb{R}^{d-1})\big)}.\qquad{(4)}\]
Comments on Theorem 2:
(i) The special case \(n=2\) of ?? gives \[\left\||\nabla_x^2 v| + |\partial_t v|\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)} \lesssim \left\|g\right\|_{\accentset{\large\bfseries .}{F}^{1-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)} +\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{2-\frac{1}{q}}(\mathbb{R}^{d-1})\big)}.\] For \(1<p<\infty\), it recovers 5 , proved in Denk–Hieber–Prüss [8], with weaker homogeneous norms on the right-hand side, and adds the borderline exponent \(q=1\).
(ii) For \(n=2\), \(p=1\), ?? recovers 6 , proved in Ogawa–Shimizu [11], and the second lower indices of both terms on the right-hand side are weakened from 1 to \(q\).
(iii) In the special case \(p=q\) and \(n>\frac{1}{q}\), the right-hand side of ?? is equivalent to the anisotropic Besov norm \(\left\|g\right\|_{\accentset{\large\bfseries .}{B}^{n-\frac{1}{q}, \frac{n}{2} - \frac{1}{2q}}_{q,q} (\mathbb{R}^{d-1}\times \mathbb{R})}\) by Chang-Kang [17]. Hence the combination of Theorems 2 and 3 recovers \(d\ge2\) case of [15] when \(n>\frac{1}{q}\).
(iv) Estimates ?? with \(n=0\) and ?? with \(n=1\) seem to be new.
Theorem 3 (Caloric trace estimates in higher dimensions). Let \(d\geq 2\). Suppose that \(g(x',t)\in C_c^\infty(\mathbb{R}^{d-1}\times \mathbb{R})\), and \(v(x,t)\) is given by 3 . For \(1<p<\infty\) and \(1\leq q\leq \infty\), as well as \(p=q=1\) and \(p=q=\infty\), and the integer \(n>\frac{1}{q}\), we have \[\label{thm:dd-trace-eq1} \left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)}+\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{n-\frac{1}{q}}(\mathbb{R}^{d-1})\big)} \lesssim\left\|\partial_{x_d}^nv\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}.\qquad{(5)}\]
Comments on Theorem 3:
(i) This trace estimate shows the optimality of ?? and the equivalence of the two sides when \(1< p<\infty\), \(1\le q <\infty\) (or \(p=q=1\)) and \(n>\frac{1}{q}\).
(ii) Unlike Theorems 1 and 2, this theorem assumes the condition \(n>\frac{1}{q}\). This condition is also assumed in the time-independent case [4] and is valid in 7 .
(iii) We have actually proved more general estimates than ?? . We can bound \[\left\|2^{(\frac{n}{2}-\frac{1}{2q})k}\|\Delta_{\leq \frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(k))} +\left\|2^{(n-\frac{1}{q})m}\|\Delta_{m}^h\Delta_{<2m}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(m))}\] by \(\left\|\partial_{x_d}^nv\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}\) without the condition \(n>\frac{1}{q}\), and bound \[\left\|2^{(\frac{n}{2}-\frac{1}{2q})k}\|\Delta_{>\frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(k))}+ \left\|2^{(n-\frac{1}{q})m}\|\Delta_{m}^h\Delta_{\geq 2m}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(m))}\] assuming \(n>\frac{1}{q}\). See Lemmas 19 and 23.
(iv) For \(1\leq p\leq \infty\), \(1\leq q <\infty\) and \(n>\frac{1}{q}\), it follows from taking \(L^p\)-norm in time of the classical trace estimate of Uspenskiı̆ [19], for general functions in \(\mathbb{R}^d_+\) given in Mironescu-Russ [4], that \[\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{n-\frac{1}{q}}(\mathbb{R}^{d-1})\big)} \lesssim\sum_{|\alpha|=n} \left\|\partial_{x}^\alpha v\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}.\] It includes borderline cases \(p=1,\infty\), \(1<q<\infty\), and \((p,q)=(\infty,1)\) not covered in ?? .
The paper is organized as follows. In Section 2, we introduce the notation used throughout the paper and collect some preliminary lemmas. In Section 3, we prove Theorem 1. In Section 4, we prove Theorems 2 and 3.
For two quantities \(X\) and \(Y\), we write \(X \lesssim_{\sigma} Y\) if \(X \leq C(\sigma)\, Y\) for some constant \(C(\sigma)>0\) depending on parameters \(\sigma=(\sigma_1,\sigma_2,\dots)\). We often suppress this dependence and simply write \(X \lesssim Y\) when it is clear from the context.
Denote by \(\mathcal{F}\) the Fourier transform, with subscripts indicating the variables when necessary. For \(x\in \mathbb{R}^d_+\), we write \(x=(x',x_d)\) with \(x'\in \mathbb{R}^{d-1}\) and \(x_d\in \mathbb{R}_+\), and \[\hat{f}\overset{\mathrm{def}}{=\joinrel\!\!=}\mathcal{F}_{x',t} f(\xi',x_d,\tau) = \frac{1}{(2\pi)^{\frac{d}{2}}} \int_{\mathbb{R}^{d-1}\times\mathbb{R}} e^{-i(x'\cdot\xi' + t\tau)} f(x,t)\,dx'\,dt,\] that is, \(\hat{f}\) denotes the Fourier transform of \(f\) in the \((x',t)\) variables.
Denote \[\label{def-delta} \Delta_{m}^{h}\overset{\mathrm{def}}{=\joinrel\!\!=}\mathcal{F}_{x'}^{-1}\varphi(2^{-m}\xi')\mathcal{F}_{x'} \quadand\quad\Delta_{k}^{\mathcal{T}}\overset{\mathrm{def}}{=\joinrel\!\!=}\mathcal{F}_{t}^{-1}\theta(2^{-k}\tau)\mathcal{F}_{t},\quad m,k\in \mathbb{Z}\tag{8}\] as the homogeneous Littlewood-Paley operators in the horizontal and temporal directions, respectively. Here \(\theta\in C_c^{\infty}\Big(\big(\frac{3}{4},\frac{8}{3}\big)\cup\big(-\frac{8}{3},-\frac{3}{4}\big)\Big)\) is an even function satisfying \[\forall \tau\in \mathbb{R}\setminus \{0\}, \quad \sum_{j\in \mathbb{Z}}\theta(2^{-j}\tau)=1,\] and \(\varphi(\xi')=\theta(|\xi'|)\). Denote (note the slight asymmetry in the definitions) \[\varphi_0(\xi')\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{j\leq0}\varphi(2^{-j}\xi'), \quad \theta_0(\tau)\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{j<0}\theta(2^{-j}\tau),\] with the convention that \(\varphi_0(0)=\theta_0(0)=1\). Denote \[\Delta_{\leq m}^{h}\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{j\leq m}\Delta_{j}^{h}=\mathcal{F}_{x'}^{-1}\varphi_0(2^{-m}\xi')\mathcal{F}_{x'}, \quad \Delta_{<k}^{\mathcal{T}}\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{j<k}\Delta_{j}^{\mathcal{T}}=\mathcal{F}_{t}^{-1}\theta_0(2^{-k}\tau)\mathcal{F}_{t}.\] We further set \(\Delta_{\leq \frac{k}{2}}^{h}\overset{\mathrm{def}}{=\joinrel\!\!=}\Delta_{\leq \lfloor\frac{k}{2}\rfloor }^{h}\), where \(\lfloor \cdot \rfloor\) denotes the floor function, and define \[\Delta_{> \frac{k}{2}}^{h}\overset{\mathrm{def}}{=\joinrel\!\!=}1-\Delta_{\leq \frac{k}{2}}^{h}, \quad \Delta_{\geq k}^{\mathcal{T}}\overset{\mathrm{def}}{=\joinrel\!\!=}1-\Delta_{<k}^{\mathcal{T}}.\] Denote \[\tilde{\varphi}(\xi') \overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{j=-1}^{1} \varphi(2^{-j}\xi'), \quad \tilde{\theta}(\tau) \overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{j=-1}^{1} \theta(2^{-j}\tau), \quad \tilde{\theta}_0(\tau) \overset{\mathrm{def}}{=\joinrel\!\!=}\theta_0(\tau) + \theta(\tau),\] which satisfy \[\label{def-tilde} \left\{ \begin{align} \;& \tilde{\varphi}=1 &&\text{on}\quad\operatorname{supp}\varphi, \\ \;& \tilde{\theta}=1 &&\text{on}\quad\operatorname{supp}\theta,\\ \;& \tilde{\theta}_0=1&&\text{on}\quad\operatorname{supp}\theta_0. \end{align}\right.\tag{9}\]
As in [20], we denote by \(\mathcal{S}'_h(\mathbb{R}^d)\) the space of tempered distributions \(f\) in \(\mathbb{R}^d\) such that \(\lim_{\lambda\to\infty}\| \mathcal{F}^{-1} (\theta(\lambda \xi) \cdot\mathcal{F}f)\|_{L^\infty}=0\) for any \(\theta \in C^\infty_c(\mathbb{R}^d)\). This space excludes nonzero polynomials, as well as distributions whose Fourier transforms contain the Dirac mass \(\delta_0\) or its derivatives. Its advantage is that homogeneous semi-norms, such as \(\accentset{\large\bfseries .}{B}_{p,q}^{s}\), become norms on this space. Moreover, for every \(f\in S'_h(\mathbb{R}^d)\), one has \(f=\sum_{j\in \mathbb{Z}}\Delta_j f\), where \(\Delta_j\) denotes the Littlewood-Paley operator in \(\mathbb{R}^d\). In Definition 4 below, we implicitly assume that the functions under consideration belong to the corresponding space \(S'_h\) in \(x'\) or \(t\) direction, and satisfy the relevant homogeneous Littlewood-Paley reconstruction formula \[f=\sum_{m\in \mathbb{Z}}\Delta^h_m f,\quad or \quad f=\sum_{k\in \mathbb{Z}}\Delta^{\mathcal{T}}_k f.\]
Here and throughout, \(\left\|\cdot\right\|_{\ell^q(m)}\) denotes the \(\ell^q\) norm over \(m\in\mathbb{Z}\), and \(\left\|\cdot\right\|_{L^{q}(\mathbb{R}^d)}\) denotes the Lebesgue norm.
Definition 4. Let \(s\in \mathbb{R}\). For \(1\leq p,q,r\leq \infty\), we define the homogeneous Besov space \(\accentset{\large\bfseries .}{B}_{p,q}^{s}(\mathbb{R}^{d-1})\) by the norm \[\left\|f\right\|_{\accentset{\large\bfseries .}{B}_{p,q}^{s}(\mathbb{R}^{d-1})} \overset{\mathrm{def}}{=\joinrel\!\!=} \left\|\,2^{ms}\left\|\Delta_{m}^{h}f\right\|_{L^{p}(\mathbb{R}^{d-1})}\,\right\|_{\ell^{q}(m)},\] and the Bochner-Besov space \(L^{r}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{p,q}^{s}(\mathbb{R}^{d-1})\big)\) by the norm \[\left\|f\right\|_{L^{r}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{p,q}^{s}(\mathbb{R}^{d-1})\big)} \overset{\mathrm{def}}{=\joinrel\!\!=} \left\|\,\left\|f(\cdot,t)\right\|_{\accentset{\large\bfseries .}{B}_{p,q}^{s}(\mathbb{R}^{d-1})}\,\right\|_{L^{r}(\mathbb{R})}.\] For \(1\leq p<\infty\) and \(1\leq q \leq \infty\), we define the homogeneous Lizorkin-Triebel space \(\accentset{\large\bfseries .}{F}_{p,q}^{s}(\mathbb{R})\) by the norm \[\left\|f\right\|_{\accentset{\large\bfseries .}{F}_{p,q}^{s}(\mathbb{R})}\overset{\mathrm{def}}{=\joinrel\!\!=}\left\|\,2^{ks}\Delta_{k}^{\mathcal{T}}f\,\right\|_{L^{p}(\mathbb{R};\ell^{q}(k))},\] and the Bochner-Lizorkin-Triebel space \(\accentset{\large\bfseries .}{F}^{s}_{p,q}(\mathbb{R};X)\) for a Banach space \(X\) by the norm \[\left\|f\right\|_{\accentset{\large\bfseries .}{F}^{s}_{p,q}(\mathbb{R};X)} \overset{\mathrm{def}}{=\joinrel\!\!=} \left\|\,2^{ks}\left\|\Delta_{k}^{\mathcal{T}}f\right\|_{X}\,\right\|_{L^{p}(\mathbb{R};\ell^{q}(k))}.\]
Note that \(\accentset{\large\bfseries .}{F}_{p,q}^{s}(\mathbb{R})\) needs a different definition for \(p=\infty>q\), see [2]. We will use \(\accentset{\large\bfseries .}{F}_{\infty,\infty}^{s}=\accentset{\large\bfseries .}{B}_{\infty,\infty}^{s}\).
