Proof of Lemma 1: . We proceed by induction on \(N\). For the base case \(N=2\), the lemma can be verified by a direct computation: \[\mathcal{D}_{b_2}\mathcal{D}_{b_1} w = \left( \frac{d}{dr} + \frac{b_2}{r} \right) \left( w' + \frac{b_1}{r}w \right) = w'' + \frac{b_1+b_2}{r} w' + \frac{b_1(b_2-1)}{r^2} w.\] Thus, given \((a_1, a_2) \in \mathbb{C}^2\), it suffices to choose \(b_1, b_2 \in \mathbb{C}\) satisfying the system \[b_1 + b_2 = a_1, \qquad b_1(b_2 - 1) = a_2.\]

Now, assume the lemma holds for an operator of order \(N-1\). To prove it for order \(N\), we first seek coefficients \((c_1, \ldots, c_N) \in \mathbb{C}^N\) such that \[\begin{align} & w^{(N)} + \frac{a_1}{r}w^{(N-1)} +\cdots + \frac{a_N}{r^N} w \\= &\mathcal{D}_{c_N} \left( w^{(N-1)} + \frac{c_1}{r}w^{(N-2)} + \cdots + \frac{c_{N-1}}{r^{N-1}} w \right) \\ = &\frac{d}{dr}\left( w^{(N-1)} + \frac{c_1}{r}w^{(N-2)} + \cdots + \frac{c_{N-1}}{r^{N-1}} w \right) + \frac{c_N}{r} \left( w^{(N-1)} + \frac{c_1}{r}w^{(N-2)} + \cdots + \frac{c_{N-1}}{r^{N-1}} w \right) . \end{align}\] Matching coefficients on both sides leads to the system \[\label{sys} \begin{cases} c_1 + c_N = a_1, \\ c_{j+1} + c_j(c_N - j) = a_{j+1}, & 1 \leq j \leq N-2, \\ c_{N-1}(c_N - N + 1) = a_N. \end{cases}\tag{4}\]

The first \(N-1\) equations allow us to express \(c_1, \ldots, c_{N-1}\) recursively in terms of \(c_N\). Specifically, they take the form \[\label{sys2}c_{j} = a_{j} - P_{j}(a_1, \ldots, a_{j-1}, c_N), \qquad 1\le j \le N-1,\tag{5}\] where \(P_{j}\) is a polynomial in \(c_N\) of degree \(j\). Substituting the resulting expression for \(c_{N-1}\) into the final equation of 4 yields a single polynomial equation in \(c_N\) of degree \(N\). By the Fundamental Theorem of Algebra, this equation admits at least one solution in \(\mathbb{C}\). For any such \(c_N\), the coefficients \(c_1, \ldots, c_{N-1}\) are uniquely determined by 5 and the operator factors as \[w^{(N)} + \frac{a_1}{r}w^{(N-1)} + \cdots + \frac{a_N}{r^N} w = \mathcal{D}_{c_N} \left( w^{(N-1)} + \frac{c_1}{r}w^{(N-2)} + \cdots + \frac{c_{N-1}}{r^{N-1}} w \right).\]

By the inductive hypothesis, there exist constants \((b_1, \dots, b_{N-1}) \in \mathbb{C}^{N-1}\) such that the \((N-1)\)-th order operator in the parentheses satisfies \[w^{(N-1)} + \frac{c_1}{r}w^{(N-2)} + \cdots + \frac{c_{N-1}}{r^{N-1}} w = \mathcal{D}_{b_{N-1}} \cdots \mathcal{D}_{b_1} w.\]

Setting \(b_N = c_N\), we conclude that \[w^{(N)} + \frac{a_1}{r}w^{(N-1)} + \cdots + \frac{a_N}{r^N} w = \mathcal{D}_{b_N} \mathcal{D}_{b_{N-1}} \cdots \mathcal{D}_{b_1} w,\] which completes the induction and the proof. ◻

Proof of Theorem 11:. Since the differential inequality ?? can be rewritten as \[|w^{(N)}(r) | \leq \tilde{C} \sum_{k=0}^{N-1} \frac{|w^{(k)}(r)|}{r^{N-k}}\] for some \(\tilde{C}\) dependent only on \(C\) and \(b_j, j=1, \ldots, N\), in view of Theorem 9 it suffices to show that \(w\) vanishes to infinite order at the origin. Suppose this is not the case, and let \(p_0 \geq N\) be the smallest integer such that \(w^{(p_0)}(0) \neq (0, \dots, 0).\) It follows that there exists a vector \(c \neq (0, \dots, 0)\) such that \[w(r) = c r^{p_0} + O(r^{p_0+1}) \quad \text{as } \,\, r \to 0^+.\] Since \(w\) is \(C^{\infty}\), as proved in the Appendix to [21], we may formally differentiate this Taylor expansion of each component \(w_i\) consecutively to obtain \[\begin{align} w'(r) & = c p_0 r^{p_0-1} + O(r^{p_0}); \\ \vdots & \\ w^{(N)} (r) & = c\prod_{j=1}^N \left (p_0-(j-1) \right ) \cdot r^{p_0-N} + O( r ^{p_0-N+1}). \end{align}\]

Substituting these expansions into ?? , a direct computation yields \[\label{fti} \begin{align} & \;|c|\prod_{j=1}^N |p_0-(j-1)+b_j| \cdot r^{p_0-N} + O( r ^{p_0-N+1}) \\ \quad \leq&\; C|c| \sum_{j=0}^{N-1} \prod_{k=1}^j \left |p_0-(k-1) \right|\cdot r ^{p_0-N} + O( r ^{p_0-N+1}) \\ \quad \leq & \;C|c|N \prod_{k=1}^{N-1} \left |p_0-(k-1) \right|\cdot r ^{p_0-N} + O( r ^{p_0-N+1}) \end{align}\tag{8}\] as \(r \to 0^+.\) In view of 7 , we can divide both sides of 8 by \(|c|\prod_{j=1}^N |p_0-(j-1)+b_j| \cdot r^{p_0-N}\). Applying the definition of \(M_b\) in 6 , we get \[1 < CNM_b + O( r )\quad \text{as} \,\, r \to 0^+ .\] Taking the limit as \(r \to 0^+\), we obtain \(1 \leq CNM_b\). This contradicts the assumption ?? , thereby proving that \(w^{(p)}(0)=0\) for all \(p \geq 0\). ◻

Proof of Theorem 10: . Since \(h\in C([0, r_0))\) and \(h(0) =0\), there exists some positive \(r_1<r_0\) such that \[h(r)< \frac{1}{NM_b}, \;\; \quad r \in (0, r_1).\] Thus \(w\) satisfies ?? on \((0, r_1)\). Applying Theorem 11 yields \(w\equiv 0\) on \([0, r_1)\). When \(r\ge r_1\), since the weights \(r^{-(N-j)}\) are bounded on the interval \([r_1, r_0)\), Corollary 1.3 in [20] further ensures that \(w \equiv 0\) on the entire domain \([0, r_0)\). ◻

Proof. We first observe that \(w\) satisfies \[\left|w^{(N)} + \frac{h_1(0)}{r}w^{(N-1)} + \cdots + \frac{h_N(0)}{r^N} w \right|\le \sum_{j=0}^{N-1}|h_{N-j}(r)-h_{N-j}(0)|\cdot \frac{| w^{(j)}(r) |}{r^{N-j}}, \quad r \in (0, r_0).\] Since \(h_j\in C([0, r_0)), j =1, \ldots, N\), there exists some function \(h\in C([0, r_0))\) with \(h(0)=0\), such that for all \(j =0, \ldots, N-1\), \[|h_{N-j}(r)-h_{N-j}(0)|\le h(r), \quad r \in (0, r_0).\] On the other hand, let \((b_1, \ldots, b_N)\in P_{h_1(0), \ldots, h_N(0)} \cap B_1\). Then \((b_1, \ldots, b_N)\) satisfies ?? , and \[|\mathcal{D}_{b_N} \cdots \mathcal{D}_{b_1} w(r) |=\left| w^{(N)} + \frac{h_1(0)}{r}w^{(N-1)} + \cdots + \frac{h_N(0)}{r^N} w\right|\le h(r)\sum_{j=0}^{N-1} \frac{| w^{(j)}(r) |}{r^{N-j}}.\] By Theorem 10, \(w\equiv 0\). ◻

Proof. We begin by considering the case \(N=1\), where \(w\) satisfies the differential inequality \[\left| \mathcal{D}_{b_1}w(r) \right| \le C \frac{|w(r)|}{r}.\] Let \(g := \mathcal{D}_{b_1} w\). By definition, we have \[\label{g1} g(r) = r^{-b_1} \frac{d}{dr} \left( r^{b_1} w(r) \right) = w'(r)+ \frac{b_1}{r} w(r).\tag{11}\] In particular, \(g\in C((0, r_0))\) and the original inequality becomes \[\label{n1} |g(r)| \le \frac{C|w(r)|}{r}.\tag{12}\]

Given that \(w(0)=0\) and \(w \in C^1([0, r_0))\), applying L’Hôpital’s rule and setting \(g(0) = (1+b_1)w'(0)\) extends \(g\) continuously to the origin. Consequently, the maximum function \[M(r) := \max_{s \in [0, r]} |g(s)|\] is well-defined and finite, and monotone non-decreasing for all \(r \in [0, r_0)\).

The condition \(\mathrm{Re}(b_1) > -1\) ensures that \[\lim_{r \to 0^+} r^{b_1} w(r) = 0.\] In fact this follows immediately from the continuity of \(w\) if \(\mathrm{Re}(b_1) \ge 0\), or from L’Hôpital’s rule if \(-1 < \mathrm{Re}(b_1) < 0\). Multiplying both sides of 11 by \(r^{b_1}\) and integrating it from \(0\) to \(r\) then yields \[w(r) = r^{-b_1} \int_0^r s^{b_1} g(s) \, ds.\]

Using the monotonicity of \(M(r)\) and the fact that \(\mathrm{Re}(b_1) > -1\), one gets \[\begin{align} \label{w11} |w(r)| &\le r^{-\mathrm{Re}(b_1)} \int_0^r s^{\mathrm{Re}(b_1)} |g(s)| \, ds \le M(r) r^{-\mathrm{Re}(b_1)} \int_0^r s^{\mathrm{Re}(b_1)} \, ds = \frac{r M(r)}{\mathrm{Re}(b_1) + 1}. \end{align}\tag{13}\]

Substituting this estimate back into 12 , we obtain \[|g(r)| \le \frac{C M(r)}{ \mathrm{Re}(b_1) + 1 } , \quad r \in [0, r_0).\] Now, for each \(r_1<r_0\), take the maximum on both sides over \([0, r_1]\) and make use of the monotonicity of \(M(r)\) to get \[M(r_1) \le \frac{C M(r_1)}{ \mathrm{Re}(b_1) + 1 }.\] If \[\label{C1} C<\mathrm{Re}(b_1) + 1,\tag{14}\] this would be impossible unless \(M(r_1)\equiv 0\). It follows from 13 and the arbitrariness of \(r_1<r_0\) that \(w \equiv 0\) on \([0, r_0)\).