Note that, in proving the trace estimate 1 , Weidemaier used the finite-difference definition of Bochner-Lizorkin-Triebel spaces in [7]. This definition is equivalent to the above Littlewood-Paley type definition in a certain range of the indices \(p\) and \(q\); see [2].
If \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d)\), the Hardy-Littlewood maximal function is defined by \[Mf(x) \overset{\mathrm{def}}{=\joinrel\!\!=}\sup_{r>0} \frac{1}{|B_r(x)|}\int_{B_r(x)} |f(y)|\,dy,\] where \(B_r(x)\subset \mathbb{R}^d\) is the ball centered at \(x\) with radius \(r\). It is well known [21] that for \(1<p\leq \infty\), \[\label{Maximal1} \left\|Mf\right\|_{L^p(\mathbb{R}^d)} \lesssim\left\|f\right\|_{L^p(\mathbb{R}^d)},\tag{10}\] while for \(p=1\) only the weak-type estimate holds. Moreover, we have \[(Mf)^r\leq M(|f|^r) \,\, ifr\geq 1,\quad M(|f|^r)\leq (Mf)^r \,\, if0<r< 1.\] These inequalities will be used frequently without further mention.
We recall the following classical theorem of maximal function, originally due to Fefferman-Stein [22].
Lemma 5. Let \(1<p<\infty\) and \(1<q\leq \infty\), or \(p=q=\infty\). There exists a constant \(C=C(p,q,d)\) such that for any sequence \(\{f_k\}_{k\in\mathbb{Z}} \subset L^1_{\mathrm{loc}}(\mathbb{R}^d)\), \[\label{eq:FS} \| M f_k \|_{L^{p}(\ell^{q}(k))} \leq C\, \| f_k \|_{L^{p}(\ell^{q}(k))}.\qquad{(6)}\]
Remark 6. The case \(1<p\leq \infty\) and \(q=\infty\) in ?? is not covered in [22], but it follows directly from the pointwise inequality \(\left\|Mf_k\right\|_{\ell^\infty(k)}\leq M\left\|f_k\right\|_{\ell^\infty(k)}\). However, it fails for \(p=\infty\) and \(1\le q<\infty\). For example, for fixed \(1\le q<\infty\), let \(f_k=\chi_{I_k}\), where \(I_k=(2^{k-1},2^k)\). Then \(\left(\sum_{k}|f_k(x)|^q\right)^{\frac{1}{q}}\leq 1\) for all \(x\in\mathbb{R}\). On the other hand, \[\begin{align} Mf_k(0)\ge \frac{1}{2\cdot2^k}\int_{0}^{2^k}f_k(y)\,{\rm d}y= \frac{1}{4}, \end{align}\] and hence \(\left(\sum_{k}|Mf_k(0)|^q\right)^{\frac{1}{q}}=\infty\).
The following lemma provides a scale-uniform pointwise bound for convolutions in terms of the Hardy-Littlewood maximal function.
Lemma 7. Let \(h:\mathbb{R}^d\to \mathbb{R}\) be a function satisfying \(|h(x)|\lesssim(1+|x|)^{-N}\) for some \(N>d\), and \(h_j(x)\overset{\mathrm{def}}{=\joinrel\!\!=}2^{jd}h(2^j x)\) for \(j\in \mathbb{Z}\). Then we have \[\label{eq:maximal-uniform} \sup_{j\in\mathbb{Z}} |h_j *f(x)| \lesssim Mf(x).\qquad{(7)}\]
Proof. The proof follows from Stein [21] as \(\Phi(x)= \sup_{|y|>|x|} |h(y)|\) is an integrable, radially non-increasing function. ◻
Remark 8. From Lemma 7, it follows that \[\sup_{j\in\mathbb{Z}}|\Delta_j f(x)| \lesssim Mf(x),\quad \sup_{j\in\mathbb{Z}}|\Delta_{\leq j}f(x)| \lesssim M f(x).\] In contrast, the estimate \[|h *f(x)| \lesssim\, (M|f|^r(x))^{\frac{1}{r}}\] cannot hold when \(0<r<1\). A counterexample is \(f=\chi_{B(y_0,\varepsilon)}\), where \(y_0\neq0\) is chosen so that \(h(-y_0)\neq 0\), and \(0<\varepsilon\ll|y_0|\). Then, at \(x=0\), we have \((M|f|^r(0))^{\frac{1}{r}}\lesssim\varepsilon^{d/r}\) while \(\varepsilon^d\lesssim|h *f(0)|\).
The following Lemma 9 states that Peetre’s maximal function (the left-hand side of ?? ) is bounded pointwise by a power of the Hardy-Littlewood maximal function. Its idea originates from Fefferman-Stein [23] and Peetre [24]. A detailed proof under the assumption \(f\in \mathcal{S}\) can be found in Triebel [2], while a version valid for more general \(f\in \mathcal{S}'\) is given in Li [25]. Here, \(\mathcal{S}\) denotes the Schwartz space and \(\mathcal{S}'\) its dual space of tempered distributions. In Lemma 9, we record a variant of [2], [25], originally stated for functions of \(t\in\mathbb{R}^n\), in a form suitable for functions of \((x,t)\in\mathbb{R}^d\times\mathbb{R}\), with the absolute value \(|\cdot|\) replaced by a Banach norm \(\left\|\cdot\right\|_X\) for functions on \(\mathbb{R}^d\).
Lemma 9. Assume that the Fourier transform of a function \(f(x,t)\) in the \(t\)-variable satisfies \[\operatorname{supp}\mathcal{F}_t f \subset \mathbb{R}^d\times B_R, \qquad B_R\overset{\mathrm{def}}{=\joinrel\!\!=}\{\tau\in\mathbb{R}:|\tau|<R\}.\] Let \(\left\|\cdot\right\|_{X}\) be a Banach norm acting on the \(x\)-variable, and set \(F(t)=\left\|f(\cdot,t)\right\|_{X}\). Then, we have that for fixed \(r\in(0,\infty)\), \[\label{eq:peetre} \sup_{s\in \mathbb{R}}\frac{F(t-s)}{1+|Rs|^{\frac{1}{r}}}\lesssim\left(M|F|^{r}(t)\right)^{\frac{1}{r}},\qquad{(8)}\] provided that the left-hand side of ?? is finite in the case \(0<r<1\).
The boundedness of the left-hand side of ?? is an essential condition and should be verified when the lemma is applied with \(0<r<1\). When we bound \(\sup_{s\in \mathbb{R}}\frac{F(t-s)}{1+|Rs|^{2}}\lesssim MF(t)\) in later sections, we can drop the exponent \(2\) and use \(r=1\) case (since \(\frac{1}{1+|Rs|^2}\lesssim\frac{1}{1+|Rs|}\)), avoiding checking the condition. However, when we use the vector-valued version ?? of Lemma 10, to cover the endpoint cases \(p=1\) or \(q=1\), it is sometimes necessary to take \(r<1\).
Proof. We follow the argument in [25]. By translation it suffices to prove the case \(t=0\) and by scaling one can assume \(R=1\). Since \(\operatorname{supp}\mathcal{F}_t f \subset \mathbb{R}^d\times B_R\), we may choose \(h\in \mathcal{S}(\mathbb{R})\) such that \(\mathcal{F}_th\) is equal to the appropriate nonzero constant on \(B_1\), so that \[\begin{align} f(x,s)=\int_\mathbb{R}h(s-u)f(x,u)\,{\rm d}u,\quad |h(s)|\leq \frac{C(h,N)}{(1+|s|)^N}, \end{align}\] for any \(N>1\).
Consider first the case \(r\geq 1\). For fixed \(s\), we decompose \(\mathbb{R}=\bigcup_{m=0}^\infty A_m\), where \(A_0=\{u:\,|s-u|\leq (1+|s|)\}\) and \(A_{m}=\{u:\,2^{m-1}(1+|s|)< |s-u|\leq 2^{m}(1+|s|)\}\) for \(m\geq 1\). Also let \(B_{m}=\{u:\,|u|\leq 2^{m+1}(1+|s|)\}\) for \(m\geq 0\). Note \(A_m\) and \(B_m\) have different centers, and \(A_m\subset B_m\). Then \[\begin{align} \label{eq:lem:dyadic} F(s)&\leq \sum_{m=0}^{\infty}\int_{A_m} |h(s-u)|F(u)\,{\rm d}u\notag\\[3pt] &\leq \int_{B_0}|h(s-u)|F(u)\,{\rm d}u+\sum_{m=1}^{\infty}\frac{C_N}{2^{N(m-1)}(1+|s|)^{N}}\int_{B_m} F(u)\,{\rm d}u\notag\\[3pt] &\lesssim\left(\int_{B_0}F(u)^r\,{\rm d}u\right)^{\frac{1}{r}}+ \sum_{m=1}^{\infty}\frac{2^{m+2}(1+|s|)}{2^{N(m-1)}(1+|s|)^{N}} M|F|(0)\notag\\[3pt] &\lesssim(1+|s|)^{\frac{1}{r}} \left(M|F|^{r}(0)\right)^{\frac{1}{r}}+M|F|(0), \end{align}\tag{11}\] where we choose \(N\) large enough so that the series converges. Since \(M|F|(0)\leq \left(M|F|^{r}(0)\right)^{\frac{1}{r}}\) for \(r\geq 1\), we obtain \[\frac{F(s)}{(1+|s|)^{\frac{1}{r}}}\lesssim\left(M|F|^{r}(0)\right)^{\frac{1}{r}},\] which proves ?? .
Next we consider the case \(0<r<1\). \[\begin{align} F(s)&\leq \int_\mathbb{R}|h(s-u)|F(u)^r\cdot F(u)^{1-r}\,{\rm d}u\notag\\[3pt] &\leq \int_\mathbb{R}|h(s-u)|F(u)^r\cdot (1+|u|)^{\frac{1-r}{r}}\,{\rm d}u\cdot \left(\sup_{u\in \mathbb{R}}\frac{F(u)}{(1+|u|)^{\frac{1}{r}}}\right)^{1-r}\notag\\[3pt] &\leq(1+|s|)^{\frac{1-r}{r}}\int_\mathbb{R}|h(s-u)|F(u)^r\cdot (1+|s-u|)^{\frac{1-r}{r}}\,{\rm d}u\cdot \left(\sup_{u\in \mathbb{R}}\frac{F(u)}{(1+|u|)^{\frac{1}{r}}}\right)^{1-r}. \end{align}\] By the same dyadic annular decomposition as in 11 , we have \[\begin{align} &\,\quad \int_\mathbb{R}|h(s-u)|F(u)^r\cdot (1+|s-u|)^{\frac{1-r}{r}}\,{\rm d}u\\[3pt] &\lesssim\int_{B_0}F(u)^r\,{\rm d}u+\sum_{m=1}^{\infty}\frac{1}{2^{(N-\frac{1-r}{r})(m-1)}(1+|s|)^{(N-\frac{1-r}{r})}}\int_{B_m} F(u)^r\,{\rm d}u\\[3pt] &\lesssim(1+|s|)M|F|^{r}(0)+M|F|^r(0). \end{align}\] Hence we obtain \[F(s)\lesssim(1+|s|)^{\frac{1}{r}}M|F|^{r}(0)\cdot \left(\sup_{u\in \mathbb{R}}\frac{F(u)}{(1+|u|)^{\frac{1}{r}}}\right)^{1-r}.\] The desired inequality then follows provided that \(\sup_{u\in \mathbb{R}}\frac{F(u)}{(1+|u|)^{\frac{1}{r}}}\) is finite. ◻
The following lemma combines Lemmas 5 and 9, and extends [2].