In the case \(N=2\), the inequality is given by \[\left|\mathcal{D}_{b_2} \mathcal{D}_{b_1} w(r)\right|\le C\left(\frac{|w(r)|}{r^{2}}+\frac{|w'(r)|}{r}\right).\]

Let \(g: = \mathcal{D}_{b_2} \mathcal{D}_{b_1} w\). The inequality then becomes \[\label{eqn:m61295first95integration} \left | g(r) \right | \leq C\left(\frac{|w(r)|}{r^{2}}+\frac{|w'(r)|}{r}\right).\tag{15}\] By the definition of the operators, \(g(r)\) can be expressed as \[\label{g2} g(r) = r^{-b_2}\frac{d}{dr}\left ( r^{b_2-b_1} \frac{d}{dr}\left(r^{b_1} w(r) \right) \right ),\;\; r\in (0, r_0).\tag{16}\] Given \(w(0) = w'(0) = 0\) and \(w \in C^2([0, r_0))\), \(g\) admits a continuous extension to the origin, so \(g \in C([0, r_0))\).

Under the conditions \(\mathrm{Re}(b_1) > -2\) and \(\mathrm{Re}(b_2) > -1\), it follows that \[\lim_{r\to 0^+} r^{b_1} w(r)=0\] and \[\lim_{r\to 0^+} r^{b_2-b_1} \frac{d}{dr}\left(r^{b_1} w(r) \right) =\lim_{r\to 0^+} r^{b_2}\left( b_1\frac{w(r)}{r}+w'(r)\right)= 0.\]

Thus, we can multiply both sides of 16 by \(r^{b_2}\) and perform successive integration from \(0\) to \(r\) to obtain \[\begin{align} w(r) &= r^{-b_1} \int_0^r s_2^{b_1-b_2} \int_0^{s_2} s_1^{b_2} g(s_1) ds_1 ds_2. \end{align}\] Differentiating the above further gives \[\begin{align} w'(r) = -b_1 r^{-b_1-1}\int_0^rs_2^{b_1-b_2}\int_0^{s_2} s_1^{b_2} g(s_1) \,ds_1 \, ds_2 + r^{-b_2}\int_0^r s_1^{b_2} g(s_1) \,ds_1. \end{align}\]

Let \[M(r) := \max_{s \in [0, r]} |g(s)|.\] Using the monotonicity of \(M(r)\), we estimate \(|w(r)|\): \[\begin{align} \label{wb} |w(r)| & \le M(r) r^{-{\mathrm{Re}(b_1)}}\int_0^rs_2^{{\mathrm{Re}(b_1)}-{\mathrm{Re}(b_2)}}\int_0^{s_2} s_1^{{\mathrm{Re}(b_2)}} \,ds_1 \, ds_2 \nonumber \\ & = \frac{ r^2M(r)}{\left ({\mathrm{Re}(b_2)}+1 \right )({\mathrm{Re}(b_1)}+2)}. \end{align}\tag{17}\] Similarly, for the derivative \(|w'(r)|\): \[\begin{align} |w'(r)| & \le \frac{|b_1|M(r)}{{\mathrm{Re}(b_2)}+1}r^{-{\mathrm{Re}(b_1)}-1}\int_0^r s_2^{\mathrm{Re}(b_1)+1} \, ds_2 + \frac{rM(r)}{ {\mathrm{Re}(b_2)}+1 } \nonumber \\ & = \left(\frac{|b_1|}{({\mathrm{Re}(b_2)}+1)({\mathrm{Re}(b_1)}+2)} + \frac{1}{ {\mathrm{Re}(b_2)}+1 }\right)rM(r). \end{align}\]

Substituting these estimates into 15 , the inequality reduces to: \[\begin{align} |g(r)| \le C \left(\frac{|b_1|+3 + {\mathrm{Re}(b_1)}}{({\mathrm{Re}(b_2)}+1)({\mathrm{Re}(b_1)}+2)} \right) M(r), \quad \;r\in(0, r_0). \end{align}\] As before, for each \(r_1<r_0\), taking the maximum over \([0, r_1]\) yields \[M(r_1)\le C \left(\frac{|b_1|+3 + {\mathrm{Re}(b_1)}}{({\mathrm{Re}(b_2)}+1)({\mathrm{Re}(b_1)}+2)} \right) M(r_1).\] If \[\label{C2} C< \frac{({\mathrm{Re}(b_2)}+1)({\mathrm{Re}(b_1)}+2)}{|b_1|+3 + {\mathrm{Re}(b_1)}} ,\tag{18}\] then \(M(r_1) = 0\), which in turn forces \(g \equiv 0\) on \([0, r_1)\). Consequently, by 17 we have \(w \equiv 0\) on \([0, r_1]\).

For the general case \(N \geq 3\), define \[\begin{align} \label{eqn:gN} g(r) &:= \mathcal{D}_{b_N} \cdots \mathcal{D}_{b_1} w(r) \nonumber \\ &= r^{-b_N} \frac{d}{dr} \left ( r^{b_N-b_{N-1}} \frac{d}{dr} \left( r^{b_{N-1}-b_{N-2}} \frac{d}{dr} \left (\cdots r^{b_2-b_1} \frac{d}{dr}\left(r^{b_1} w(r) \right) \right )\cdots \right) \right). \end{align}\tag{19}\]

The inequality is given by \[\label{wn} \left| g(r) \right| \le C\sum_{j=0}^{N-1}\frac{|w^{(j)}(r)|}{r^{N -j}}.\tag{20}\]

Given the vanishing conditions \(w^{(k)}(0) = 0\) for \(k = 0, \dots, N-1\) and the constraints ?? on \(b_j\), a recursive application of the L’Hôpital’s rule ensures that the following limits vanish at the origin: \[\lim_{r \to 0^+} r^{b_{j}-b_{j-1}} \frac{d}{dr} \left ( r^{b_{j-1}-b_{j-2}}\frac{d}{dr} \left (\cdots r^{b_2-b_1} \frac{d}{dr}\left(r^{b_1} w(r) \right) \right )\cdots \right ) =0 \quad \text{for} \,\, j=1, \dots, N.\] Thus performing \(N\) successive integration on 19 yields the integral representation of \(w(r)\): \[\begin{align} \label{we} w(r) = r^{-b_1} \int_0^r s_N^{b_{1}-b_{2}} \int_0^{s_{N}} s_{N-1}^{b_{2}-b_{3}} \cdots \int_0^{s_3}s_2^{b_{N-1}-b_N}\int_0^{s_2} s_1^{b_N}g(s_1)\, ds_1 \,ds_2 \cdots ds_{N}. \end{align}\tag{21}\] Differentiating this expression inductively, the \(j\)-th order derivative satisfies \[\label{wj} \begin{align} w^{(j)}(r) = &r^{-b_{j+1}} \int_0^{r} s_{N-j}^{\,b_{j+1}-b_{j+2}} \int_0^{s_{N-j}} s_{N-j-1}^{\,b_{j+2}-b_{j+3}} \cdots \int_0^{s_2} s_1^{b_N} g(s_1)\, ds_1\, ds_2 \cdots ds_{N-j} \\ &+ \sum_{m=0}^{j-1} \frac{C_{j,m}}{r^{\,j-m}}\, w^{(m)}(r),\;\;j=1, \ldots, N-1, \end{align}\tag{22}\] where the constants \(C_{j,m}\) are dependent only on \((b_1, \ldots, b_N)\).

Let \[M(r) := \max_{s \in [0, r]} |g(s)|.\] Utilizing the monotonicity of \(M(r)\), we obtain from 21 that \[\begin{align} \label{es} |w(r)| \leq & M(r) r^{-\mathrm{Re}(b_1)} \int_0^r s_N^{\mathrm{Re}(b_{1})-\mathrm{Re}(b_{2})} \cdots \int_0^{s_3}s_2^{\mathrm{Re}(b_{N-1})-\mathrm{Re}(b_N)}\int_0^{s_2} s_1^{\mathrm{Re}(b_N)}\, ds_1 \,ds_2 \cdots ds_{N} \nonumber \\ = &\frac{M(r)}{\mathrm{Re}(b_N)+1} r^{-\mathrm{Re}(b_1)} \int_0^r s_N^{-\mathrm{Re}(b_{1})-\mathrm{Re}(b_{2})} \cdots \int_0^{s_3}s_2^{\mathrm{Re}(b_{N-1})-\mathrm{Re}(b_N)+\mathrm{Re}(b_N)+1} \,ds_2 \cdots ds_{N} \nonumber \\ & \vdots \nonumber \\ =& \frac{r^NM(r)}{\prod_{j=1}^N\left(\mathrm{Re}(b_j)+N-j+1\right)}. \end{align}\tag{23}\]

Similarly, through induction on \(j\), it follows that there exist constants \(C_j > 0\) dependent only on \((b_1, \ldots, b_N)\) such that \[\label{wjn} |w^{(j)}(r)|\le C_j r^{N-j}M(r), \quad j=0, \ldots, N-1 .\tag{24}\]

Substituting the estimates 24 into the right-hand side of 20 , the inequality reduces to \[|g(r)|\leq C \left (\sum_{j=0}^{N-1}C_j \right ) M(r).\] Finally, taking the maximum over \([0, r_1]\) for each \(r_1<r_0\), we find \[M(r_1) \leq C \left (\sum_{j=0}^{N-1}C_j \right ) M(r_1).\] If \[\label{CN} C <\frac{1}{ \sum_{j=0}^{N-1}C_j },\tag{25}\] then \(M(r_1) = 0\) necessarily. Consequently, \(w \equiv 0\) on the interval \([0, r_1]\). The proof is complete. ◻

Proof of Theorem 13: . The proof follows from Theorem 14 by an argument similar to that used in Theorem 10, and is therefore omitted. ◻

Proof. For each \(\lambda\in A\), \(r^\lambda\) satisfies the initial conditions. To determine if \(r^\lambda\) is a solution, we substitute \(u = r^\lambda\) into the homogeneous equation. A straightforward computation yields \[\mathcal{D}_{b_N} \cdots \mathcal{D}_{b_1} (r^\lambda) = \left[ \prod_{j=1}^N (\lambda - (j-1) + b_j) \right] r^{\lambda-N}.\]

Consequently, \(r^\lambda\) satisfies the homogeneous equation if and only if \[\prod_{j=1}^N (\lambda - (j-1) + b_j) = 0,\] i.e., if and only if \(\lambda \in \{j - 1 - b_j : j = 1, \ldots, N\}\). This completes the proof. ◻

Proof. As seen in the proof of Theorem 13, the admissibility condition ?? ensures that the iterated integral in the definition of \(\mathcal{T}\) is well-defined. Furthermore, \(\mathcal{T}g \in C^N((0, r_1))\) and satisfies the system ?? . The bound ?? follows directly from 24 . Consequently, it remains only to show that \((\mathcal{T}g)^{(N)}\) is continuous at \(0\) and satisfies ?? .

Recall that 22 with \(j=N-1\) yields \[\label{tgn-1} (\mathcal{T} g)^{(N-1)}(r) = r^{-b_{N}} \int_0^{r} s_1^{b_N} g(s_1)\, ds_1 + \sum_{m=0}^{N-2} \frac{C_{N-1,m}}{r^{\,N-1-m}}\, (\mathcal{T} g)^{(m)}(r).\tag{28}\] Before proceeding to the \(N\)-th derivative of \(\mathcal{T}g\), we first claim that \[\label{cla} \lim_{r\rightarrow 0^+} \frac{(\mathcal{T} g)^{(m)}(r)}{{r^{\,N-m}}} = a_m g(0), \;\;m=0, \ldots, N-1,\tag{29}\] where \[\label{am} a_m: = \frac{(\mathcal{T} 1)^{(m)}(r)}{r^{N-m}} = \frac{N!}{(N-m)!\cdot \prod_{j=1}^N\left( b_j +N-j+1\right)}.\tag{30}\] Indeed, to see this, we first estimate \(\mathcal{T}\left(g-g(0)\right)\) by replacing \(g\) in the expression of 21 by \(g-g(0)\). The estimate 23 yields \[\frac{|\mathcal{T}\left(g-g(0)\right)(r)|}{r^N}\le \frac{\max_{s\in [0,r]}|g(s)-g(0)|}{\prod_{j=1}^N\left(\mathrm{Re}(b_j)+N-j+1\right)}.\] Since \(g\) is continuous at \(r=0\), the right-hand side goes to zero as \(r\rightarrow 0^+\). Hence \[\lim_{r\rightarrow 0^+}\frac{\mathcal{T} g(r)}{r^N} =\frac{ \mathcal{T}(g(0))}{r^N} = g(0)\cdot \frac{\mathcal{T}1(r) }{r^N}= \frac{g(0)}{\prod_{j=1}^N\left( b_j +N-j+1\right)} =a_0g(0).\] A similar argument as above can further show that \[\frac{|\left(\mathcal{T}\left(g-g(0)\right)\right)^{(m)}(r)|}{r^{N-m}} \rightarrow 0, \;\;m=0, \ldots, N-1\] as \(r\rightarrow 0^+\) as well. Thus \[\lim_{r\rightarrow 0^+} \frac{(\mathcal{T} g)^{(m)}(r)}{{r^{\,N-m}}} = \frac{(\mathcal{T} (g(0))(r))^{(m)}(r)}{{r^{\,N-m}}} = g(0) \cdot \frac{(\mathcal{T} 1(r))^{(m)}(r)}{{r^{\,N-m}}} = a_m g(0).\] The claim 29 is proved.

We now verify the continuity of the \(N\)-th derivative of \(\mathcal{T}g\) at \(0\). Since \((\mathcal{T}g )^{(N-1)}(0)=0,\) we use the definition of derivative along with 28 , L’Hôpital’s Rule, and the limit established in 29 to obtain \[\begin{align} \label{22} (\mathcal{T}g)^{(N)}(0) & = \lim_{r\to 0^+}\frac{(\mathcal{T}g)^{(N-1)}(r) - (\mathcal{T}g)^{(N-1)}(0)}{r} \nonumber \\ & = \lim_{r\to 0^+} \left (r^{-b_{N}-1} \int_0^{r} s_1^{b_N} g(s_1)\, ds_1 + \sum_{m=0}^{N-2} \frac{C_{N-1,m}}{r^{\,N-m}}\, (\mathcal{T} g)^{(m)}(r) \right )\nonumber \\ & = \frac{g(0)}{b_N+1} + \sum_{m=0}^{N-2}C_{N-1,m}a_mg(0). \end{align}\tag{31}\]

On the other hand, for \(r\in (0, r_0)\), differentiating 28 with respect to \(r\) yields \[\begin{align} \label{tN} (\mathcal{T} g)^{(N)}(r) & = g(r) -\frac{b_N}{r} \left( r^{-b_N} \int_0^r s_1^{b_N} g(s_1) \, ds_1 \right) \nonumber \\ & \quad + \sum_{m=0}^{N-2} \left (\frac{-(N-1-m) C_{N-1,m}}{r^{N-m}} (\mathcal{T} g)^{(m)}(r) + \frac{C_{N-1,m}}{r^{N-1-m}} (\mathcal{T} g)^{(m+1)}(r) \right ) \nonumber \\ & = g(r) -\frac{b_N}{r} \left( (\mathcal{T} g)^{(N-1)}(r) - \sum_{m=0}^{N-2} \frac{C_{N-1,m}}{r^{N-1-m}} (\mathcal{T} g)^{(m)}(r) \right) \nonumber \\ & \quad + \sum_{m=0}^{N-2} \frac{-(N-1-m) C_{N-1,m}}{r^{N-m}} (\mathcal{T} g)^{(m)}(r) + \sum_{m=1}^{N-1}\frac{C_{N-1,m-1}}{r^{N-m}} (\mathcal{T} g)^{(m)}(r) \nonumber \\ & = g(r) +\sum_{m=0}^{N-1}C_{N,m} \frac{ (\mathcal{T} g)^{(m)}(r)}{r^{\,N-m}}, \end{align}\tag{32}\] where the coefficients \(\{C_{N,m}\}_{0 \leq m \leq N-1}\) are determined by \(\{C_{N-1,m}\}_{0 \leq m \leq N-2}\), via the following relations: \[\begin{align} \label{cm} \begin{aligned} &C_{N,0}=\big (b_N-(N-1) \big )C_{N-1,0},\\ &C_{N,m}= \big (b_N-(N-1-m) \big )C_{N-1,m}+C_{N-1,m-1},\quad 1\le m\le N-2,\\ & C_{N,N-1}= C_{N-1,N-2}-b_N. \end{aligned} \end{align}\tag{33}\]

Taking the limit as \(r \to 0^+\) in 32 and applying the limit established in 29 , we find \[\begin{align} \label{11} \lim_{r\to 0^+}(\mathcal{T} g)^{(N)}(r) & = g(0) + \sum_{m=0}^{N-1}C_{N,m}\lim_{r\to 0^+}\frac{ (\mathcal{T} g)^{(m)}(r)}{r^{\,N-m}} \nonumber \\ & = g(0) + \sum_{m=0}^{N-1}C_{N,m}a_mg(0). \end{align}\tag{34}\]

In view of 34 and 31 , the continuity of \((\mathcal{T} g)^{(N)}\) at \(0\) is therefore equivalent to the identity \[\frac{1}{b_N+1}+\sum_{m=0}^{N-2} C_{N-1,m} a_m = 1+\sum_{m=0}^{N-1} C_{N,m} a_m .\] By substituting the recurrence relations 33 for \(C_{N,m}\) and using the relation \[a_{m+1} = (N-m) a_m\] from 30 , we observe that \[\begin{align} \sum_{m=0}^{N-1} C_{N,m} a_m =&\sum_{m=1}^{N-2} \big((b_N-(N-1-m))C_{N-1,m}+C_{N-1,m-1}\big)a_m \\ &+\;(b_N-(N-1))C_{N-1,0}a_0 +(C_{N-1,N-2}-b_N)a_{N-1}\\ =& \sum_{m=0}^{N-2} C_{N-1,m}\bigg(\big(b_N-(N-1-m)\big)a_m+a_{m+1}\bigg) - b_N a_{N-1} \\ =& (b_N+1)\sum_{m=0}^{N-2} C_{N-1,m} a_m - b_N a_{N-1}. \end{align}\]

Substituting this back into our equivalence condition, we see that the continuity holds if \[\frac{1}{b_N+1}+\sum_{m=0}^{N-2} C_{N-1,m} a_m =1+ (b_N+1)\sum_{m=0}^{N-2} C_{N-1,m} a_m - b_N a_{N-1},\] which simplifies to \[\begin{align} \label{eq} 0 & =\frac{b_N}{b_N+1} + b_N\sum_{m=0}^{N-2}C_{N-1,m}a_m - b_N a_{N-1} \nonumber \\ & = b_N\!\left( \frac{1}{b_N+1} + \sum_{m=0}^{N-2}C_{N-1,m}a_m-a_{N-1}\right). \end{align}\tag{35}\]

On the other hand, from 28 and the definition of \(a_m\) in 30 it follows immediately that \[\begin{align} a_{N-1} &= \frac{(\mathcal{T} 1)^{(N-1)}(r)}{r} = \frac{r^{-b_N} \int_0^r s^{b_N} \, ds + \sum_{m=0}^{N-2} \frac{C_{N-1,m}}{r^{N-1-m}} (\mathcal{T} 1)^{(m)}(r)}{r} \\ &= \frac{1}{b_N+1} + \sum_{m=0}^{N-2} C_{N-1,m} \frac{(\mathcal{T} 1)^{(m)}(r)}{r^{N-m}} = \frac{1}{b_N+1} + \sum_{m=0}^{N-2} C_{N-1,m} a_m. \end{align}\]

Therefore, 35 holds and we thus conclude that \((\mathcal{T} g)^{(N)}\in C([0, r_1])\). Finally, the estimate ?? follows from 32 and 24 . ◻

Proof of Theorem 18: . Let the operator \(\mathcal{T}\) and the constant \(C\) be given as in Lemma 3, satisfying ?? and ?? . Define \[\label{M} M: = \sup_{\substack{r\in [0, 1], |p_j|\leq 1, \\\\ j=0, \ldots, N-1 }}|f(r, p_0, \ldots, p_{N-1})|,\tag{36}\] and choose \(0 < {r_1} < \min\{1, r_0\}\) such that \[MC{r_1} < 1.\]

It suffices to show that there exists a \(C^N([0, {r_1}])\) solution to the system 27 on \([0,{r_1})\). Indeed, for \({r_1}<r <r_0\), the system is non-singular. Thus any local solution \(u\) extends uniquely to a \(C^N([0, r_0))\) solution by the standard continuation theorem for initial value problems of ordinary differential equations.

We construct a sequence of functions \((u_l)_{l \ge 1}\) inductively as follows. For each \(l \in \mathbb{Z}^+\), define\[u_l(r) := 0, \quad r \in \left[0, \frac{{r_1}}{l}\right],\]and for each \(j = 2, \ldots, l\), set \[\label{lim} u_l(r) := \mathcal{T}\left( f(\cdot, u_l, \ldots, u_l^{(N-1)})\right)\left(r - \frac{{r_1}}{l}\right), \quad r \in \left(\frac{(j-1){r_1}}{l}, \frac{j{r_1}}{l}\right].\tag{37}\] From ?? , it is straightforward to verify that \(u_l \in C^{N-1}([0, {r_1}])\).

We now estimate the \(C^{N-1}\) norm of \(u_l\). Clearly, \[\|u_l\|_{C^{N-1}([0, \frac{{r_1}}{l}])} = 0 < 1.\] Proceeding inductively for each \(j = 2, \ldots, l\), we apply ?? to obtain \[\|u_l\|_{C^{N-1}([\frac{{r_1}}{l} , \frac{j{r_1}}{l}])} \le C {r_1}\sup_{r\in [0, \frac{(j-1){r_1}}{l}]}|f(\cdot, u_l, \ldots, u_l^{(N-1)})|\le MC{r_1}<1.\] Combining these estimates, we find that \[\|u_l\|_{C^{N-1}([0 , {r_1}])}<1.\] Thus, the sequence \((u_l)\) is uniformly bounded in \(C^{N-1}([0, {r_1}])\).