Lemma 10. Let \(0<p<\infty\) and \(0<q\leq\infty\), or \(p=q=\infty\), and \(0<r<\min (p,q)\). Suppose that \(f_j\), for \(j\in\mathbb{Z}\), satisfy the assumptions of Lemma 9 with radius \(R_j\), and \(F_j(t)=\left\|f_j(\cdot,t)\right\|_{X}\) for the same \(X\)-norm. Then, we have \[\label{eq:tri83-2} \left\|\sup_{s\in \mathbb{R}}\frac{F_j(t-s)}{1+|R_js|^{\frac{1}{r}}}\right\|_{L^{p}(\mathbb{R};\ell^q(j))} \lesssim\left\|F_j(t)\right\|_{L^{p}(\mathbb{R};\ell^q(j))},\qquad{(9)}\] provided that \(\sup_{s\in \mathbb{R}}\frac{F_j(t-s)}{1+|R_js|^{\frac{1}{r}}}\) is finite for each \(j\) in the case \(0<r<1\).
Proof. By ?? , we have \[\left\|\sup_{s\in \mathbb{R}}\frac{F_j(t-s)}{1+|R_js|^{\frac{1}{r}}}\right\|_{L^{p}(\mathbb{R};\ell^q(j))} \lesssim\left\|\left(M|F_j|^{r}(t)\right)^{\frac{1}{r}}\right\|_{L^{p}(\mathbb{R};\ell^q(j))}\\[3pt] =\left\|M|F_j|^{r}(t)\right\|_{L^{\frac{p}{r}}(\mathbb{R};\ell^\frac{q}{r}(j))}^{\frac{1}{r}},\] which together with ?? yields \[\left\|\sup_{s\in \mathbb{R}}\frac{F_j(t-s)}{1+|R_js|^{\frac{1}{r}}}\right\|_{L^{p}(\mathbb{R};\ell^q(j))} \lesssim\left\||F_j|^{r}(t)\right\|_{L^{\frac{p}{r}}(\mathbb{R};\ell^\frac{q}{r}(j))}^{\frac{1}{r}}\\[3pt] =\left\|F_j(t)\right\|_{L^{p}(\mathbb{R};\ell^q(j))}.\] Hence we get ?? . ◻
The following calculus lemma will be useful in later sections.
Lemma 11. Let \(1\leq q<\infty\), \(R>1\), \(a,b\in\mathbb{R}\), and \(c>0\). We have that for \(a+\frac{1}{q^2}>0\) and any sequence \(A_j\), \[\label{eq:int-xd1} \int_0^\infty\Big(\sum_{j\in\mathbb{Z}} x^{a}R^{bj}e^{-cxR^{j}}|A_j|\Big)^q \,{\rm d}x\lesssim\sum_{j\in\mathbb{Z}}R^{(b-a)jq-j}|A_j|^q.\qquad{(10)}\]
Proof. It is straightforward to verify that for \(r>0\), \[\label{eq:lem-xd4} \sup_{x>0} \sum_{j\in\mathbb{Z}}\left[x R^{j}e^{-cx R^{j}} \right]^{r}\lesssim 1,\tag{12}\] and for \(\sigma>-1\), \[\label{eq:lem-xd5} \int_0^\infty x^{\sigma} e^{-cxR^{j}}\,{\rm d}x\lesssim R^{-j(\sigma+1)}.\tag{13}\]
For \(q=1\), ?? is trivial, so in the following we assume \(1<q<\infty\). By using Hölder’s inequality, we have \[\begin{align} \Big( \sum_{j\in\mathbb{Z}}R^{bj}e^{-cxR^{j}}|A_j|\Big)^q&=\Big(\sum_{j\in\mathbb{Z}}\Big[R^{(b-1)j}\cdot \left(R^{j}e^{-cx R^{j}} \right)^{1-\frac{q-1}{q^2}} |A_j|\Big]\cdot\left[R^{j}e^{-c xR^{j}} \right]^{\frac{q-1}{q^2}}\Big)^q\\[3pt] &\lesssim\Big(\sum_{j\in\mathbb{Z}}\Big[R^{(b-1)j}\cdot \left(R^{j}e^{-cx R^{j}} \right)^{1-\frac{q-1}{q^2}} |A_j|\Big]^q\Big)\cdot\Big(\sum_{j\in\mathbb{Z}}\left[R^{j}e^{-cx R^{j}} \right]^{\frac{1}{q}}\Big)^{q-1}, \end{align}\] which together with 12 and 13 yields (using \(aq^2+1>0\)) \[\begin{align} \int_0^\infty\Big( \sum_{j\in\mathbb{Z}}x^{a}R^{bj}e^{-cxR^{j}}|A_j|\Big)^q\,{\rm d}x&\lesssim \sum_{j\in\mathbb{Z}}\int_{0}^{+\infty} R^{(bq-1+\frac{1}{q})j} x^{aq-1+\frac{1}{q}}e^{-cxR^j(q-1+\frac{1}{q})} \,{\rm d}x\cdot |A_j|^q \\[3pt] &\lesssim \sum_{j\in\mathbb{Z}}\;R^{(b-a)jq-j} |A_j|^q. \end{align}\] Hence ?? . ◻
In this section, we prove Theorem 1, the one-dimensional case. We first establish the maximal regularity estimate ?? in Lemma 12, and then prove the trace estimate ?? in Lemmas 13-15.
Lemma 12. Under the assumptions of Theorem 1, estimate ?? holds.
Proof. For \(v\) defined in 4 , we have (see [16]) \[\label{eq:1D-v} \mathcal{F}_t v= e^{-x\sqrt{i\tau}} \cdot \mathcal{F}_tg.\tag{14}\] Here we take the principal branch of the square root, so that the real part \(Re \sqrt{i \tau} = \sqrt{|\tau|/2}\) for \(\tau \in \mathbb{R}\). Hence \[\begin{align} \label{eq:1D-0} \partial_x^{\alpha}\partial_t^{\beta}v=\sum_{k\in \mathbb{Z}}\Delta_{k}^{\mathcal{T}}\partial_x^{\alpha}\partial_t^{\beta}v&=\sum_{k\in \mathbb{Z}} \mathcal{F}_{t}^{-1}\left(P(x,\tau) \cdot\theta(2^{-k}\tau)\cdot \mathcal{F}_t g\right)\notag\\[3pt] &=\sum_{k\in \mathbb{Z}} \mathcal{F}_{t}^{-1}\left(P(x,\tau) \cdot\tilde{\theta}(2^{-k}\tau)\cdot\theta(2^{-k}\tau)\cdot\mathcal{F}_t g\right), \end{align}\tag{15}\] where \(\Delta_{k}^{\mathcal{T}}\) and \(\tilde{\theta}\) are defined in 8 and 9 respectively, and \[P(x,\tau)\overset{\mathrm{def}}{=\joinrel\!\!=}(-\sqrt{i\tau})^{\alpha}\cdot(i \tau)^\beta e^{-x\sqrt{i\tau}}.\]
Now we claim that there exists a constant \(c>0\) such that \[\label{eq:1d-1} |\mathcal{F}_{t}^{-1}\left(P(x,\tau) \cdot \tilde{\theta}(2^{-k}\tau)\right)(t)|\lesssim\frac{2^k\cdot 2^{\frac{nk}{2}}}{\left(1+|2^k t|^2\right)^2}e^{-cx 2^{\frac{k}{2}}}.\tag{16}\] The reader may refer to [20] for the underlying idea of this estimate. We include a detailed derivation here, and will use similar estimates repeatedly later without further comment. Notice that \[\begin{align} \mathcal{F}_{t}^{-1}\left(P(x,\tau) \cdot \tilde{\theta}(2^{-k}\tau)\right)(t)&=\frac{1}{(2\pi)^{1/2}}\int_{\mathbb{R}} e^{i t\cdot\tau}\cdot P(x,\tau)\cdot \tilde{\theta}(2^{-k}\tau) \,{\rm d}\tau\\[3pt] &=\frac{2^k}{(2\pi)^{1/2}}\int_{\mathbb{R}} e^{i2^k t\cdot\tau}\cdot P(x,2^k\tau)\cdot \tilde{\theta}(\tau) \,{\rm d}\tau. \end{align}\] By repeated integration by parts using \(e^{i2^k t\cdot\tau} = (1+|2^k t|^2)^{-1}(1-\partial_{\tau}^2)e^{i2^k t\cdot\tau}\), we obtain \[\label{claim1} |\mathcal{F}_{t}^{-1}\left(P(x,\tau) \cdot \tilde{\theta}(2^{-k}\tau)\right)(t)|\lesssim\frac{2^k}{\left(1+|2^k t|^2\right)^2}\cdot \left\|\left(1-\partial_{\tau}^2\right)^2\left(P(x,2^k\tau)\cdot \tilde{\theta}(\tau)\right)\right\|_{L^{1}(\mathbb{R})}.\tag{17}\] Since the real part \(Re(\sqrt{i2^k\tau})=\frac{2^{\frac{k}{2}}}{\sqrt 2}\sqrt {|\tau|}\) no matter \(\tau\) is positive or negative, we deduce \[\begin{align} |\left(1-\partial_{\tau}^2\right)^2\left(P(x,2^k\tau)\cdot \tilde{\theta}(\tau)\right)|\lesssim 2^{\frac{nk}{2}}e^{-cx 2^{\frac{k}{2}}}. \end{align}\] Given the compact support of \(\tilde{\theta}\), the corresponding \(L^1\) norm satisfies the same bound, which together with 17 leads to 16 .
Therefore, in view of 15 and 16 , we have \[|\Delta_{k}^{\mathcal{T}}\partial_x^{\alpha}\partial_t^{\beta}v| \lesssim\int_{\mathbb{R}}\frac{2^k\cdot 2^{\frac{nk}{2}}}{\left(1+|2^k u|^2\right)^2}e^{-cx 2^{\frac{k}{2}}}\cdot|\Delta_k^{\mathcal{T}}g(t-u)|\,{\rm d}u\lesssim 2^{\frac{nk}{2}}e^{-cx 2^{\frac{k}{2}}}\cdot \sup_{u\in \mathbb{R}} \frac{|\Delta_k^{\mathcal{T}}g(t-u)|}{1+|2^ku|^2},\] which together with ?? implies that \[\begin{align} \int_0^\infty |\partial_x^{\alpha}\partial_t^{\beta}v|^q \,{\rm d}x&\lesssim\int_0^\infty \Big(\sum_{k\in \mathbb{Z}}2^{\frac{nk}{2}}e^{-cx 2^{\frac{k}{2}}}\cdot \sup_{u\in \mathbb{R}} \frac{|\Delta_k^{\mathcal{T}}g(t-u)|}{1+|2^ku|^2}\Big)^q \,{\rm d}x\\[3pt] &\lesssim\sum_{k\in \mathbb{Z}} 2^{\frac{nkq-k}{2}}\cdot \Big(\sup_{u\in \mathbb{R}} \frac{|\Delta_k^{\mathcal{T}}g(t-u)|}{1+|2^ku|^2}\Big)^q. \end{align}\] Since \(g\in \accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}(\mathbb{R})\), \(\Delta_k^{\mathcal{T}}g\) is bounded for each \(k\) (by the Bernstein inequality \(\left\|\Delta_k^{\mathcal{T}}g\right\|_{L^\infty}\lesssim_k \left\|\Delta_k^{\mathcal{T}}g\right\|_{L^p}\)), thus the quantity \(\sup_{u\in \mathbb{R}} \frac{|\Delta_k^{\mathcal{T}}g(t-u)|}{1+|2^ku|^2}\) is finite. By applying ?? with \(r=\frac{1}{2}\), we arrive at ?? . ◻
In this subsection, we present the proof of the caloric trace estimate ?? in the following three lemmas (Lemma 13-15), each corresponding to a different range of the indices \(p,q\).
Lemma 13. Under the assumptions of Theorem 1 with \(1<p,q<\infty\), estimate ?? holds.