Furthermore, by construction, \(u_l^{(N)}\) is continuous on \([0, {r_1}]\) except possibly at \(r = \frac{{r_1}}{l}\). By ?? , we have \[\|u_l^{(N)}\|_{L^\infty([0, {r_1}])}\le C \sup_{r\in [0, {r_1}]}|f(\cdot, u_l, \ldots, u_l^{(N-1)})|\le C M.\] The equicontinuity of \((u_l)\) and its derivatives up to order \(N-1\) follows from the inequality \[\begin{align} |u^{(j)}_l(t_1) - u^{(j)}_l(t_2)| &\le \max\{\|u_l\|_{C^{N-1}([0, {r_1}])}, \|u_l^{(N)}\|_{L^\infty([0, {r_1}])}\} |t_1 - t_2| \\ &= \max\{1, CM\} \cdot |t_1 - t_2|, \quad j = 0, \ldots, N-1. \end{align}\]

Therefore, by the Arzelà-Ascoli theorem, there exists a subsequence, still denoted by \((u_l)\), such that \(u_l \to u\) uniformly in \(C^{N-1}([0, {r_1}])\) norm. Taking the limit as \(l \to \infty\) in 37 and invoking the continuity of \(f\), we obtain \[u = \mathcal{T}\left( f(\cdot, u, \ldots, u^{(N-1)})\right).\]

Since \(f(\cdot, u, \ldots, u^{(N-1)})\in C([0,{r_1}])\), Lemma 3 implies that \(u\in C^N([0, {r_1}])\). In addition, by ?? , \(u\) satisfies the initial value problem 27 on \([0, {r_1}]\). The proof is complete. ◻

Proof. Let \(r_1\in (0, r_0)\) be a sufficiently small number to be determined later. As in the proof of Theorem 18, we only need to show the existence of \(C^N\) solutions on \([0, r_1]\). Let \(X\) be the closed subset of \(C^{N}([0,r_1])\) defined by \[X := \left \{ u \in C^{N}([0,r_1]) : u^{(j)}(0)=0 \text{ for } j=0,\dots,N-1, \;\text{and}\;\|u\|_{C^{N-1}([0, r_1])}\le 1 \right \}.\] Then \(X\) is a complete metric space with respect to the induced metric \[d(u, v): = \|u-v\|_{C^{N-1}([0, r_1])} \;\;\text{for}\;\; u, v\in X.\] We shall establish the existence of \(C^N\) solutions by applying the contraction mapping principle (see [22]) to a suitably defined operator on \(X\).

Given \(u\in X\), define \[\Phi(u):=\mathcal{T}\big(f(\,\cdot\,,u,\ldots,u^{(N-1)})\big).\] By Lemma 3 and the fact that \(f(\cdot, u, \dots, u^{(N-1)}) \in C([0, r_1])\), we know \(\Phi(u) \in C^{N}([0,r_1])\) with \(u^{(j)}(0)=0\) for \(j=0,\dots,N-1\). Moreover, by ?? , \[\|\mathcal{T}\big(f(\,\cdot\,,u,\ldots,u^{(N-1)})\big)\|_{C^{N-1}([0, r_1])}\le Cr_1\| f(\,\cdot\,,u,\ldots,u^{(N-1)}) \|_{C([0, r_1])}\le Cr_1M,\] where \(M\) is defined in 36 . Thus, by choosing \(r_1\) such that \(Cr_1M\le 1\), one has \(\Phi\) maps \(X\) into itself.

Furthermore, since \(f\) is locally Lipschitz continuous with respect to \((u, \dots, u^{(N-1)})\), for any \(u, v \in X\), the functions \(f(\cdot, u, \dots, u^{(N-1)})\) and \(f(\cdot, v, \dots, v^{(N-1)})\) belong to \(C([0, r_1])\) and satisfy the estimate \[\|f(\cdot,u,\ldots,u^{(N-1)})-f(\cdot,v,\ldots,v^{(N-1)})\|_{C([0,r_1])} \le L \|u-v\|_{C^{N-1}([0, r_1])},\] where \(L>0\) is the Lipschitz constant of \(f(r, p_0, \ldots, p_{N-1})\) on the compact set \(\{r\in [0, 1], |p_j|\leq 1, j=0, \ldots, N-1 \}\). Consequently, by the estimate in ?? , we have \[\begin{align} d(\Phi(u), \Phi(v)) &= \|\Phi(u)-\Phi(v) \|_{C^{N-1}([0, r_1])}\\ & \le C\, r_1\, \|f(\cdot,u,\ldots,u^{(N-1)})-f(\cdot,v,\ldots,v^{(N-1)})\|_{C([0,r_1])} \\ & \le C L \, r_1 \|u-v\|_{C^{N-1}([0, r_1])} =(CLr_1) d(u, v). \end{align}\] By further shrinking \(r_1\) such that \(C L r_1 < 1\), the mapping \(\Phi\) becomes a contraction on the complete metric space \(X\).

According to the contraction mapping principle, \(\Phi\) possesses a unique fixed point \(u \in X \subset C^N([0, r_1])\). By construction, this fixed point satisfies the homogeneous initial conditions and solves the differential system \[\mathcal{D}_{b_N}\cdots \mathcal{D}_{b_1}u =\mathcal{D}_{b_N}\cdots \mathcal{D}_{b_1}\left( \Phi( u) \right)= f(r,u,\ldots,u^{(N-1)}) \quad\text{on }(0,r_1).\] ◻

Proof of Theorem 16: . First, by Lemma 2, the function \[h: = \sum_{j\in \Lambda} \frac{\zeta_j}{j!}r^j\] satisfies \[\left\{ \begin{align} & \mathcal{D}_{b_N} \cdots\mathcal{D}_{b_1} u = 0, \;\;r\in (0, r_0)\\ & u^{(j)}(0) = \zeta_j, \;\;j\in \Lambda;\\ & u^{(j)}(0) = 0, \;\;j\in A\setminus \Lambda. \end{align}\right.\]

We seek a solution to ?? of the form \(u={\tilde{u}}+h\). Substituting this into the equation yields the following initial value problem \[\left\{ \begin{align} & \mathcal{D}_{b_N} \cdots\mathcal{D}_{b_1}{\tilde{u}}(r) = \tilde{f} (r, {\tilde{u}}, \ldots, {\tilde{u}}^{(N-1)}), \quad r\in (0, r_0) \\ & \tilde{u}(0) = \tilde{u}'(0) = \cdots = \tilde{u}^{(N-1)}(0), \end{align} \right.\] where the nonlinearity is defined as \[\tilde{f}(r,{\tilde{u}}, \ldots, {\tilde{u}}^{(N-1)}): = f(r, {\tilde{u}}+h, \ldots, ({\tilde{u}}+h) ^{(N-1)}).\] According to the existence result in Theorem 18, this homogeneous initial value problem admits a solution \(\tilde{u} \in C^N([0, r_0))\). Consequently, \(u = \tilde{u} + h\) is a \(C^N\) solution to ?? .

Next, we establish uniqueness. Suppose \(u\) and \(v\) are both \(C^{N}\) solutions to the system ?? . Then \(w: = u - v\) and its derivatives up to order \(N-1\) vanish at the origin. Moreover, \(w\) satisfies \[\begin{align} \mathcal{D}_{b_N} \cdots\mathcal{D}_{b_1} {w} & = {f}(r, { {u}, \dots, {u}^{(N-1)}} ) - {f}(r, { {v}, \dots, {v}^{(N-1)}} ) . \end{align}\] Taking the absolute value of both sides and applying the Lipschitz continuity of \(f\), we obtain that for any \(r_1 < r_0\), there exists a constant \(L> 0\) such that \[| \mathcal{D}_{b_N} \cdots\mathcal{D}_{b_1} w (r)| \leq L \sum_{j=0}^{N-1} \left | w^{(j)} (r)\right | \;\;\text{for}\;\;r \in (0, r_1).\] It follows from Theorem 13 with \(h(r)=Lr\) that \(w \equiv 0\) on the interval \([0, r_1]\). Since \(r_1\) was chosen arbitrarily, we conclude that \(u \equiv v\) on \([0, r_0)\). This completes the proof. ◻

Proof. Since \[g(\nabla_{X_i} X_j, X_l) = \sum_{k=0}^{n-1} \omega_{ij}^k \tilde{g}_{kl},\] multiplying both sides by the inverse matrix \((\tilde{g}^{kl})_{n \times n}\) yields \[\omega_{ij}^k = \sum_{l=0}^{n-1} \tilde{g}^{kl} g(\nabla_{X_i} X_j, X_l).\] Due to the uniform boundedness of \((\tilde{g}^{kl})_{n \times n}\), it suffices to show that the terms \(g(\nabla_{X_i} X_j, X_l)\) are all \(O(r^{-1})\). By the Koszul formula, \[\begin{align} 2 g(\nabla_{X_i} X_j, X_l) &= \underbrace{X_i \left(g(X_j, X_l) \right) + X_j \left( g(X_i, X_l) \right) - X_l \left( g(X_i, X_j) \right)}_{\text{part I}} \\ &\quad + \underbrace{g([X_i, X_j], X_l) - g([X_i, X_l], X_j) - g([X_j, X_l], X_i)}_{\text{part II}}. \end{align}\]

We first analyze the derivative terms in part I. Since \(\gamma\) is smooth at \(r=0\), \(\partial_r \gamma_{ab}\) and \(\partial_{\theta^c} \gamma_{ab}\) (where \(c=1, \dots, n-1\)) are all bounded. Consequently, the radial derivatives \(X_0 \left( g(X_a, X_b) \right) = \partial_r \gamma_{ab}\) are \(O(1)\), and the angular derivatives \(X_c \left( g(X_a, X_b) \right) = r^{-1} \partial_{\theta^c} \gamma_{ab}\) are \(O(r^{-1})\). Thus, all terms in part I are at most \(O(r^{-1})\).

For terms in part II, the only non-vanishing commutators for the frame elements are \[[X_0, X_a] = \partial_r(r^{-1}\partial_{\theta^a}) - r^{-1}\partial_{\theta^a}\partial_r = -r^{-2}\partial_{\theta^a} = -r^{-1}X_a,\] while \([X_a, X_b] = 0\) for all \(a,b=1, \dots, n-1\). Therefore, all bracket terms are either \(0\) or of the form \(g(\pm r^{-1} X_a, X_l)\). Given that \(g(X_i, X_j)\) is bounded, those terms are all \(O(r^{-1})\). Hence, all terms in part II are \(O(r^{-1})\).

Combining the two parts, we have \(g(\nabla_{X_i} X_j, X_l) = O(r^{-1})\). This completes the proof. ◻

Proof. Since the frame \(\{X_0,\dots,X_{n-1}\}\) is uniformly equivalent to an orthonormal frame, there exists \(C>0\) such that \[|\nabla_g^k f| \le C \sum_{i_1,\dots,i_k=0}^{n-1} \big| (\nabla_g^k f)(X_{i_1},\dots,X_{i_k}) \big|.\] Thus it suffices to estimate the components \[T_I^{(k)} := (\nabla_g^k f)(X_{i_1},\dots,X_{i_k}).\] We claim that for each multi-index \(I = (i_1, \ldots, i_k)\) of length \(k\), \[\label{eqn:component95expansion} T_I^{(k)} = \sum_{\alpha+|\beta|\le k} A_{I,\alpha,\beta}(r,\theta)\, r^{\alpha-k}\,\partial_r^\alpha \partial_\theta^\beta f,\tag{40}\] where \(A_{I,\alpha,\beta}\) are smooth and uniformly bounded functions. This claim immediately implies the desired estimate.

We prove 40 by induction on \(k\). For \(k=1\), the components of the differential are simply the directional derivatives: \[(\nabla_g f)(X_0) = X_0 f = \partial_r f, \qquad (\nabla_g f)(X_a) = X_a f = r^{-1}\partial_{\theta^a} f.\] Both expressions match the form of 40 with constant scalar coefficients, satisfying the base case.

Assume the claim holds for some \(k\ge 1\). For \(j\in\{0,\dots,n-1\}\), the definition of the covariant derivative gives \[\begin{align} T_{jI}^{(k+1)} &:= (\nabla_g^{k+1} f)(X_j, X_{i_1}, \dots, X_{i_k}) \\ &= X_j \left ( T_I^{(k)} \right ) - \sum_{s=1}^k (\nabla_g^k f) \left (X_{i_1}, \dots, \nabla_{X_j}X_{i_s}, \dots, X_{i_k}\right ) \\ &= \underbrace{ X_j \left ( T_I^{(k)} \right ) }_{\text{Derivative Term}} - \underbrace{ \sum_{s=1}^k \sum_{l=0}^{n-1} \omega_{j i_s}^l T_{i_1, \dots, l, \dots, i_k}^{(k)} }_{\text{Connection Terms}}. \end{align}\]

For the derivative term, if \(j=0\), then \(X_0=\partial_r\), and differentiating 40 gives \[\begin{align} & X_0 \left ( T_I^{(k)} \right ) = \partial_r \left ( \sum_{\alpha+|\beta| \leq k} A_{I,\alpha,\beta}(r,\theta) r^{\alpha-k} \partial_r^\alpha \partial_\theta^{\beta} f \right ) \\ & = \sum_{\alpha+|\beta| \leq k} \Big[ (\partial_r A_{I,\alpha,\beta}) r^{\alpha-k} \partial_r^\alpha \partial_\theta^\beta f + A_{I,\alpha,\beta}(\alpha-k) r^{\alpha-k-1}\partial_r^\alpha \partial_\theta^{\beta} f + A_{I,\alpha,\beta} r^{\alpha-k} \partial_r^{\alpha+1} \partial_\theta^{\beta} f \Big]. \end{align}\] By consolidating terms and shifting the indices of summation as needed to absorb the higher derivative, we can rewrite the expression using the original dummy variables \(\alpha\) and \(\beta\) to obtain \[X_0 \left ( T_I^{(k)} \right ) = \sum_{\alpha+|\beta|\le k+1} \widetilde{A}_{I,\alpha,\beta}(r,\theta)\, r^{\alpha-(k+1)}\, \partial_r^{\alpha} \partial_\theta^\beta f,\] where \(\widetilde{A}_{I,\alpha,\beta}\) are smooth and bounded.

If \(j=a\) for \(a=1, \dots, n-1\), then \(X_a=r^{-1}\partial_{\theta^a}\), and similar differentiation yields \[\begin{align} X_a \left ( T_I^{(k)} \right ) & = r^{-1}\partial_{\theta^a} \left ( \sum_{\alpha+|\beta| \leq k} A_{I,\alpha,\beta}(r,\theta) r^{\alpha-k} \partial_r^\alpha \partial_\theta^\beta f \right ) \\ & = \sum_{\alpha+|\beta| \leq k} \left [ (\partial_{\theta^a} A_{I,\alpha,\beta}) r^{\alpha-(k+1)} \partial_r^\alpha \partial_\theta^\beta f + A_{I,\alpha,\beta} r^{\alpha-(k+1)} \partial_r^\alpha \partial_\theta^{\beta+e_a} f \right ]. \end{align}\] Again, by shifting the multi-index in the second term to absorb \(e_a\) (the \(a\)-th standard basis vector) and relabeling the dummy variables back to \(\alpha\) and \(\beta\), we find \[X_a \left ( T_I^{(k)} \right ) = \sum_{\alpha+|\beta|\le k+1} \widetilde{A}_{I,\alpha,\beta}(r,\theta)\, r^{\alpha-(k+1)}\, \partial_r^\alpha \partial_{\theta}^{\beta} f,\] with smooth, bounded coefficients \(\widetilde{A}_{I,\alpha,\beta}\). Hence, the desired structure is preserved for the derivative term.

For the connection terms, Lemma 4 implies \[\omega_{j i_s}^{l} = r^{-1} B_{j i_s}^{l}(r,\theta),\] where \(B_{j i_s}^{l}\) is smooth and bounded. Combining this with the inductive expression for \(T_{i_1,\dots,l,\dots,i_k}^{(k)}\), we obtain a linear combination of terms of the form \[\widetilde{A}_{I,\alpha,\beta}(r,\theta)\, r^{\alpha-(k+1)}\, \partial_r^\alpha \partial_\theta^\beta f,\] where \(\widetilde{A}_{I,\alpha,\beta}\) is again smooth and bounded. Therefore, the desired structure is preserved for the connection terms as well. Altogether, the claim holds for \(k+1\), completing the induction. ◻

Proof. We proceed by induction on \(m\). The base case \(m=1\) follows directly from 43 . Since \(u=\phi(r)\) is a radial function, the angular derivatives all vanish. Thus \[\label{eqn:Delta95g951} \Delta_g u =\mathcal{D}_{n-1}\mathcal{D}_0 \phi(r) + r\Psi(r,\theta) \phi'(r) = \mathcal{D}_{n-1}\mathcal{D}_0 \phi(r) + E_{1,1}(r,\theta) \frac{\phi'(r)}{r^{-1}},\tag{44}\] where \(E_{1,1}:=\Psi\) is a smooth function on \(\overline{V}\).

Now, assume the expansion ?? holds for \(\Delta_g^m\). We will show that it holds for \(\Delta_g^{m+1}\). By the inductive hypothesis, \[\label{eqn:Delta95m43195decomp} \Delta_g^{m+1} u = \Delta_g \left ( \Delta_g^{m} u \right ) = \Delta_g \big(\left( \mathcal{D}_{n-1}\mathcal{D}_0\right)^m \phi(r) \big ) + \sum_{k=1}^{2m-1} \Delta_g \left ( E_{m,k}(r,\theta) \frac{\phi^{(k)}(r)}{r^{2m-k-2}} \right ) ,\tag{45}\] where \(E_{m,k}\) are all smooth functions on \(\overline{V}\). Recall that by Lemma 1 there exist constants \(a_1, \dots, a_{2m}\) such that \[\left( \mathcal{D}_{n-1}\mathcal{D}_0\right)^m \phi(r)= \sum_{i=1}^{2m} a_i \frac{\phi^{(i)}(r)}{r^{2m-i}}.\] (The sum starts at \(i=1\) because of the initial \(\mathcal{D}_0\) operator). Applying 44 to \(\left( \mathcal{D}_{n-1}\mathcal{D}_0\right)^m \phi(r)\) yields \[\begin{align} \Delta_g \left( \left( \mathcal{D}_{n-1}\mathcal{D}_0\right)^m \phi(r) \right) = \left( \mathcal{D}_{n-1}\mathcal{D}_0\right)^{m+1} \phi(r) + E_{1,1}(r,\theta) \sum_{i=1}^{2m+1} \tilde{a}_i \frac{\phi^{(i)}(r)}{r^{2m-i}} \end{align}\] for some constants \(\tilde{a}_1, \dots, \tilde{a}_{2m+1}\). Hence, 45 becomes \[\begin{align} \Delta_g^{m+1} u - \left( \mathcal{D}_{n-1}\mathcal{D}_0\right)^{m+1} \phi(r) & = E_{1,1}(r,\theta) \sum_{i=1}^{2m+1} \tilde{a}_i \frac{\phi^{(i)}(r)}{r^{2m-i}} + \sum_{k=1}^{2m-1} \Delta_g \left( E_{m,k}(r,\theta) \frac{\phi^{(k)}(r)}{r^{2m-k-2}} \right) \\ & = \sum_{i=1}^{2m+1} \tilde{a}_i E_{1,1}(r,\theta)\frac{\phi^{(i)}(r)}{r^{2(m+1)-i-2}} + \sum_{k=1}^{2m-1} \Delta_g \left( E_{m,k}(r,\theta) \frac{\phi^{(k)}(r)}{r^{2m-k-2}} \right) . \end{align}\]

It remains to show that each term \(\Delta_g \left( E_{m,k}(r,\theta) \frac{\phi^{(k)}(r)}{r^{2m-k-2}} \right)\) can be expressed as a linear combination of the form \[E_j(r,\theta) \frac{\phi^{(j)}(r)}{r^{2(m+1)-j-2}} = E_j(r,\theta) \frac{\phi^{(j)}(r)}{r^{2m-j}},\] where \(j=1, \ldots, 2m+1\) and \(E_j(r,\theta)\) are smooth functions on \(\overline{V}\). By the product rule for Laplacian and applying 44 to \(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\), for each \(k=1, \ldots, 2m-1\), we have \[\begin{align} \label{eqn:Delta95g95E95induction} & \quad \Delta_g \left( E_{m,k} \frac{\phi^{(k)}(r)}{r^{2m-k-2}} \right) \nonumber \\ &= E_{m,k} \Delta_g \left( \frac{\phi^{(k)}(r)}{r^{2m-k-2}}\right) + \frac{\phi^{(k)}(r)}{r^{2m-k-2}} \Delta_g E_{m,k} + 2\left\langle \nabla_g E_{m,k}, \nabla_g \left(\frac{\phi^{(k)}(r)}{r^{2m-k-2}} \right)\right \rangle_g \nonumber \\ &= E_{m,k} \mathcal{D}_{n-1}\mathcal{D}_0 \left(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\right) + E_{m,k} E_{1,1} \frac{1}{r^{-1}} \frac{d}{dr}\left(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\right)\\ &\quad + \frac{\phi^{(k)}(r)}{r^{2m-k-2}} \Delta_g E_{m,k} + 2 \left (\partial_r E_{m,k} \right ) \frac{d}{dr}\left(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\right). \nonumber \end{align}\tag{46}\] In the last step, we used the facts that \(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\) is a purely radial function, the radial vector field \(\partial_r\) has unit norm, and \(\partial_r\) is orthogonal to the geodesic spheres.

For the first term of 46 , using \(\mathcal{D}_{n-1}\mathcal{D}_0 = \partial_r^2 + \frac{n-1}{r}\partial_r\), we obtain \[E_{m,k}\mathcal{D}_{n-1}\mathcal{D}_0 \left(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\right) = E_{m,k} \sum_{j=k}^{k+2} c_j \frac{\phi^{(j)}(r)}{r^{2m-j}}\] for some constants \(c_k, c_{k+1}, c_{k+2}\). Similarly, the second and fourth terms can be written as \[\begin{align} E_{m,k}E_{1,1} \frac{1}{r^{-1}} \frac{d}{dr}\left(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\right) &= r^2E_{m,k}E_{1,1} \sum_{j=k}^{k+1} \tilde{c}_j \frac{\phi^{(j)}(r)}{r^{2m-j}},\\ 2\left (\partial_r E_{m,k} \right ) \frac{d}{dr}\left(\frac{\phi^{(k)}(r)}{r^{2m-k-2}}\right) &= \left ( 2r \partial_r E_{m,k} \right ) \sum_{j=k}^{k+1} \tilde{c}_j \frac{\phi^{(j)}(r)}{r^{2m-j}} \end{align}\] for some constants \(\tilde{c}_k, \tilde{c}_{k+1}\).