Proof. In view of 14 , we have \[\label{eq:3-5} \mathcal{F}_tg(\tau)=4\int_{0}^\infty x(-\sqrt{i\tau})^{2-n}\cdot e^{-x\sqrt{i\tau}}\cdot \partial_x^n\mathcal{F}_tv(x,\tau)\,{\rm d}x.\tag{18}\] By a similar argument of 16 , we get \[\begin{align} \label{eq:lem-trace-2} \left|\Delta_{k}^{\mathcal{T}}g\right| &\lesssim\left|\int_{0}^\infty \mathcal{F}_t^{-1}\left(x(-\sqrt{i\tau})^{2-n}\cdot e^{-x\sqrt{i\tau}}\cdot\theta(2^{-k}\tau)\cdot \partial_x^n\mathcal{F}_tv(x,\tau)\right)\,{\rm d}x\right|\notag\\[3pt] &\lesssim\int_{\mathbb{R}}\int_{0}^\infty\frac{x 2^{\frac{4-n}{2}k}\cdot e^{-cx 2^{\frac{k}{2}}}}{\left(1+|2^k (t-s)|^2\right)^2}\cdot\left|\partial_x^nv(x,s)\right|\,{\rm d}x\,{\rm d}s. \end{align}\tag{19}\]
We next use a duality argument. For any sequence \(\{\psi_k(t)\}_{k\in\mathbb{Z}}\) satisfying \[\|\psi_k(t)\|_{L^{p'}(\mathbb{R};\ell^{q'}(k))}<\infty, \qquad \frac{1}{p}+\frac{1}{p'}=\frac{1}{q}+\frac{1}{q'}=1,\] we have \[\label{Dual1} \int_{\mathbb{R}} \Delta_{k}^{\mathcal{T}}g\cdot\psi_{k}(t)\,{\rm d}t=\int_{\mathbb{R}} \Delta_{k}^{\mathcal{T}}g\cdot\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(t)\,{\rm d}t,\tag{20}\] where \[\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}\overset{\mathrm{def}}{=\joinrel\!\!=}\mathcal{F}_t^{-1}\left(\tilde{\theta}(2^{-k}\tau)\cdot\mathcal{F}_t\psi_{k}\right).\] By virtue of 19 and 20 , we infer \[\begin{align} K_1&\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{k\in\mathbb{Z}}2^{(\frac{n}{2}-\frac{1}{2q})k}\int_{\mathbb{R}} \Delta_{k}^{\mathcal{T}}g\cdot\psi_{k}(t)\,{\rm d}t\notag\\[3pt] &\lesssim\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{0}^{\infty} \frac{x2^{\frac{4-\frac{1}{q}}{2}k}\cdot e^{-cx 2^{\frac{k}{2}}}}{\left(1+|2^k (t-s)|^2\right)^2}\cdot\left|\partial_x^nv(x,s)\right|\cdot\left|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(t)\right|\,{\rm d}x\,{\rm d}s\,{\rm d}t\notag\\[3pt] &\lesssim\int_{\mathbb{R}}\int_{0}^{\infty}\sum_{k\in\mathbb{Z}}\Big( x2^{\frac{2-\frac{1}{q}}{2}k}\cdot e^{-cx 2^{\frac{k}{2}}}\cdot\sup_{s\in \mathbb{R}} \frac{|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(t-s)|}{1+|2^ks|^2}\Big)\cdot\left|\partial_x^nv(x,t)\right|\,{\rm d}x\,{\rm d}t.\label{M1} \end{align}\tag{21}\] By Hölder’s inequality and ?? with \(1<q<\infty\), we deduce \[\begin{align} K_1&\lesssim\int_{\mathbb{R}}\Big(\sum_{k\in\mathbb{Z}}\Big(\sup_{s\in \mathbb{R}} \frac{|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(t-s)|}{1+|2^ks|^2}\Big)^{q'}\Big)^{\frac{1}{q'}}\cdot\left\|\partial_x^nv(\cdot,t)\right\|_{L^q\left(\mathbb{R}_+\right)}\,{\rm d}t\\[3pt] &\lesssim\left\|\sup_{s\in \mathbb{R}} \frac{|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(t-s)|}{1+|2^ks|^2}\right\|_{L^{p'}(\mathbb{R};\ell^{q'}(k))}\cdot\left\|\partial_x^nv\right\|_{L^{p}(\mathbb{R};L^q\left(\mathbb{R}_+\right))}. \end{align}\] Thus, applying ?? with \(r=1\), ?? and ?? in succession, we obtain \[K_1 \lesssim\left\|\psi_{k}\right\|_{L^{p'}(\mathbb{R};\ell^{q'}(k))}\cdot\left\|\partial_x^nv\right\|_{L^{p}(\mathbb{R};L^q\left(\mathbb{R}_+\right))}.\] By a duality argument, it follows that for \(1<p<\infty\) and \(1< q<\infty\), \[\left\|2^{(\frac{n}{2}-\frac{1}{2q})k} \Delta_{k}^{\mathcal{T}}g\right\|_{L^{p}(\mathbb{R};\ell^{q}(k))}\lesssim\left\|\partial_x^nv\right\|_{L^{p}(\mathbb{R};L^q\left(\mathbb{R}_+\right))},\] which leads to ?? . ◻
We next deal with two endpoint cases.
Lemma 14. Under the assumptions of Theorem 1 with \(1\leq p<\infty\) and \(q=1\), estimate ?? holds.
Proof. The estimate 21 is still valid for \(q=1\). Hence \[\begin{align} K_1&\leq \int_{\mathbb{R}}\int_{0}^{\infty}\sum_{k\in\mathbb{Z}}\Big( x2^{\frac{k}{2}}\cdot e^{-cx 2^{\frac{k}{2}}}\Big)\cdot \sup_{k}\Big(\sup_{s\in \mathbb{R}} \frac{|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(t-s)|}{1+|2^ks|^2}\Big)\cdot\left|\partial_x^nv(x,t)\right|\,{\rm d}x\,{\rm d}t\\[3pt] &\lesssim\int_{\mathbb{R}} \sup_{k}\Big(\sup_{s\in \mathbb{R}} \frac{|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(t-s)|}{1+|2^ks|^2}\Big)\cdot\left\|\partial_x^nv(\cdot,t)\right\|_{L^1(\mathbb{R}_+)}\,{\rm d}t, \end{align}\] which together with ?? with \(r=1\), ?? and ?? yields that the inequality \[K_1 \lesssim\left\|\psi_{k}\right\|_{L^{p'}(\mathbb{R};\ell^{\infty}(k))}\cdot\left\|\partial_x^nv\right\|_{L^{p}(\mathbb{R};L^1\left(\mathbb{R}_+\right))}\] holds for any \(\psi_{k}(t)\in L^{p'}(\mathbb{R};\ell^{\infty}(k))\). Thus for \(1\leq p<\infty\) and \(q=1\), we achieve \[\left\|2^{(\frac{n}{2}-\frac{1}{2q})k} \Delta_{k}^{\mathcal{T}}g\right\|_{L^{p}(\mathbb{R};\ell^{q}(k))}\lesssim\left\|\partial_x^nv\right\|_{L^{p}(\mathbb{R};L^q\left(\mathbb{R}_+\right))},\] which leads to ?? . ◻
Lemma 15. Under the assumptions of Theorem 1 with \(1<p\leq\infty\) and \(q= \infty\), estimate ?? holds.
Proof. By a similar argument of 19 , we have that for \(q=\infty\), \[\begin{align} \label{eq:lem-trace-3} \left|\Delta_{k}^{\mathcal{T}}g\right| &\lesssim\left|\int_{0}^\infty \mathcal{F}_t^{-1}\left(x(-\sqrt{i\tau})^{2-n}\cdot e^{-x\sqrt{i\tau}}\cdot\tilde{\theta}(2^{-k}\tau)\cdot\theta(2^{-k}\tau)\cdot \partial_x^n\mathcal{F}_tv(x,\tau)\right)\,{\rm d}x\right|\notag\\[3pt] &\lesssim\int_{\mathbb{R}}\int_{0}^\infty\frac{ x2^{\frac{4-n}{2}k}\cdot e^{-cx 2^{\frac{k}{2}}}}{\left(1+|2^k s|^2\right)^2}\cdot\left|\Delta_k^{\mathcal{T}}\partial_x^nv(x,t-s)\right|\,{\rm d}x\,{\rm d}s\notag\\[3pt] &\lesssim\int_{0}^\infty x2^{\frac{2-n}{2}k}\cdot e^{-cx 2^{\frac{k}{2}}} \left(\sup_{s\in\mathbb{R}}\frac{ \left|\Delta_k^{\mathcal{T}}\partial_x^nv(x,t-s)\right|}{1+|2^k s|^2}\right)\,{\rm d}x\notag\\[3pt] &\lesssim 2^{-\frac{n}{2}k}\left\|\sup_{s\in\mathbb{R}}\frac{ \left|\Delta_k^{\mathcal{T}}\partial_x^nv(\cdot,t-s)\right|}{1+|2^k s|^2}\right\|_{L^\infty(\mathbb{R}_+)}. \end{align}\tag{22}\] Applying ?? with \(r=1\) and ?? , we obtain \[\label{Dual2} \sup_{s\in \mathbb{R}}\frac{|\Delta_k^{\mathcal{T}}\partial_{x}^nv(x,t-s)|}{1+|2^{k}s|^2}\lesssim M^{\mathcal{T}}M^{\mathcal{T}} \partial_x^n v(x,t),\tag{23}\] where \(M^{\mathcal{T}}\) is the Hardy–Littlewood maximal operator acting on the temporal variable \(t\). By substituting 23 into 22 , we achieve \[\left\|2^{\frac{n}{2}k} \Delta_{k}^{\mathcal{T}}g\right\|_{\ell^{\infty}(k)} \lesssim\left\|M^{\mathcal{T}}M^{\mathcal{T}} \partial_x^n v(\cdot,t)\right\|_{L^\infty(\mathbb{R}_+)}\lesssim M^{\mathcal{T}}M^{\mathcal{T}} \left\|\partial_x^n v(\cdot,t)\right\|_{L^\infty(\mathbb{R}_+)},\] which together with 10 implies that for \(1<p\leq\infty\), \[\begin{align} \left\|2^{\frac{n}{2}k} \Delta_{k}^{\mathcal{T}}g\right\|_{L^p(\mathbb{R};\ell^{\infty}(k))}\lesssim\left\|\partial_x^nv\right\|_{L^{p}(\mathbb{R};L^\infty\left(\mathbb{R}_+\right))}. \end{align}\] Hence ?? holds for \(1<p\leq\infty\) and \(q=\infty\). ◻
In this section we prove Theorem 2 and Theorem 3.