It remains to consider the third term. Applying 43 to \(E_{m,k}\) gives \[\begin{align} \frac{\phi^{(k)}(r)}{r^{2m-k-2}}\Delta_g E_{m,k} & = \left (\partial_r^2 E_{m,k} + \frac{n-1}{r}\partial_r E_{m,k} + r \Psi \partial_r E_{m,k}+ \frac{1}{r^2}\Delta_{\gamma(r,\theta)}E_{m,k} \right ) \frac{\phi^{(k)}(r)}{r^{2m-k-2}}\\ & = \left ( r^2\partial_r^2 E_{m,k} + (n-1) r\partial_r E_{m,k} + r^3 \Psi \partial_r E_{m,k}+ \Delta_{\gamma(r,\theta)}E_{m,k} \right ) \frac{\phi^{(k)}(r)}{r^{2m-k}}. \end{align}\] Thus each contribution has the required form, with all the resulting coefficients smooth on \(\overline{V}\). Hence the decomposition ?? holds with \(m\) replaced by \(m+1\). This completes the proof by induction. ◻

Proof. Fix a unit vector \(\theta\in S^{n-1} \subset T_{{p}}M\), and consider the geodesic \(\gamma_{\theta}(t)=\exp_{{p}}(t\theta)\), \(|t|<r_\theta.\) Define \[\widetilde{\phi}(t):=u(\gamma_{\theta}(t)).\] Since \(u\in C^{2m}(U_p, \mathbb{R}^\nu)\) and \(\gamma_{\theta}\) is smooth, we have \(\widetilde{\phi}\in C^{2m}((-r_\theta,r_\theta), \mathbb{R}^\nu).\) Moreover, by local geodesic radial symmetry, \[\widetilde{\phi}(t) = u(\exp_{{p}}(t\theta)) = \phi(d_g({p},\exp_{{p}}(t\theta))) = \phi(|t|).\] Hence \(\widetilde{\phi}\) is an even function. Therefore, \[\widetilde{\phi}^{(q)}(0)=0\] for every odd integer \(q\le 2m-1\). Since the \(q\)-th radial derivative of \(u\) at \({p}\) is identified with \(\phi^{(q)}(0)=\widetilde{\phi}^{(q)}(0)\), we obtain \[\partial_r^{2j-1}u({p})=0, \qquad j=1,\ldots,m.\] This proves the lemma. ◻

Proof of Theorem 1: . Let \(u\in C^{2m} (U_p, \mathbb{R}^\nu)\) satisfy the inequality ?? and be geodesically radially symmetric in \(U_p\) about \(p\). To establish the theorem, it suffices to show that if \(\partial^{2j}_{r} u(p) = 0\) for all \(j = 0, \dots, m-1\), then \(u\equiv 0\) on \(U_p\).

First, it follows from Lemma 8 that all the radial derivatives of \(u\) up to order \(2m-1\) vanish at \(p\). Let \(\phi(r)\) be the associated radial profile such that \(u = \phi(r)\) for \(r \in [0, r_\theta)\). Then \[\phi^{(j)}(0) = 0, \qquad j = 0, \dots, 2m-1.\] We claim that these vanishing jets up to order \(2m-1\), under the constraint of ?? , imply that \(u \equiv 0\) in \(U_p\).

By Lemma 6, the covariant derivatives of \(u\) satisfy the bound \[|\nabla^{j}_g u| \leq C_0 \sum_{k=1}^j |\phi^{(k)}(r)| \, r^{k-j} \;\;\text{on}\;U_p\setminus \{p\}\] for \(j\ge 1\). Thus, there exists a constant \(C_1 > 0\) such that \[\label{eqn:nabla95j95u95bound95Riem} \sum_{j=0}^{2m-1} |\nabla_g ^{j} u| \leq C_0 \sum_{j=0}^{2m-1}\sum_{k=0}^{j}\frac{| \phi^{(k)}(r)|}{r^{j-k}} \leq C_1\sum_{k=0}^{2m-1}\frac{| \phi^{(k)}(r)|}{r^{2m-1-k}} \;\;\text{on}\;U_p\setminus \{p\}.\tag{47}\]

On the other hand, by Lemma 7, \[\Delta_g^m u = \left( \mathcal{D}_{n-1}\mathcal{D}_0\right)^{m} \phi(r) + \sum_{k=1}^{2m-1} E_{m,k}(r, \theta) \frac{ \phi^{(k)}(r)}{r^{2m-k-2}} \;\;\text{on}\;U_p\setminus \{p\},\]with \[|E_{m,k}(r, \theta)| \leq C_2 \;\;\text{on} \;V\] for a constant \(C_2 > 0\). Therefore, \[\begin{align} \left | \left (\mathcal{D}_{n-1} \mathcal{D}_0 \right)^m \phi (r) \right | & \leq \left | \Delta_g^m u \right | + C_2\sum_{k=1}^{2m-1} \frac{\left | \phi^{(k)}(r) \right |}{r^{2m-k-2}} \\ & \leq C \sum_{j=0}^{2m-1} | \nabla_g^j u | + C_2\sum_{k=1}^{2m-1} \frac{\left | \phi^{(k)}(r) \right |}{r^{2m-k-2}} \\ & \leq C \cdot C_1 \sum_{k=0}^{2m-1}\frac{| \phi^{(k)}(r)|}{r^{2m-1-k}} + C_2\sum_{k=1}^{2m-1} \frac{\left | \phi^{(k)}(r) \right |}{r^{2m-k-2}} \\ &= \left(CC_1 r + C_2 r^2\right) \sum_{k=0}^{2m-1} \frac{\left | \phi^{(k)}(r) \right |}{r^{2m-k}} \end{align}\] for \(r\in (0, r_0)\), where \(r_0: = \sup_{\theta\in S^{n-1}} r_\theta<\infty\). By Theorem 13, we know \(\phi \equiv 0\) for \(r\in [0, r_0)\), and equivalently, \(u \equiv 0\) in \(U_p\). ◻

Proof of Theorem 2: . Let \(w := u - v\). Since \(u\) and \(v\) both solve ?? , we have \[\Delta_g^m w = f(\cdot, u, \nabla_g u, \dots, \nabla_g^{2m-1}u) - f(\cdot, v, \nabla_g v, \dots, \nabla_g^{2m-1}v).\] Because \(f\) is locally Lipschitz and \(u,v \in C^{2m}\), it follows that for every \(\tilde{U}_p \Subset U_p\), there exists a constant \(C>0\) such that \[|\Delta_g^m w | \le C \sum_{j=0}^{2m-1} |\nabla_g^j w | \quad \text{in }\; \tilde{U}_p.\] Moreover, by assumption \(w\) is geodesically radially symmetric in \(U_p\) about \(p\), and all even-order radial derivatives of \(w\) up to order \(2m-2\) vanish at \(p\). It then follows from Theorem 1 that \(w \equiv 0\) in \(\tilde{U}_p\). That is, \(u \equiv v\) in \(\tilde{U}_p\). Since \(\tilde{U}_p\) is arbitrary, \(u \equiv v\) in \(U_p\). ◻

Proof. In holomorphic normal coordinates, \(r(z)=|z|+O(|z|^2)\). Therefore \[r_i=\partial_i r=\frac{\bar \xi^i}{2}+O(r),\] which proves ?? .

It remains to prove ?? . In geodesic polar coordinates \((r,\theta)\) for the Riemannian metric \(g\) induced by \(\omega\), the metric has the form \[g=dr^2+r^2\gamma_{ab}(r,\theta)\,d\theta^a d\theta^b,\] where \(\gamma_{ab}(r,\theta)\) is smooth and uniformly positive for \(r\) small, and \(\gamma_{ab}(0,\theta)\) is the standard metric on \(S^{2n-1}\). Hence \[\nabla^2 r(\partial_r,\cdot)=0, \qquad \nabla^2 r(\partial_{\theta^a},\partial_{\theta^b}) =\frac{1}{2} \partial_r (r^2\gamma_{ab}) = r \gamma_{ab}(0, \theta) + O(r^2).\] See, for instance, [24]. This gives the intrinsic expansion \[\label{exp1} \nabla^2 r = \frac{1}{r}\bigl(g-dr\otimes dr\bigr)+O(1).\tag{48}\] Taking the \((1,\bar 1)\)-component of 48 and making use of ?? further give \[r_{i\bar j} = \frac{1}{2r}\delta_{ij} - \frac{1}{r}r_i r_{\bar j} + O(1) = \frac{1}{2r}\delta_{ij} - \frac{1}{4r}\bar\xi^i\xi^j + O(1).\] This proves the lemma. ◻

Proof of Theorem 8 part 2:. Define \[F(\varphi):=\log\frac{\omega_\varphi^n}{\omega^n}, \;\; \text{where}\;\; \omega_\varphi:=\omega+\sqrt{-1}\,\partial\bar\partial\varphi.\] Then equation 1 is equivalent to \[F(\varphi)=f.\] Let \(\varphi_0\) and \(\varphi_1\) be two solutions and set \(u:=\varphi_1-\varphi_0\). Letting \[\varphi_t:=\varphi_0+tu, \qquad 0\leq t\leq 1,\] the fundamental theorem of calculus gives \[\label{fun} 0=F(\varphi_1)-F(\varphi_0) =\int_0^1 D_{\varphi_t}F(u)\,dt .\tag{49}\] The linearization \(D_{\varphi_t}F\) is computed as follows. Since \[\omega_{\varphi_t+su} =\omega_{\varphi_t}+s\sqrt{-1}\,\partial\bar\partial u,\] we have \[D_{\varphi_t}F (u) = \left.\frac{d}{ds}\right|_{s=0} \log\frac{ (\omega_{\varphi_t}+s\sqrt{-1}\,\partial\bar\partial u)^n }{\omega^n}.\] Let \(g_t\) be the Kähler metric associated to \(\omega_{\varphi_t}\). In local holomorphic coordinates, \(g_{t,i\bar j}=g_{i\bar j}+(\varphi_t)_{i\bar j}\). Hence \[D_{\varphi_t} F (u) = \left.\frac{d}{ds}\right|_{s=0} \log\det (g_{t, i\bar j}+s\,u_{i\bar j}) = g_t^{i\bar j}u_{i\bar j} \left(= \Delta_{\omega_{\varphi_t}}u\right),\] where \(g_t^{i\bar j}\) denotes the inverse matrix of \(g_{t,i\bar j}\) and the last step has used the matrix identity \(\left.\frac{d}{ds}\right|_{s=0} \log\det(A+sB) = \operatorname{tr}(A^{-1}B)\) for two matrices \(A\) and \(B\) with \(A\) invertible. Combining this with 49 gives \[0=\left(\int_0^1 g_t^{i\bar j}\,dt\right)u_{i\bar j}.\] Set \(a^{i\bar j}:=\int_0^1 g_t^{i\bar j}\,dt\). Since each \(g_t^{i\bar j}\) is positive definite, \(a^{i\bar j}\) is positive definite, and \(u\) satisfies \[\label{linear-eq} a^{i\bar j}u_{i\bar j}=0.\tag{50}\]

Assume now that \(u\) is geodesically radial with respect to the induced Riemannian metric \(g\) in a normal neighborhood \(U_p\) of \(p\), so that \(u(z)=\phi(r(z))\) for some single variable function \(\phi\), where \(r(z)=d_g(z, p)\). All derivatives \(u_{i\bar j}\) are taken with respect to local holomorphic coordinates. By the chain rule in holomorphic coordinates, \[u_i=\phi'(r)r_i \qquad \text{and}\qquad u_{i\bar j} = \phi''(r)r_i r_{\bar j} + \phi'(r)r_{i\bar j}.\] Substituting this into 50 , we obtain \[\label{en1} a^{i\bar j}r_i r_{\bar j}\,\phi''(r) + a^{i\bar j}r_{i\bar j}\,\phi'(r) = 0.\tag{51}\] Since \(r\) is the distance function with respect to \(g\), \(|\nabla r|_g=1\) for \(r\in (0, r_\theta)\), where \(r_\theta\) is the radial bound in the direction \(\theta\). By the uniform ellipticity of \(a^{i\bar j}\), we have \(0<c\leq a^{i\bar j}r_i r_{\bar j}\leq C\). Therefore, after dividing 51 by \(a^{i\bar j}r_i r_{\bar j}\), we get \[\label{en3} \phi''(r)+\widetilde{a}(r,\theta)\phi'(r)=0, \;\; \text{where}\;\; \widetilde{a}(r,\theta):= \frac{a^{i\bar j}r_{i\bar j}}{a^{i\bar j}r_i r_{\bar j}}.\tag{52}\] Here \(\widetilde{a}\) is smooth on \(0< r<r_\theta,\theta\in S^{2n-1}\).