In this subsection we prove Theorem 2 only in the case \(\beta=0\) and \(n=|\alpha|\), while for \(\beta>0\), the same conclusion follows by the fact that \(\partial_t v-\Delta v=0\). For the caloric extension \(v\) defined in 3 , as in 14 , we have by [16] \[\label{eq:v-g-hat} \hat{v}(\xi',x_d,\tau)= e^{-x_d\sqrt{|\xi'|^2+i\tau}} \cdot \hat{g}(\xi',\tau),\tag{24}\] and hence \[\partial_x^{\alpha}\Delta_{m}^h\Delta_{k}^{\mathcal{T}}v=\mathcal{F}_{x',t}^{-1}\left(P(\xi',x_d,\tau) \cdot\varphi(2^{-m}\xi')\cdot\theta(2^{-k}\tau)\cdot \hat{g}\right),\] where \[P(\xi',x_d,\tau)\overset{\mathrm{def}}{=\joinrel\!\!=}\prod_{j=1}^{d-1}(i\xi_j)^{\alpha_j} \cdot(-\sqrt{|\xi'|^2+i\tau})^{\alpha_d}\cdot e^{-x_d\sqrt{|\xi'|^2+i\tau}}.\] Now, we decompose \[\label{eq:vmk1} \partial_x^{\alpha}v=\sum_{m,k\in\mathbb{Z}}\partial_x^{\alpha}\Delta_{m}^h\Delta_{k}^{\mathcal{T}}v=I_1+I_2,\tag{25}\] where \[\begin{align} \label{eq:vmk2} I_1&\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{m,k\in\mathbb{Z},k<2m}\partial_x^{\alpha}\Delta_{m}^h\Delta_{k}^{\mathcal{T}}v =\sum_{m\in\mathbb{Z}}\partial_x^{\alpha}\Delta_{m}^h\Delta_{<2m}^{\mathcal{T}}v\notag\\[3pt] &=\sum_{m\in\mathbb{Z}}\,\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{\mathbb{R}^{d-1}\times\mathbb{R}}\Phi_{m}^{h}(x'-y',x_d,t-s)\cdot \Delta_{m}^h\Delta_{< 2m}^{\mathcal{T}} g(y',s)\,{\rm d}y'\,{\rm d}s, \end{align}\tag{26}\] with \[\Phi_{m}^{h}(x,t)\overset{\mathrm{def}}{=\joinrel\!\!=}\mathcal{F}_{x',t}^{-1}\left(P(\xi',x_d,\tau)\cdot\tilde{\varphi}(2^{-m}\xi')\cdot\tilde{\theta}_0(2^{-2m}\tau)\right),\] and \[\begin{align} \label{eq:vmk3} I_2&\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{m,k\in\mathbb{Z},k\geq 2m}\partial_x^{\alpha}\Delta_{m}^h\Delta_{k}^{\mathcal{T}}v =\sum_{k\in\mathbb{Z}}\partial_x^{\alpha}\Delta_{\leq \lfloor k /2\rfloor}^h\Delta_{k}^{\mathcal{T}}v\notag\\[3pt] &=\sum_{k\in\mathbb{Z}}\,\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{\mathbb{R}^{d-1}\times\mathbb{R}}\Phi_{k}^{\mathcal{T}}(x'-y',x_d,t-s)\cdot \Delta_{k}^{\mathcal{T}} g(y',s)\,{\rm d}y'\,{\rm d}s, \end{align}\tag{27}\] with \[\Phi_{k}^{\mathcal{T}}(x,t)\overset{\mathrm{def}}{=\joinrel\!\!=}\mathcal{F}_{x',t}^{-1}\left(P(\xi',x_d,\tau)\cdot\varphi_0(2^{-\lfloor k /2\rfloor}\xi')\cdot\tilde{\theta}(2^{-k}\tau)\right).\]
Lemma 16. There exists a constant \(c>0\) such that \[\label{eq:est-Phih} \left\|\Phi_{m}^{h}(\cdot,x_d,t)\right\|_{L^{1}(\mathbb{R}^{d-1})}\lesssim\frac{2^{m(2+n)}}{ (1+|2^{2m} t|^2)^2} \cdot e^{-cx_d2^{m}},\qquad{(11)}\] and \[\label{eq:est-Phit} \left\|\Phi_{k}^{\mathcal{T}}(\cdot,x_d,t)\right\|_{L^{1}(\mathbb{R}^{d-1})}\lesssim\frac{2^{\frac{k(2+n)}{2}}}{ (1+|2^{k} t|^2)^2} \cdot e^{-cx_d2^{\frac{k}{2}}}.\qquad{(12)}\]
Proof. The proof is similar to 16 . By scaling, we have \[\begin{align} \Phi_{m}^h(x,t)&=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} e^{i(x'\cdot\xi'+t\cdot\tau)}P(\xi',x_d,\tau)\cdot\tilde{\varphi}(2^{-m}\xi')\cdot\tilde{\theta}_0(2^{-2m}\tau)\,{\rm d}\xi'\,{\rm d}\tau\\[3pt] &=\frac{2^{m(d+1)}}{(2\pi)^{\frac{d}{2}}}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} e^{i(2^{m}x'\cdot\xi'+2^{2m}t\cdot\tau)}P(2^{m}\xi',2^{2m}\tau)\cdot\tilde{\varphi}(\xi')\cdot\tilde{\theta}_0(\tau)\,{\rm d}\xi'\,{\rm d}\tau. \end{align}\] By \(E\overset{\mathrm{def}}{=\joinrel\!\!=}e^{i(2^{m}x'\cdot\xi'+2^{2m}t\cdot\tau)} =\left(\frac{1-\Delta_{\xi'}}{1+|2^m x'|^{2}} \right)^d\left(\frac{1-\partial_{\tau}^2}{1+|2^{2m} t|^2}\right)^2E\) and repeated integration by parts, we obtain \[\begin{align} |\Phi_{m}^h(x,t)|&\lesssim\frac{2^{m(d+1)}}{\left(1+|2^m x'|^{2}\right)^d\cdot\left(1+|2^{2m} t|^2\right)^2}\notag\\[3pt] &\quad\times \left\|\left(1-\Delta_{\xi'}\right)^d\left(1-\partial_{\tau}^2\right)^2\left(P(2^{m}\xi',2^{2m}\tau)\cdot\tilde{\varphi}(\xi')\cdot\tilde{\theta}_0(\tau)\right)\right\|_{L^{1}(\mathbb{R}^{d-1}\times\mathbb{R})}. \end{align}\] Notice that there exists a constant \(c>0\) such that \[\begin{align} Re(\sqrt{|\xi'|^2+i\tau})\geq \frac{1}{\sqrt 2}(|\xi'|^4+|\tau|^2)^{1/4}\geq 2c \end{align}\] on \(\operatorname{supp}\tilde{\varphi}\times\operatorname{supp}\tilde{\theta}_0\). Therefore, we infer that \[|\Phi_{m}^h(x,t)| \lesssim\frac{2^{m(d+1+n)}}{\left(1+|2^m x'|^{2}\right)^d \cdot \left(1+|2^{2m} t|^2\right)^2} \cdot e^{-cx_d2^{m}},\] which leads to ?? . Analogously, we can derive ?? . ◻
Lemma 17 (The estimate of \(I_1\)). Under the assumptions of Theorem 2, for \(I_1\) defined in 26 , we have that for \(n-\frac{1}{q}=0\) \((\)i.e. \(n=q=1)\), \[\label{I10} \left\|I_1\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}\lesssim\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{n-\frac{1}{q}}(\mathbb{R}^{d-1})\big)},\qquad{(13)}\] and for \(n-\frac{1}{q}<0\) \((\)i.e. \(n=0)\), \[\label{eq:I11} \left\|I_1\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}\lesssim\left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)},\qquad{(14)}\] and for \(n-\frac{1}{q}>0\), \[\label{eq:I12} \left\|I_1\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}\lesssim\left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)}+\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{n-\frac{1}{q}}(\mathbb{R}^{d-1})\big)}.\qquad{(15)}\]
Proof. We first deal with the case \(n-\frac{1}{q}=0\) (i.e. \(n=q=1\)) by a duality argument. As in 20 , we get that for any function \(\psi(x,t)\in L^{p'}\big(\mathbb{R};\,L^{\infty}(\mathbb{R}_+^{d})\big)\), \[\begin{align} K_2&\overset{\mathrm{def}}{=\joinrel\!\!=}\left|\int_{\mathbb{R}^d_+\times\mathbb{R}}I_1\cdot\psi\,{\rm d}x\,{\rm d}t\right|\notag\\[3pt] &\lesssim\left|\sum_{m\in\mathbb{Z}}\int_{\mathbb{R}^d_+\times\mathbb{R}}\left(\int_{\mathbb{R}^{d-1}\times\mathbb{R}}\Phi_{m}^{h}(x'-y',x_d,t-s)\cdot \Delta_{m}^h g(y',s)\,{\rm d}y'\,{\rm d}s\right) \cdot\Delta_{< 2m}^{\mathcal{T}}\psi\,{\rm d}x\,{\rm d}t\right|, \end{align}\] which together with ?? yields (denoting \(s=t'\), \(t-s=-s'\), and dropping prime) \[\begin{align} K_2&\lesssim\sum_{m\in\mathbb{Z}}\int_{\mathbb{R}_+\times\mathbb{R}}2^{m} e^{-cx_d2^{m}}\cdot\left\|\Delta_{m}^h g\right\|_{L^{1}(\mathbb{R}^{d-1})}\cdot \sup_{s\in\mathbb{R}}\frac{\left\|\Delta_{< 2m}^{\mathcal{T}}\psi(\cdot,x_d,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}}{1+|2^{2m} s|^2}\,{\rm d}x_d\,{\rm d}t\notag\\[3pt] &\lesssim\sum_{m\in\mathbb{Z}}\int_{\mathbb{R}}\left\|\Delta_{m}^h g\right\|_{L^{1}(\mathbb{R}^{d-1})}\cdot\sup_{s\in\mathbb{R}}\frac{\left\|\Delta_{< 2m}^{\mathcal{T}}\psi(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d}_+)}}{1+|2^{2m} s|^2}\,{\rm d}t.\label{Dual3} \end{align}\tag{28}\] By ?? and ?? with \(r=1\), we have \[\label{Dual4} \sup_{s\in\mathbb{R}}\frac{\left\|\Delta_{< 2m}^{\mathcal{T}}\psi(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d}_+)}}{1+|2^{2m} s|^2}\lesssim M^{\mathcal{T}}M^{\mathcal{T}}\left\|\psi(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{d}_+)}.\tag{29}\] By substituting 29 into 28 and then using 10 , we achieve that for \(1\leq p<\infty\), \[\begin{align} K_2&\lesssim\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{1,1}^{0}(\mathbb{R}^{d-1})\big)}\cdot\left\|M^{\mathcal{T}}M^{\mathcal{T}} \left\|\psi(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{d}_+)}\right\|_{L^{p'}(\mathbb{R})}\\[3pt] &\lesssim\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{1,1}^{0}(\mathbb{R}^{d-1})\big)}\cdot\left\|\psi\right\|_{L^{p'}\big(\mathbb{R};\,L^{\infty}(\mathbb{R}_+^{d})\big)}, \end{align}\] which leads to ?? by duality.
Now we assume \(n-\frac{1}{q}\neq0\). By 26 , ?? and Young’s inequality, we obtain \[\begin{align} \left\|I_1(\cdot,x_d,t)\right\|_{L^q(\mathbb{R}^{d-1})}&\lesssim\sum_{m\in\mathbb{Z}}\int_{\mathbb{R}}\left\|\Phi_{m}^{h}(\cdot,x_d,s)\right\|_{L^{1}(\mathbb{R}^{d-1})}\cdot\left\|\Delta_{m}^{h}\Delta_{< 2m}^{\mathcal{T}}g(\cdot,t-s)\right\|_{L^{q}(\mathbb{R}^{d-1})} \,{\rm d}s\notag\\[3pt] &\lesssim\sum_{m\in\mathbb{Z}}2^{mn} e^{-cx_d2^{m}} \cdot \sup_{s\in\mathbb{R}}\frac{\left\|\Delta_{m}^{h}\Delta_{< 2m}^{\mathcal{T}}g(\cdot,t-s)\right\|_{L^{q}(\mathbb{R}^{d-1})}}{1+|2^{2m} s|^2}.\label{eq:I951-est1} \end{align}\tag{30}\] The above sup is bounded for each \(m\), since \(g\in L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{n-\frac{1}{q}}(\mathbb{R}^{d-1})\big)\) if \(n-\frac{1}{q}>0\) and \(g\in \accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)\) if \(n-\frac{1}{q}<0\). Indeed, if \(n-\frac{1}{q}>0\), by Young’s inequality, \[\left\|\left\|\Delta^{h}_{m}\Delta^{\mathcal{T}}_{<2m} g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{\infty}(\mathbb{R})}\lesssim_{m} \left\|\left\|\Delta^{h}_{m} g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{p}(\mathbb{R})}\lesssim_{m}\left\|g\right\|_{L^{p}\big(\mathbb{R};\,\accentset{\large\bfseries .}{B}_{q,q}^{n-\frac{1}{q}}(\mathbb{R}^{d-1})\big)},\] and if \(n-\frac{1}{q}<0\), applying Young’s inequality and Hölder’s inequality in \(\ell^1\) gives \[\begin{align} \left\|\left\|\Delta^{h}_{m}\Delta^{\mathcal{T}}_{<2m} g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{\infty}(\mathbb{R})}&\lesssim_{m} \left\|\sum_{j<2m} 2^{-(\frac{n}{2}-\frac{1}{2q})j}\cdot 2^{(\frac{n}{2}-\frac{1}{2q})j}\left\|\Delta^{\mathcal{T}}_{j} g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{p}(\mathbb{R})}\\[3pt] &\lesssim_{m}\left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)}. \end{align}\] Estimate 30 , together with ?? with \(r=\frac{1}{2}\) and ?? , implies that \[\begin{align} \label{eq:lem-I1} \left\|I_1\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}&= \left\|\left(\int_0^{\infty}\left\|I_1(\cdot,x_d,t)\right\|_{L^{q}(\mathbb{R}^{d-1})}^{q}\,{\rm d}x_d\right)^{\frac{1}{q}}\right\|_{L^p(\mathbb{R})}\notag\\[3pt] &\lesssim\left\|\left(\sum_{m\in\mathbb{Z}}2^{mnq-m} \Big(\sup_{s\in\mathbb{R}}\frac{\left\|\Delta_{m}^{h}\Delta_{< 2m}^{\mathcal{T}}g(\cdot,t-s)\right\|_{L^{q}(\mathbb{R}^{d-1})}}{1+|2^{2m} s|^2}\Big)^q\right)^{\frac{1}{q}}\right\|_{L^p(\mathbb{R})}\notag\\[3pt] &\lesssim\left\|2^{m(n-\frac{1}{q})}\left\|\Delta^{h}_{m}\Delta^{\mathcal{T}}_{<2m} g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{p}(\mathbb{R};\ell^q(m))}. \end{align}\tag{31}\] In what follows, we shall bound the RHS of 31 separately in the case \(n-\frac{1}{q}<0\) and \(n-\frac{1}{q}>0\).
Case 1. If \(n-\frac{1}{q}<0\), we get \[\begin{align} 2^{m(n-\frac{1}{q})}\left\|\Delta^{h}_{m}\Delta^{\mathcal{T}}_{<2m} g\right\|_{L^q(\mathbb{R}^{d-1})}&\lesssim 2^{m(n-\frac{1}{q})}\left\|\Delta^{\mathcal{T}}_{<2m}g\right\|_{L^q(\mathbb{R}^{d-1})}\notag\\[3pt] &\lesssim\sum_{j<2m}2^{(m-\frac{j}{2})(n-\frac{1}{q})}\cdot2^{\frac{j}{2}(n-\frac{1}{q})}\left\|\Delta^{\mathcal{T}}_{j}g\right\|_{L^q(\mathbb{R}^{d-1})}, \end{align}\] which together with 31 and the Young’s convolution inequality for \(\ell^q\) implies that \[\left\|I_1\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)} \lesssim\left\|2^{\frac{j}{2}(n-\frac{1}{q})}\left\|\Delta^{\mathcal{T}}_j g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{p}(\mathbb{R};\ell^q(j))}.\] Hence ?? .
Case 2. If \(n-\frac{1}{q}>0\), we get \[\begin{align} &\quad\;2^{m(n-\frac{1}{q})}\left\|\Delta^{h}_{m}\Delta^{\mathcal{T}}_{<2m} g\right\|_{L^q(\mathbb{R}^{d-1})}\notag\\[3pt] &\lesssim 2^{m(n-\frac{1}{q})}\Big(\left\|\Delta^{h}_{m}g\right\|_{L^q(\mathbb{R}^{d-1})}+\left\|\Delta^{h}_{m}\Delta^{\mathcal{T}}_{\geq 2m}g\right\|_{L^q(\mathbb{R}^{d-1})}\Big)\notag\\[3pt] &\lesssim 2^{m(n-\frac{1}{q})}\Big(\left\|\Delta^{h}_{m}g\right\|_{L^q(\mathbb{R}^{d-1})}+\left\|\Delta^{\mathcal{T}}_{\geq 2m}g\right\|_{L^q(\mathbb{R}^{d-1})}\Big)\notag\\[3pt] &\lesssim 2^{m(n-\frac{1}{q})}\left\|\Delta^{h}_{m}g\right\|_{L^q(\mathbb{R}^{d-1})}+\sum_{j\geq2m}2^{(m-\frac{j}{2})(n-\frac{1}{q})}\cdot2^{\frac{j}{2}(n-\frac{1}{q})}\left\|\Delta^{\mathcal{T}}_{j}g\right\|_{L^q(\mathbb{R}^{d-1})}, \end{align}\] which together with 31 and Young’s convolution inequality for \(\ell^q\) implies that \[\begin{align} \left\|I_1\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)} &\lesssim\left\|2^{m(n-\frac{1}{q})}\left\|\Delta^{h}_m g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{p}(\mathbb{R};\ell^q(m))}\notag\\[3pt] &\quad\;+\left\|2^{\frac{j}{2}(n-\frac{1}{q})}\left\|\Delta^{\mathcal{T}}_j g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{p}(\mathbb{R};\ell^q(j))}. \end{align}\] Hence ?? . ◻
Lemma 18 (The estimate of \(I_2\)). Under the assumptions of Theorem 2, for \(I_2\) defined in 27 , we have \[\label{eq:I2} \left\|I_2\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}\lesssim\left\|g\right\|_{\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)}.\qquad{(16)}\]
Proof. By 27 , ?? and Young’s inequality, we obtain \[\begin{align} \left\|I_2(\cdot,x_d,t)\right\|_{L^q(\mathbb{R}^{d-1})}&\lesssim\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}}\left\|\Phi_{k}^{\mathcal{T}}(\cdot,x_d,s)\right\|_{L^{1}(\mathbb{R}^{d-1})}\cdot\left\|\Delta_{k}^{\mathcal{T}}g(\cdot,t-s)\right\|_{L^{q}(\mathbb{R}^{d-1})} \,{\rm d}s\\[3pt] &\lesssim\sum_{k\in\mathbb{Z}}2^{\frac{kn}{2}} e^{-cx_d2^{\frac{k}{2}}} \cdot \sup_{s\in\mathbb{R}}\frac{\left\|\Delta_{k}^{\mathcal{T}}g(\cdot,t-s)\right\|_{L^{q}(\mathbb{R}^{d-1})}}{1+|2^{k} s|^2}. \end{align}\] Since \(g\in\accentset{\large\bfseries .}{F}^{\frac{n}{2}-\frac{1}{2q}}_{p,q}\big(\mathbb{R};\,L^{q}(\mathbb{R}^{d-1})\big)\), the above sup is bounded for each \(k\). The above estimate together with ?? with \(r=\frac{1}{2}\) and ?? implies that \[\begin{align} &\,\quad \left\|\left(\int_0^{\infty}\left\|I_2(\cdot,x_d,t)\right\|_{L^{q}(\mathbb{R}^{d-1})}^{q}\,{\rm d}x_d\right)^{\frac{1}{q}}\right\|_{L^p(\mathbb{R})}\\[3pt] &\lesssim\left\|\left(\sum_{k\in\mathbb{Z}}2^{\frac{knq-k}{2}} \Big(\sup_{s\in\mathbb{R}}\frac{\left\|\Delta_{k}^{\mathcal{T}}g(\cdot,t-s)\right\|_{L^{q}(\mathbb{R}^{d-1})}}{1+|2^{k} s|^2}\Big)^q\right)^{\frac{1}{q}}\right\|_{L^p(\mathbb{R})}\\[3pt] &\lesssim\left\|2^{k(\frac{n}{2}-\frac{1}{2q})}\left\|\Delta^{\mathcal{T}}_{k} g\right\|_{L^q(\mathbb{R}^{d-1})}\right\|_{L^{p}(\mathbb{R};\ell^q(k))}. \end{align}\] Hence ?? . ◻
We are now in a position to prove Theorem 2.
Proof of Theorem 2. By 25 , Lemma 17, and Lemma 18, we obtain ?? and ?? . This completes the proof. ◻
In this subsection we present the proof of Theorem 3, which follows from Lemma 19 and Lemma 23, bounding the two terms in the left-hand side of ?? separately.
Lemma 19. Let \(d\geq 2\). Suppose that \(g(x',t)\in C_c^\infty(\mathbb{R}^{d-1}\times \mathbb{R})\), and \(v(x,t)\) is given by 3 . Then, for \(1<p<\infty\) and \(1\leq q\leq \infty\), as well as for \(p=q=1\) and \(p=q=\infty\), and for every integer \(n\geq 0\), we have \[\label{eq:trace1-1} \left\|2^{(\frac{n}{2}-\frac{1}{2q})k}\|\Delta_{\leq \frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(k))} \lesssim\left\|\partial_{x_d}^nv\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}.\qquad{(17)}\] Moreover, if \(n>\frac{1}{q}\), then \[\label{eq:trace1-2} \left\|2^{(\frac{n}{2}-\frac{1}{2q})k}\|\Delta_{>\frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(k))} \lesssim\left\|\partial_{x_d}^nv\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}.\qquad{(18)}\]
As in the one-dimensional case, we divide the proof into the following three lemmas (Lemma 20-22) corresponding to different ranges of the indices \(p,q\).
Lemma 20. Under the assumptions of Lemma 19 with \(1<p,q<\infty\), estimates ?? and ?? hold.
Proof. Let \(\gamma=\lfloor \frac{1}{q}\rfloor\in\{0,1\}\) for \(1\leq q \leq \infty\) (\(\gamma=0\) here and in Lemma 22, \(\gamma=1\) in Lemma 21). In view of 24 , we have \[\label{eq:lem:trace-d} \hat{g}(\xi',\tau)=(-2)^{1+\gamma}\int_{0}^{+\infty}x_d^{\gamma} (-\sqrt{|\xi'|^2+i\tau})^{1+\gamma-n}\cdot e^{-x_d\sqrt{|\xi'|^2+i\tau}}\cdot \partial_{x_d}^n\hat{v}(\xi',x_d,\tau) \,{\rm d}x_d.\tag{32}\]
We now use a duality argument. For any sequence \(\{\psi_k(x',t)\}_{k\in\mathbb{Z}}\) satisfying \[\big\|\|\psi_k(\cdot,t)\|_{L^{q'}(\mathbb{R}^{d-1})}\big\|_{L^{p'}(\mathbb{R};\ell^{q'}(k))}<\infty,\] we define \[\begin{align} K_{31}&\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{k\in\mathbb{Z}}2^{(\frac{n}{2}-\frac{1}{2q})k}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \Delta_{\leq \frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}}g\cdot\psi_{k}\,{\rm d}x'\,{\rm d}t\notag\\[3pt] &=\sum_{k\in\mathbb{Z}}2^{(\frac{n}{2}-\frac{1}{2q})k}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \Delta_{\leq \frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}}g\cdot\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}\,{\rm d}x'\,{\rm d}t,\label{K31} \end{align}\tag{33}\] and \[\begin{align} K_{32}&\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{k\in\mathbb{Z}}2^{(\frac{n}{2}-\frac{1}{2q})k}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \Delta_{>\frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}}g\cdot\psi_{k}\,{\rm d}x'\,{\rm d}t\notag\\[3pt] &=\sum_{k\in\mathbb{Z}}2^{(\frac{n}{2}-\frac{1}{2q})k}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \sum_{j>\frac{k}{2}}\Delta_{j}^{h}\Delta_{k}^{\mathcal{T}}g\cdot\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}\,{\rm d}x'\,{\rm d}t.\label{K32} \end{align}\tag{34}\]
We estimate \(K_{31}\) first. By a similar argument as in Lemma 16, we have \[\begin{align} &\quad\;\left\|\mathcal{F}_{x',t}^{-1}\Big(x_d^{\gamma}(-\sqrt{|\xi'|^2+i\tau})^{1+\gamma-n}\cdot e^{-x_d\sqrt{|\xi'|^2+i\tau}}\cdot\theta(2^{-k}\tau)\cdot \varphi_0(2^{-\lfloor \frac{k}{2}\rfloor }\xi')\Big)\right\|_{L^{1}(\mathbb{R}^{d-1})}\\[3pt] &\lesssim\frac{x_d^{\gamma}2^{\frac{3+\gamma-n}{2}k}}{\left(1+|2^{k} t|^2\right)^2}\cdot e^{-cx_d 2^{\frac{k}{2}}}, \end{align}\] which together with 32 , Minkowski and Young’s inequality implies that \[\label{eq:hg1} \|\Delta_{\leq \frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\lesssim\int_{0}^{\infty}\int_{\mathbb{R}} \frac{ x_d^{\gamma}2^{\frac{3+\gamma-n}{2}k} e^{-cx_d 2^{\frac{k}{2}}}}{\left(1+|2^k (t-s)|^2\right)^2}\left\|\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d.\tag{35}\] As in the proof of Lemma 13, we get that, by 35 , \[\begin{align} K_{31}&\lesssim\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}}\Big(\int_{0}^{\infty}\int_{\mathbb{R}}\frac{ x_d^{\gamma}2^{\frac{3+\gamma-\frac{1}{q}}{2}k} e^{-cx_d 2^{\frac{k}{2}}}}{\left(1+|2^k (t-s)|^2\right)^2}\cdot\left\|\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d\Big)\notag\\[3pt] &\quad\quad \times\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})}\,{\rm d}t\notag \\[3pt] &\lesssim\int_{\mathbb{R}}\int_{0}^{\infty}\sum_{k\in\mathbb{Z}}\Big( x_d^{\gamma}2^{\frac{1+\gamma-\frac{1}{q}}{2}k} e^{-cx_d 2^{\frac{k}{2}}}\cdot\Big(\sup_{s\in \mathbb{R}} \frac{\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t-s)\right\|_{L^{q'}(\mathbb{R}^{d-1})}}{1+|2^ks|^2}\Big)\Big)\notag\\[3pt] &\quad\quad \times\left\|\partial_{x_d}^nv(\cdot,x_d,t)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}x_d\,{\rm d}t.\label{eq:K31} \end{align}\tag{36}\] Applying Hölder’s inequality and then ?? , we obtain \[\label{eq:K31-bound} K_{31}\lesssim\int_{\mathbb{R}}\Big(\sum_{k\in\mathbb{Z}}\Big(\sup_{s\in \mathbb{R}} \frac{\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t-s)\right\|_{L^{q'}(\mathbb{R}^{d-1})}}{1+|2^ks|^2}\Big)^{q'}\Big)^{\frac{1}{q'}}\cdot\left\|\partial_{x_d}^nv(\cdot,t)\right\|_{L^{q}(\mathbb{R}_+^{d})}\,{\rm d}t.\tag{37}\] By ?? with \(r=1\), Minkowski inequality and ?? , we have \[\label{44622} \sup_{s\in \mathbb{R}} \frac{\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t-s)\right\|_{L^{q'}(\mathbb{R}^{d-1})}}{1+|2^ks|^2} \lesssim M^{\mathcal{T}}M^{\mathcal{T}}\left\|\psi_k(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})}.\tag{38}\] Hence by ?? , \[K_{31}\lesssim\left\|\left\|\psi_{k}(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})}\right\|_{L^{p'}(\mathbb{R};\ell^{q'}(k))}\cdot\left\|\partial_{x_d}^nv\right\|_{L^{p}(\mathbb{R};L^q\left(\mathbb{R}_+^d\right))}.\] By a duality argument, it follows that for \(1<p<\infty\) and \(1< q<\infty\), estimate ?? holds.