We next determine the leading behavior of \(\widetilde{a}\) as \(r\to 0\). Indeed, it follows from ?? and ?? that \[a^{i\bar j}r_i r_{\bar j} = \frac{1}{4} a^{i\bar j}(p)\bar\xi^i\xi^j + O(r),\] and \[a^{i\bar j}r_{i\bar j} = \frac{1}{2r}\sum_i a^{i\bar i}(p) - \frac{1}{4r}a^{i\bar j}(p)\bar\xi^i\xi^j + O(1) = \frac{1}{r}\left(\frac{1}{2}\sum_i a^{i\bar i}(p) - \frac{1}{4}a^{i\bar j}(p)\bar\xi^i\xi^j + O(r)\right).\] Consequently, \[\label{atil-leading} \widetilde{a}(r,\theta) = \frac{1}{r} \left( \frac{2\sum_i a^{i\bar i}(p)}{a^{i\bar j}(p)\bar\xi^i\xi^j} -1+ O(r) \right) .\tag{53}\] Since \(a^{i\bar j}(p)\) is positive definite, all the eigenvalues \(\lambda_i\) of \((a^{i\bar j}(p))\) are positive. We have \[\frac{\sum_i a^{i\bar i}(p)}{a^{i\bar j}(p)\bar\xi^i\xi^j} \ge \frac{\sum_i \lambda_i}{\max_{i} \lambda_i} \geq 1.\] Hence \[\label{ct} c_\theta:= \frac{2\sum_i a^{i\bar i}(p)}{a^{i\bar j}(p)\bar\xi^i\xi^j} -1\ge 1.\tag{54}\] Combined with 53 , we have for each fixed \(\theta\in S^{2n-1}\), there exists a continuous function \(h_\theta\) on \([0,r_\theta)\), with \(h_\theta(0)=0\) such that \[\widetilde{a}(r,\theta)=\frac{c_\theta-h_\theta(r)}{r}.\] Therefore 52 can be rewritten as \[\label{radial-ode} \phi''(r)+\frac{c_\theta}{r}\phi'(r) = h_\theta(r)\frac{\phi'(r)}{r}.\tag{55}\] Finally, recalling the operator \(\mathcal{D}_\alpha:=\frac{d}{dr}+\frac{\alpha}{r}\), we have \(\mathcal{D}_{c_\theta}\mathcal{D}_0\phi = \phi''(r)+\frac{c_\theta}{r}\phi'(r).\) Thus 55 becomes \[\mathcal{D}_{c_\theta}\mathcal{D}_0\phi = h_\theta(r)\frac{\phi'(r)}{r}.\] Since \(c_\theta>0\) by 54 , condition ?? is satisfied. Therefore Theorem 13 applies to give \(\phi\equiv 0\). ◻

Proof of Theorem 21: . Without loss of generality, let \(p=0\). For each \(M \in \text{Herm}(n)\) satisfying \(g_{i\bar j}(0)+M_{i\bar j}>0\), define \[\mathcal{F}(z,M) := \log\det(g_{i\bar j}(z)+M_{i\bar j}) - \log\det(g_{i\bar j}(z))- f(z).\] Then the Calabi equation 1 reduces to \(\mathcal{F}(\cdot,\partial\bar\partial\varphi)=0\). For every \(K\in\Sigma (:=\Sigma_0)\), set \[Q_K(z):=K_{i\bar j}z^i\bar z^j.\] Then \((Q_K)_{i\bar j}=K_{i\bar j}\). We seek a solution of the form \[\varphi_K=Q_K+w_K,\] where \(w_K\) is a smooth function on \(B_\rho\) with \(w_K|_{\partial B_\rho}=0\). Thus \(w:= w_K\) should solve \[\label{eq:wK-equation} \mathcal{F}(z,K+\partial\bar\partial w)=0 \qquad\text{in }B_\rho.\tag{56}\]

Rewrite \(\mathcal{F}(z,K+\partial\bar\partial w)\) as the sum of the following three components: \[\mathcal{F}(z,K+\partial\bar\partial w) = \mathcal{F}(z,K) +L_K(w) + N_K(w),\] where \(L_K\) is the linearization of \(\mathcal{F}\) in the matrix variable at \(K\), and \(N_K\) is the quadratic remainder after subtracting the linear part of \(\mathcal{F}\) in the matrix variable. Namely, \[\label{aij} L_Kw:=A_K^{i\bar j}(z)w_{i\bar j}, \;\;\text{with}\;\;A_K^{i\bar j}(z) := D_{M_{i\bar j}} \mathcal{F}(z,K) = (g(z)+K)^{i\bar j},\tag{57}\] and \[N_K(w) := \mathcal{F}(z,K+\partial\bar\partial w) - \mathcal{F}(z,K) - A_K^{i\bar j}(z)w_{i\bar j}.\] Thus, 56 can be rewritten as \[\label{eq:fixed-point-equation} L_Kw=- \mathcal{F}(\cdot,K)-N_K(w).\tag{58}\] Note that since \(g(0)+K>0\), after shrinking \(\rho\) if necessary, the operator \(L_K\) is uniformly elliptic on \(B_\rho\) and for \(K\) in a small neighborhood of \(H\) inside \(\Sigma\).

We next estimate the scaled Hölder norms of \(\mathcal{F}(\cdot,K)\) and \(N_K(w)\). The admissibility condition \(K\in\Sigma\) ensures that \(\mathcal{F}(0,K)=0\). By the smoothness of the data, \[\label{eq:RK-scaled} \| \mathcal{F}(\cdot,K)\|_{C^\alpha_\rho } \leq C\rho^\alpha\tag{59}\] for \(K\) near \(H\). For \(N_K(w)\), Taylor’s formula in the matrix variable gives \[N_K(w) = \int_0^1(1-s)\, D_{M_{i\bar j}M_{k\bar \ell}}\mathcal{F} \bigl(z,K+s\partial\bar\partial w\bigr) w_{i\bar j}w_{k\bar \ell}\,ds .\] Since the matrices involved remain in a fixed compact subset of the admissible cone, the second derivatives \(D_{M_{i\bar j}M_{k\bar \ell}}\mathcal{F}\) are uniformly bounded. Hence by the scaled Hölder product estimate, \[\label{eq:NK-quadratic} \|N_K(w)\|_{C^\alpha_\rho} \leq C\|\partial\bar\partial w\|_{C^\alpha_\rho}^2 \le C\rho^{-4}\|w\|_{C^{2,\alpha}_\rho}^2,\tag{60}\] where the last inequality made use of the fact \(\|\partial\bar\partial w\|_{C^\alpha_\rho} \leq C\rho^{-2}\|w\|_{C^{2,\alpha}_\rho}\) by definition.

Let \(T_K\) denote the inverse operator of the linear Dirichlet problem \[L_Kw=h, \qquad w|_{\partial B_\rho}=0\] for \(h\in C^\alpha(B_\rho)\). By the scaled Schauder estimate (see [25]), \(T_Kh\in C^{2, \alpha}(B_\rho)\) with \[\label{eq:scaled-schauder} \|T_Kh\|_{C^{2,\alpha}_\rho} \leq C\rho^2\|h\|_{C^\alpha_\rho},\tag{61}\] where \(C\) is independent of sufficiently small \(\rho\) and of \(K\) near \(H\). Define \[\mathcal{T}_K(w):=T_K\bigl(- \mathcal{F}(\cdot,K)-N_K(w)\bigr).\] We shall show that, for \(\rho>0\) sufficiently small and some constant \(M_0\) sufficiently large, \(\mathcal{T}_K\) maps the set \[\mathcal{B}_\rho := \left\{ w\in C^{2, \alpha}(B_\rho): w|_{\partial B_\rho}=0\;\;\text{and}\;\; \|w\|_{C^{2,\alpha}_\rho}\leq M_0\rho^{2+\alpha} \right\}\] into itself and is a contraction on \(\mathcal{B}_\rho\).

Indeed, for \(w\in\mathcal{B}_\rho\), by 59 61 , \[\begin{align} \|\mathcal{T}_K(w)\|_{C^{2,\alpha}_\rho} &\leq C\rho^2 \left( \| \mathcal{F}(\cdot,K) \|_{C^\alpha_\rho} + \|N_K(w)\|_{C^\alpha_\rho} \right) \\ &\leq C\rho^2 \left( \rho^\alpha + \rho^{-4}\|w\|_{C^{2,\alpha}_\rho}^2 \right) \\ &\leq C\rho^{2+\alpha} + C\rho^{-2}(M_0\rho^{2+\alpha})^2 \\ &= C(1 + M_0^2\rho^\alpha)\rho^{2+\alpha}. \end{align}\] Choosing \(M_0\) sufficiently large (for instance, \(M_0\ge 2C\)) and then taking \(\rho>0\) sufficiently small (say, \(\rho\le M_0^{-\frac{2}{\alpha}}\)), we get \[\|\mathcal{T}_K(w)\|_{C^{2,\alpha}_\rho} \leq M_0\rho^{2+\alpha}.\] Thus \(\mathcal{T}_K\) maps \(\mathcal{B}_\rho\) into itself.

Next, for \(w_1,w_2\in\mathcal{B}_\rho\), set \(w_t:=w_2+t(w_1-w_2)\). Then by the fundamental theorem of calculus and 57 , \[\begin{align} N_K(w_1)-N_K(w_2) &= \mathcal{F}(z,K+\partial\bar\partial w_1)-\mathcal{F}(z,K+\partial\bar\partial w_2) - A_K^{i\bar j}(z) (w_1-w_2)_{i\bar j} \\ &= \int_0^1 \left[ D_{M_{i\bar j}}\mathcal{F}(z,K+\partial\bar\partial w_t) - D_{M_{i\bar j}}\mathcal{F}(z,K) \right] (w_1-w_2)_{i\bar j}\,dt . \end{align}\] Since the matrices remain in a compact subset of the admissible cone, \[\left|D_{M_{i\bar j}} \mathcal{F}(z,K+\partial\bar\partial w_t) - D_{M_{i\bar j}}\mathcal{F} (z,K) \right| \leq C|\partial\bar\partial w_t|.\] Hence, using the scaled Hölder product estimate, \[\label{eq:NK-lip-ddbar} \begin{align} \|N_K(w_1)-N_K(w_2)\|_{C^\alpha_\rho} \leq& C \left( \|\partial\bar\partial w_1\|_{C^\alpha_\rho} + \|\partial\bar\partial w_2\|_{C^\alpha_\rho} \right) \|\partial\bar\partial(w_1-w_2)\|_{C^\alpha_\rho}\\ \le & C\rho^{-4} \left( \|w_1\|_{C^{2,\alpha}_\rho} + \|w_2\|_{C^{2,\alpha}_\rho} \right) \|w_1-w_2\|_{C^{2,\alpha}_\rho}. \end{align}\tag{62}\] Together with 61 and linearity of the operator \(T_K\), we obtain \[\begin{align} \|\mathcal{T}_K(w_1)-\mathcal{T}_K(w_2)\|_{C^{2,\alpha}_\rho}&= \| T_K\left(N_k(w_1) -N_k(w_2)\right)\|_{C^{2,\alpha}_\rho} \\ &\leq C\rho^2 \|N_K(w_1)-N_K(w_2)\|_{C^\alpha_\rho} \\ &\leq C\rho^2\rho^{-4} \left( \|w_1\|_{C^{2,\alpha}_\rho} + \|w_2\|_{C^{2,\alpha}_\rho} \right) \|w_1-w_2\|_{C^{2,\alpha}_\rho} \\ &\leq C\rho^{-2}(2M_0\rho^{2+\alpha}) \|w_1-w_2\|_{C^{2,\alpha}_\rho} \\ &= CM_0\rho^\alpha \|w_1-w_2\|_{C^{2,\alpha}_\rho}. \end{align}\] Shrinking \(\rho>0\) sufficiently small again so that \(CM_0\rho^\alpha<1\), we conclude that \(\mathcal{T}_K\) is a contraction on \(\mathcal{B}_\rho\). By the contraction mapping principle, \(\mathcal{T}_K\) has a unique fixed point \(w_K\in\mathcal{B}_\rho\) satisfying \[\label{eq:wK-bound} \|w_K\|_{C^{2,\alpha}_\rho} \leq C\rho^{2+\alpha}.\tag{63}\] In particular, this \(w_K\) satisfies 58 , which is equivalent to 56 . By elliptic bootstrapping, \(w_K\) is smooth on \(B_\rho\). Thus \(\varphi_K:=Q_K+w_K\) is a smooth local solution to 1 for every \(K\in\Sigma\) close to \(H\). Its complex Hessian at \(0\) is \[(\varphi_K)_{i\bar j}(0) = K_{i\bar j}+(w_K)_{i\bar j}(0).\]