We now estimate \(K_{32}\). For \(j>\frac{k}{2}\), we have \[\begin{align} &\quad\;\left\|\mathcal{F}_{x',t}^{-1}\Big(x_d^{\gamma}(-\sqrt{|\xi'|^2+i\tau})^{1+\gamma-n}\cdot e^{-x_d\sqrt{|\xi'|^2+i\tau}}\cdot\theta(2^{-k}\tau)\cdot \varphi(2^{-j}\xi')\Big)\right\|_{L^{1}(\mathbb{R}^{d-1})}\\[3pt] &\lesssim\bigg\|\frac{2^{(d-1)j+k}\cdot x_d^{\gamma}2^{j(1+\gamma-n)}}{(1+|2^jx'|^2)^d(1+|2^kt|^2)^2}\left\|\left(1-\Delta_{\xi'}\right)^d\left(1-\partial_{\tau}^2\right)^2 P_1(\xi',x_d,\tau)\right\|_{L^{1}(\mathbb{R}^{d-1}\times\mathbb{R})}\bigg\|_{L^1(\mathbb{R}^{d-1})} \\[3pt] &\lesssim\frac{2^kx_d^{\gamma}2^{j(1+\gamma-n)}}{\left(1+|2^{k} t|^2\right)^2}\cdot e^{-cx_d 2^j}, \end{align}\] where \[P_1(\xi',x_d,\tau)=\Big(\sqrt{|\xi'|^2+2^{k-2j}i\tau }\Big)^{1+\gamma-n}\cdot e^{-x_d\cdot 2^j\sqrt{|\xi'|^2+2^{k-2j}i\tau}}\cdot\theta(\tau)\cdot \varphi(\xi').\] The estimate above, together with 32 , Minkowski’s and Young’s inequalities, implies that for \(j>\frac{k}{2}\), \[\label{eq:hg2} \|\Delta_{j}^{h}\Delta_{k}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\lesssim\int_{0}^{\infty} \!\!\int_{\mathbb{R}}\frac{ 2^kx_d^{\gamma}2^{j(1+\gamma-n)}e^{-cx_d 2^j}}{\left(1+|2^k (t-s)|^2\right)^2}\left\|\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d.\tag{39}\] Given \(1+\gamma-n\leq0\) (as \(n>1/q\)), we have that for \(\varepsilon>0\), \[\label{eq:gamma-q-n} \sum_{j>\frac{k}{2}} 2^{(1+\gamma-n)j}e^{-cx_d 2^j}\lesssim 2^{(1+\gamma-n)\frac{k}{2}}e^{-\frac{c}{2} x_d2^{\frac{k}{2}}}\sum_{j>\frac{k}{2}}(x_d2^j)^{-\varepsilon}\lesssim 2^{(1+\gamma-n-\varepsilon)\frac{k}{2}}x_d^{-\varepsilon}e^{-\frac{c}{2}x_d2^{\frac{k}{2}}}.\tag{40}\] Hence we obtain \[\begin{align} K_{32}&\lesssim\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}}\Big(\int_{0}^{\infty}\int_{\mathbb{R}} \frac{2^{\frac{3+\gamma-\frac{1}{q}-\varepsilon}{2}k} x_d^{\gamma-\varepsilon}e^{-\frac{c}{2}x_d 2^{\frac{k}{2}}}}{\left(1+|2^k (t-s)|^2\right)^2}\cdot\left\|\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d\Big) \notag\\[3pt] &\quad\quad \times\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})}\,{\rm d}t\notag\\[3pt] &\lesssim\int_{\mathbb{R}}\int_{0}^{\infty}\sum_{k\in\mathbb{Z}}\Big( 2^{\frac{1+\gamma-\frac{1}{q}-\varepsilon}{2}k}x_d^{\gamma-\varepsilon} e^{-\frac{c}{2}x_d 2^{\frac{k}{2}}}\cdot\Big(\sup_{s\in \mathbb{R}} \frac{\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t-s)\right\|_{L^{q'}(\mathbb{R}^{d-1})}}{1+|2^ks|^2}\Big)\Big)\notag\\[3pt] &\quad\quad \times\left\|\partial_{x_d}^nv(\cdot,x_d,t)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}x_d\,{\rm d}t. \label{eq:K32} \end{align}\tag{41}\] By applying Hölder’s inequality and then ?? with \(\varepsilon=\frac{1}{2q'^2}\), we obtain the same bound as in 37 for \(K_{32}\). The estimate ?? follows in the same way. ◻
Note that in the previous lemma, we have \(\gamma=0\), and 40 requires \(n\geq 1\). In the next lemma for the endpoint case \(q=1\), we will choose \(\gamma=1\), which produces the bounded term \[\sum_{k\in \mathbb{Z}}\left((x_d2^{\frac{k}{2}})^\frac{1}{2}e^{-\frac{c}{2}x_d 2^{\frac{k}{2}}}\right).\] If we instead choose \(\gamma=0\), then the corresponding term carries a non-positive power of \(x_d\), and the summation diverges. This also forces the condition \(n\geq 2\), since we need 40 . This explains the condition \(n>\frac{1}{q}\) in Lemma 19.
Lemma 21. Under the assumptions of Lemma 19 with \(1\leq p<\infty\) and \(q=1\), estimates ?? and ?? hold.
Proof. The proof is similar to Lemma 14. We continue to use 33 and 34 from Lemma 20. By 36 with \(q=1\), \[K_{31}\lesssim\int_{\mathbb{R}}\sup_{k}\sup_{s\in \mathbb{R}} \frac{\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}}{1+|2^ks|^2}\cdot\left\|\partial_{x_d}^nv(\cdot,t)\right\|_{L^{1}(\mathbb{R}_+^{d})}\,{\rm d}t. \label{eq:K31-q1}\tag{42}\] Hence by 38 and ?? , we derive \[K_{31}\lesssim\left\|\left\|\psi_{k}(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}\right\|_{L^{p'}(\mathbb{R};\ell^{\infty}(k))}\cdot\left\|\partial_{x_d}^nv\right\|_{L^{p}(\mathbb{R};L^1\left(\mathbb{R}_+^d\right))}.\] Thus by a duality argument, for \(1\leq p<\infty\) and \(q=1\), we obtain ?? .
For \(K_{32}\) with \(n>1/q=1\), by 41 with \(\varepsilon=\frac{1}{2}\), we get \[\begin{align} K_{32} &\lesssim\int_{\mathbb{R}}\int_0^\infty \sum_{k\in \mathbb{Z}}\left((x_d2^{\frac{k}{2}})^\frac{1}{2}e^{-\frac{c}{2}x_d 2^{\frac{k}{2}}}\right) \sup_{k}\sup_{s\in \mathbb{R}} \frac{\left\|\tilde{\Delta}_{k}^{\mathcal{T}}\psi_{k}(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}}{1+|2^ks|^2}\\[3pt] &\quad \quad \times \left\|\partial_{x_d}^nv(\cdot,x_d,t)\right\|_{L^{1}(\mathbb{R}^{d-1})}\,{\rm d}x_d\,{\rm d}t. \end{align}\] Applying 12 , we obtain the same bound 42 for \(K_{32}\). The estimate ?? follows in the same way. ◻
Lemma 22. Under the assumptions of Lemma 19 with \(1< p\leq\infty\) and \(q=\infty\), estimates ?? and ?? hold.
Proof. The proof is similar to that of Lemma 15: we insert the factor \(\tilde{\theta}(2^{-k}\tau)\theta(2^{-k}\tau)\) before \(\partial_{x_d}^n\hat{v}\) when we apply \(\Delta_k^\mathcal{T}\) to 32 . By 35 , 13 , ?? with \(r=1\) and ?? , we have that for \(q=\infty\), \[\begin{align} \|\Delta_{\leq \frac{k}{2}}^h\Delta_{k}^{\mathcal{T}} g\|_{L^{\infty}(\mathbb{R}^{d-1})}&\lesssim\int_{0}^{\infty}\int_{\mathbb{R}} \frac{ 2^{(\frac{3}{2}-\frac{n}{2})k} e^{-cx_d 2^{\frac{k}{2}}}}{\left(1+|2^k (t-s)|^2\right)^2}\left\|\Delta_{k}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d\notag \\[3pt] &\lesssim\int_{0}^{\infty} 2^{(\frac{1}{2}-\frac{n}{2})k} e^{-cx_d 2^{\frac{k}{2}}}\cdot\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{k}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}}{1+|2^ks|^2}\,{\rm d}x_d \notag\\[3pt] &\lesssim 2^{-\frac{n}{2}k}\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{k}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}}{1+|2^ks|^2} \notag \\[3pt] &\lesssim 2^{-\frac{n}{2}k} \cdot M^{\mathcal{T}}M^{\mathcal{T}} \left\|\partial_{x_d}^nv(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}. \label{44624} \end{align}\tag{43}\] Then, applying ?? , we obtain ?? .
Analogously, by 39 and 40 with \(n>1/q=0\) and \(\varepsilon=\frac{1}{2}\), we have \[\begin{align} &\quad\;\sum_{j>\frac{k}{2}}\|\Delta_{j}^h\Delta_{k}^{\mathcal{T}} g\|_{L^{\infty}(\mathbb{R}^{d-1})}\\[3pt] &\lesssim\int_{0}^{\infty}\int_{\mathbb{R}} \frac{ x_d^{-\frac{1}{2}}2^{(\frac{5}{2}-n)\frac{k}{2}}e^{-\frac{c}{2}x_d 2^{\frac{k}{2}}}}{\left(1+|2^k (t-s)|^2\right)^2}\left\|\Delta_{k}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d \notag\\[3pt] &\lesssim\int_{0}^{\infty}x_d^{-\frac{1}{2}} 2^{(\frac{1}{2}-n)\frac{k}{2}} e^{-\frac{c}{2}x_d 2^{\frac{k}{2}}}\cdot\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{k}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}}{1+|2^ks|^2}\,{\rm d}x_d. \end{align}\] Therefore, by 13 , ?? with \(r=1\) and ?? , we achieve \[\begin{align} \|\Delta_{>\frac{k}{2}}^{h}\Delta_{k}^{\mathcal{T}} g\|_{L^{\infty}(\mathbb{R}^{d-1})} &\lesssim 2^{-\frac{n}{2}k}\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{k}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}}{1+|2^ks|^2} \\ &\lesssim 2^{-\frac{n}{2}k} \cdot M^{\mathcal{T}}M^{\mathcal{T}} \left\|\partial_{x_d}^nv(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}, \end{align}\] which together with ?? yields ?? . ◻
Lemma 23. Let \(d\geq 2\). Suppose that \(g(x',t)\in C_c^\infty(\mathbb{R}^{d-1}\times \mathbb{R})\), and \(v(x,t)\) is given by 3 . Then, for \(1<p<\infty\) and \(1\leq q\leq \infty\), as well as for \(p=q=1\) and \(p=q=\infty\), and for every integer \(n\geq 0\), we have \[\begin{align} \label{eq:trace2-1} \left\|2^{(n-\frac{1}{q})m}\|\Delta_{m}^h\Delta_{<2m}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(m))} \lesssim\left\|\partial_{x_d}^nv\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}. \end{align}\qquad{(19)}\] Moreover, if \(n>\frac{1}{q}\), then \[\label{eq:trace2-2} \left\|2^{(n-\frac{1}{q})m}\|\Delta_{m}^h\Delta_{\geq 2m}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\right\|_{L^p(\mathbb{R};\ell^q(m))} \lesssim\left\|\partial_{x_d}^nv\right\|_{L^{p}\big(\mathbb{R};\,L^{q}(\mathbb{R}_+^{d})\big)}.\qquad{(20)}\]
The proof follows the same scheme as that of Lemma 19. As before, we split the argument into three lemmas (Lemma 24-26), corresponding respectively to the three ranges of the indices \(p\) and \(q\).
Lemma 24. Under the assumptions of Lemma 23 with \(1<p,q<\infty\), estimates ?? and ?? hold.
Proof. We use a duality argument. For any sequence \(\{\psi_m(x',t)\}_{m\in\mathbb{Z}}\) satisfying \[\|\|\psi_m(\cdot,t)\|_{L^{q'}(\mathbb{R}^{d-1})}\|_{L^{p'}(\mathbb{R};\ell^{q'}(m))}<\infty,\] we define \[\begin{align} K_{41}&\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{m\in\mathbb{Z}}2^{(n-\frac{1}{q})m}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \Delta_{m}^{h}\Delta_{<2m}^{\mathcal{T}}g\cdot \psi_{m}\,{\rm d}x'\,{\rm d}t\\[3pt] &=\sum_{m\in\mathbb{Z}}2^{(n-\frac{1}{q})m}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \Delta_{m}^{h}\Delta_{<2m}^{\mathcal{T}}g\cdot \Delta_{<2m+2}^{\mathcal{T}}\psi_{m}\,{\rm d}x'\,{\rm d}t, \end{align}\] and \[\begin{align} K_{42}&\overset{\mathrm{def}}{=\joinrel\!\!=}\sum_{m\in\mathbb{Z}}2^{(n-\frac{1}{q})m}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \Delta_{m}^{h}\Delta_{\geq2m}^{\mathcal{T}}g\cdot\psi_{m}\,{\rm d}x'\,{\rm d}t\\[3pt] &=\sum_{m\in\mathbb{Z}}2^{(n-\frac{1}{q})m}\int_{\mathbb{R}^{d-1}\times\mathbb{R}} \sum_{j\geq 2m}\Delta_{m}^{h}\Delta_{j}^{\mathcal{T}}g\cdot\tilde{\Delta}_{j}^{\mathcal{T}}\psi_{m}\,{\rm d}x'\,{\rm d}t. \end{align}\]
We estimate \(K_{41}\) first. By a similar argument of 35 , we get \[\begin{align} \label{eq:hg1-1} \|\Delta_{m}^h\Delta_{<2m}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}&\lesssim\int_{0}^{\infty}\int_{\mathbb{R}} \frac{ x_d^\gamma 2^{(3+\gamma-n)m} e^{-cx_d 2^{m}}}{\left(1+|2^{2m} (t-s)|^2\right)^2}\left\|\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d, \end{align}\tag{44}\] and hence \[\begin{align} K_{41} &\lesssim\int_{\mathbb{R}}\int_{0}^{\infty}\sum_{m\in\mathbb{Z}}\Big( x_d^{\gamma}2^{(1+\gamma-\frac{1}{q})m} e^{-cx_d 2^{m}}\cdot\Big(\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{<2m+2}^{\mathcal{T}}\psi_{m}(\cdot,t-s)\right\|_{L^{q'}(\mathbb{R}^{d-1})}}{1+|2^{2m}s|^2}\Big)\Big)\notag\\[3pt] &\quad\quad \times\left\|\partial_{x_d}^nv(\cdot,x_d,t)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}x_d\,{\rm d}t. \label{eq:K41} \end{align}\tag{45}\] Arguing similarly as that for 36 , and using the fact that \[\label{ML1} \sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{<2m+2}^{\mathcal{T}}\psi_{m}(\cdot,t-s)\right\|_{L^{q'}(\mathbb{R}^{d-1})}}{1+|2^{2m}s|^2} \lesssim M^{\mathcal{T}}M^{\mathcal{T}}\left\|\psi_m(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})},\tag{46}\] we get by ?? , \[K_{41}\lesssim\left\|\left\|\psi_{m}(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})}\right\|_{L^{p'}(\mathbb{R};\ell^{q'}(m))}\cdot\left\|\partial_{x_d}^nv\right\|_{L^{p}(\mathbb{R};L^q\left(\mathbb{R}_+^d\right))}.\] By a duality argument, it follows that for \(1<p<\infty\) and \(1< q<\infty\), estimate ?? holds.
We now estimate \(K_{42}\). For \(j\geq 2m\), arguing similarly as that for 39 , we get \[\label{eq:hg2-2} \|\Delta_{m}^{h}\Delta_{j}^{\mathcal{T}} g\|_{L^{q}(\mathbb{R}^{d-1})}\lesssim\int_{0}^{\infty} \!\!\int_{\mathbb{R}}\frac{ x_d^{\gamma}2^{(3+\gamma-n)\frac{j}{2}}e^{-cx_d 2^{\frac{j}{2}}}}{\left(1+|2^j (t-s)|^2\right)^2}\left\|\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d.\tag{47}\] Hence, we obtain \[\begin{align} K_{42}&\lesssim\sum_{m\in\mathbb{Z}}\int_{\mathbb{R}}\Big(\int_{0}^{\infty}\int_{\mathbb{R}}\sum_{j\geq 2m} \frac{2^{(n-\frac{1}{q})m+(3+\gamma-n)\frac{j}{2}} x_d^{\gamma}e^{-cx_d 2^{\frac{j}{2}}}}{\left(1+|2^j (t-s)|^2\right)^2}\cdot\left\|\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d\Big) \notag\\[3pt] &\quad\quad \times\left\|\tilde{\Delta}_{j}^{\mathcal{T}}\psi_{m}(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})}\,{\rm d}t\notag\\[3pt] &\lesssim\int_{\mathbb{R}}\int_{0}^{\infty}\sum_{m \in\mathbb{Z}}\sum_{j\geq 2m}\Big( 2^{(n-\frac{1}{q})m+(1+\gamma-n)\frac{j}{2}}x_d^{\gamma} e^{-cx_d 2^{\frac{j}{2}}}\cdot\Big(\sup_{s\in \mathbb{R}} \frac{\left\|\tilde{\Delta}_{j}^{\mathcal{T}}\psi_{m}(\cdot,t-s)\right\|_{L^{q'}(\mathbb{R}^{d-1})}}{1+|2^js|^2}\Big)\Big)\notag\\[3pt] &\quad\quad \times\left\|\partial_{x_d}^nv(\cdot,x_d,t)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}x_d\,{\rm d}t. \end{align}\] Bounding Peetre’s maximal function by Hardy-Littlewood maximal function, and then applying 40 with \(1+\gamma-n\leq0\) (as \(n>1/q\)), we get \[\begin{align} K_{42}&\lesssim\int_{\mathbb{R}}\int_{0}^{\infty}\sum_{m \in\mathbb{Z}}\Big( 2^{(1+\gamma-\frac{1}{q}-\varepsilon)m}x_d^{\gamma-\varepsilon} e^{-\frac{c}{2}x_d 2^{m}}\cdot M^{\mathcal{T}}M^{\mathcal{T}}\left\|\psi_m(\cdot,t)\right\|_{L^{q'}(\mathbb{R}^{d-1})}\Big)\notag\\[3pt] &\quad\quad \times\left\|\partial_{x_d}^nv(\cdot,x_d,t)\right\|_{L^{q}(\mathbb{R}^{d-1})}\,{\rm d}x_d\,{\rm d}t. \label{eq:K42} \end{align}\tag{48}\] Then applying Hölder’s inequality and then ?? with \(\varepsilon=\frac{1}{2q'^2}\), we obtain the same bound as \(K_{41}\) for \(K_{42}\). The estimate ?? follows in the same way. ◻
Lemma 25. Under the assumptions of Lemma 23 with \(1\leq p<\infty\) and \(q=1\), estimates ?? and ?? hold.
Proof. The proof is similar to Lemma 21. By 45 with \(q=1\) and 46 , we get \[K_{41}\lesssim\int_{\mathbb{R}}\sup_{m}M^{\mathcal{T}}M^{\mathcal{T}}\left\|\psi_m(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}\cdot\left\|\partial_{x_d}^nv(\cdot,t)\right\|_{L^{1}(\mathbb{R}_+^{d})}\,{\rm d}t. \label{eq:K31-q611}\tag{49}\] Hence by ?? and a duality argument, we obtain ?? .
By 48 with \(\varepsilon=\frac{1}{2}\), we obtain the same bound as \(K_{41}\) for \(K_{42}\). The estimate ?? follows in the same way. ◻
Lemma 26. Under the assumptions of Lemma 23 with \(1< p\leq\infty\) and \(q=\infty\), estimates ?? and ?? hold.
Proof. The proof is similar to Lemma 22. Analogous to 44 and 43 , it is easy to verify that \[\begin{align} &\quad \;\|\Delta_{m}^h\Delta_{<2m}^{\mathcal{T}} g\|_{L^{\infty}(\mathbb{R}^{d-1})}\\[3pt] &\lesssim\int_{0}^{\infty}\int_{\mathbb{R}} \frac{ 2^{(3-n)m} e^{-cx_d 2^{m}}}{\left(1+|2^{2m} (t-s)|^2\right)^2}\left\|\Delta_{<2m}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d\notag \\[3pt] &\lesssim\int_{0}^{\infty} 2^{(1-n)m} e^{-cx_d 2^{m}}\cdot\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{<2m}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}}{1+|2^{2m}s|^2}\,{\rm d}x_d\notag \\[3pt] &\lesssim 2^{-nm}\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{<2m}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}}{1+|2^{2m}s|^2}\notag\\[3pt] &\lesssim 2^{-nm} \cdot M^{\mathcal{T}}M^{\mathcal{T}} \left\|\partial_{x_d}^nv(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}, \end{align}\] which together with 10 yields ?? .
By 47 , 13 , ?? with \(r=1\) and ?? , we achieve \[\begin{align} &\quad\;\sum_{j\geq2m}\|\Delta_{m}^h\Delta_{j}^{\mathcal{T}} g\|_{L^{\infty}(\mathbb{R}^{d-1})}\\[3pt] &\lesssim\sum_{j\geq2m}\int_{0}^{\infty}\int_{\mathbb{R}} \frac{ 2^{(3-n)\frac{j}{2}}e^{-cx_d 2^{\frac{j}{2}}}}{\left(1+|2^j (t-s)|^2\right)^2}\left\|\Delta_{j}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}\,{\rm d}s\,{\rm d}x_d \\[3pt] &\lesssim\sum_{j\geq2m}\int_{0}^{\infty} 2^{(1-n)\frac{j}{2}} e^{-cx_d 2^{\frac{j}{2}}}\cdot\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{j}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,x_d,t-s)\right\|_{L^{\infty}(\mathbb{R}^{d-1})}}{1+|2^js|^2}\,{\rm d}x_d \notag \\[3pt] &\lesssim\sum_{j\geq2m} 2^{-n\frac{j}{2}} \cdot\sup_{s\in \mathbb{R}} \frac{\left\|\Delta_{j}^{\mathcal{T}}\partial_{x_d}^nv(\cdot,t-s)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}}{1+|2^js|^2} \notag \\[3pt] &\lesssim 2^{-nm} \cdot M^{\mathcal{T}}M^{\mathcal{T}} \left\|\partial_{x_d}^nv(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}_+^{d})}. \end{align}\] which together with 10 yields ?? . ◻
Proof of Theorem 3. The first term on the left-hand side of ?? is bounded by its right-hand side by summing the two estimates in Lemma 19. The second term on the left-hand side of ?? is treated similarly using Lemma 23. ◻
We warmly thank Professor Senjo Shimizu and Ting Zhang for fruitful discussions. TT also thanks colleagues at Kyoto University for their hospitality during his visit in June 2025.
Chen was supported in part by National Natural Science Foundation of China under grant [12101556]. The research of both Liang and Tsai was partially supported by Natural Sciences and Engineering Research Council of Canada (NSERC) under grant RGPIN-2023-04534.