Next we adjust \(K\) so that the complex Hessian of \(\varphi_K\) at the origin is exactly \(H\). Define an operator \[\label{E} \begin{align} E_\rho:\Sigma &\to \Sigma \\ K &\mapsto (\varphi_K)_{i\bar j}(0) = K+\partial\bar\partial w_K(0). \end{align}\tag{64}\] The image indeed lies in \(\Sigma\), since \(\varphi_K\) solves the equation and therefore \(\mathcal{F}(0,E_\rho(K))=0.\) Choose a local coordinate chart on \(\Sigma\) centered at \(H\), and identify \(H\) with \(0\). We use this chart to identify a sufficiently small neighborhood of \(H\) in \(\Sigma\) with a ball in \(T_H\Sigma\). Under this identification, we suppress the chart from the notation and write points of \(\Sigma\) simply as \(K\). In these coordinates, it suffices to find \(K_\rho\) such that \(E_\rho(K_\rho)=0,\) or equivalently, by 64 , \[K_\rho=-\partial\bar\partial w_{K_\rho}(0).\] Define \[R_\rho(K):=-\partial\bar\partial w_K(0).\] We will prove that \(R_\rho\) is a contraction on a fixed small ball \(B_\delta\subset T_H\Sigma\) in these local coordinates. To this end, we first establish the necessary estimates. We claim that the map \(\partial\bar\partial w_K(0)\) depends \(C^1\)-smoothly on \(K\). Moreover, for every \(\dot{K}\in T_K\Sigma\), if \[\dot{w}_K:=D_Kw_K[\dot{K}],\] then \[\label{ome} \left| \partial\bar\partial \dot{w}_K(0) \right| \leq C\rho^\alpha |\dot{K}|.\tag{65}\] Assuming this claim, it follows from 63 and 65 that \[\|R_\rho\|_{C(B_\delta)}\leq C\rho^\alpha, \qquad \|D_KR_\rho\|_{C(B_\delta)}\leq C\rho^\alpha.\] Choosing \(\rho>0\) sufficiently small so that \(C\rho^\alpha<\min\left\{\frac{\delta}{2},\frac{1}{2}\right\},\) we conclude that \(R_\rho\) maps \(B_\delta\) into itself and is a contraction. Hence, by the contraction mapping principle, there exists \(K_\rho\in B_\delta\) such that \[K_\rho=R_\rho(K_\rho).\] Therefore \[E_\rho(K_\rho) = K_\rho+\partial\bar\partial w_{K_\rho}(0) = K_\rho-R_\rho(K_\rho) = 0.\] Recalling that \(H\) is identified with \(0\), this yields \[E_\rho(K_\rho)=H.\] In particular, for \(\varphi:=\varphi_{K_\rho}\), we achieve the prescribed Hessian \(\varphi_{i\bar j}(0)=H_{i\bar j}.\)

Now it remains to prove the claim. We first show that the dependence of \(w_K\) on \(K\) is smooth in the \(C^{2,\alpha}\)-topology. Indeed, let \(\mathcal{X}_\rho := \{w\in C^{2,\alpha}({B_\rho}): w|_{\partial B_\rho}=0\}\), and consider the map \[\begin{align} \mathcal{P}:\Sigma\times \mathcal{X}_\rho&\to C^\alpha(B_\rho) \\ (K,w) &\mapsto \mathcal{F}(z,K+\partial\bar\partial w). \end{align}\] At a solution \(w=w_K\), the derivative in the \(w\)-variable is \[D_w\mathcal{P}(K,w_K)[v] = G_K^{i\bar j}v_{i\bar j},\] where \[\label{gij} G_{K,i\bar j} := g_{i\bar j}+K_{i\bar j}+(w_K)_{i\bar j},\tag{66}\] and \(G_K^{i\bar j}\) is the inverse matrix of \(G_{K,i\bar j}\). Then \(D_w\mathcal{P}(K,w_K)\) is an isomorphism \(\mathcal{X}_\rho\to C^\alpha(B_\rho)\) by the elliptic Dirichlet theory. Hence the implicit function theorem applied to \(\mathcal{P}(K, w) =0\) gives the smooth dependence \[K\mapsto w_K\] as a map into \(\mathcal{X}_\rho.\) Since the evaluation map \(w\mapsto \partial\bar\partial w(0)\) is a bounded linear map from \(C^{2,\alpha}(B_\rho)\) to the finite-dimensional space of Hermitian matrices, it follows that \[K\mapsto \partial\bar\partial w_K(0)\] is smooth.

Differentiating \(\mathcal{P}(K, w_K)=0\) in the direction \(\dot{K}\in T_K\Sigma\), we obtain \(G_K^{i\bar j}\bigl(\dot{K}_{i\bar j}+(\dot{w}_K)_{i\bar j}\bigr)=0,\) or equivalently, \(\dot{w}_K\) satisfies the linear elliptic equation \[\label{eq:dotw-equation} G_K^{i\bar j}(\dot{w}_K)_{i\bar j} = - G_K^{i\bar j} \dot{K}_{i\bar j}.\tag{67}\] Since \(\dot{K}\in T_K\Sigma\), differentiating the constraint \(\mathcal{F}(0,K)=0\) gives \[(g(0)+K)^{i\bar j}\dot{K}_{i\bar j}=0.\] Using this and 66 , together with \(g(z)-g(0)=O(\rho)\) and 63 , we get \[\begin{align} \|G_K^{i\bar j}\dot{K}_{i\bar j}\|_{C^\alpha_\rho} =& \| ((g+K+ \partial\bar\partial w_K)^{i\bar j}- \left(g(0)+K\right)^{i\bar j} )\dot{K}_{i\bar j}\|_{C^\alpha_\rho} \\ \le & \| \left(g(0)+K+ (g-g(0))+ \partial\bar\partial w_K\right)^{i\bar j}- \left(g(0)+K\right)^{i\bar j}) \|_{C^\alpha_\rho}|\dot{K}| \\ \le& C \| g-g(0) + \partial\bar\partial w_K \|_{C^\alpha_\rho} |\dot{K}|\\ \le & C\rho^\alpha |\dot{K}|. \end{align}\] Here the third line has also used the matrix identity \(A^{-1}-B^{-1} = B^{-1}(B-A)A^{-1}\) for two invertible matrices \(A\) and \(B\). In addition, since \(w_K\in \mathcal{X}_\rho\), \(\dot{w}_K|_{\partial B_\rho}=0\) as well. Applying the scaled Schauder estimate 61 to the equation 67 , we obtain \[\|\dot{w}_K\|_{C^{2,\alpha}_\rho} \leq C\rho^{2+\alpha}|\dot{K}|,\] from which and the fact that \(\left| \partial\bar\partial \dot{w}_K(0) \right|\le \rho^{-2} \|\dot{w}_K\|_{C^{2,\alpha}_\rho}\), the inequality 65 follows.

Finally, the \(0\)-jet and \(1\)-jet of \(\varphi\) may be prescribed independently. Indeed, adding a real constant and the real part of a linear function does not change \(\partial\bar\partial\varphi\). Hence it does not change the Calabi equation or the prescribed complex Hessian, but it can adjust \(\varphi(p)\) and \(d\varphi(p)\) arbitrarily. ◻

Proof. First, according to Lemma 1, there exists \((b_1, \ldots, b_{2m}) \in \mathbb{C}^{2m}\) such that the general Euler-type operator can be decomposed as a product of first-order singular operators: \[u^{(2m)} + \frac{a_1}{r}u^{(2m-1)} + \cdots + \frac{a_{2m}}{r^{2m}} u =\mathcal{D}_{b_{2m}} \cdots \mathcal{D}_{b_1} u.\]

For each \(j=1, \ldots, m\), we seek pairs \((c_{2j-1}, c_{2j})\) such that the radial component of \(\Delta_{(c_{2j-1}, c_{2j})}\) coincides with the second-order factor \(\mathcal{D}_{b_{2j}} \mathcal{D}_{b_{2j-1}}\): \[\label{bc2} \Delta^{\text{rad}}_{(c_{2j-1}, c_{2j})} = \mathcal{D}_{b_{2j}} \mathcal{D}_{b_{2j-1}}.\tag{69}\] Using the expansion in 68 , the operator is given by \[\Delta^{\text{rad}}_{(c_{2j-1}, c_{2j})} u = u'' +\frac{c_{2j-1}+c_{2j}+n-1}{r}u' +\frac{c_{2j-1}(c_{2j}+n-2)}{r^2}u.\]

On the other hand, a direct expansion of the product \(\mathcal{D}_{b_{2j}}\mathcal{D}_{b_{2j-1}}\) yields \[\mathcal{D}_{b_{2j}}\mathcal{D}_{b_{2j-1}} u = u''+ \frac{b_{2j-1}+b_{2j}}{r}u'+\frac{b_{2j-1}(b_{2j}-1)}{r^2}u.\]

To ensure 69 , we equate the corresponding coefficients: \[\begin{cases} b_{2j-1}+b_{2j} = c_{2j-1}+c_{2j}+n-1;\\ b_{2j-1}(b_{2j}-1) = c_{2j-1}(c_{2j}+n-2). \end{cases}\]

This system admits two possible solutions for the parameters \((b_{2j-1}, b_{2j})\): \[\label{b2} (b_{2j-1}, b_{2j}) = (c_{2j-1}, c_{2j} + n - 1) \quad \text{or} \quad (b_{2j-1}, b_{2j}) = (c_{2j} + n - 2, c_{2j-1} + 1).\tag{70}\] Equivalently, one may determine the parameters \((c_{2j-1}, c_{2j})\) as \[\label{c2} (c_{2j-1}, c_{2j}) = (b_{2j-1}, b_{2j} - n + 1) \quad \text{or} \quad (c_{2j-1}, c_{2j}) = (b_{2j} - 1, b_{2j-1} - n + 2).\tag{71}\]

With either choice, the \(2m\)-th order Euler-type operator is realized precisely as the radial component of the iterated divergence-form operator \(\Delta_{(c_{2m-1}, c_{2m})} \cdots \Delta_{(c_1, c_2)}\). This completes the proof. ◻

Proofs of Theorem 22 and Corollary 23:. If \(u\) is radially symmetric about \(0\) with profile function \(\phi\), we may choose \((b_1, \dots, b_N)\) according to 70 to write \[\Delta_{(c_{2m-1}, c_{2m})}\cdots \Delta_{(c_1, c_2)}u(x)=\mathcal{D}_{b_{2m}} \cdots\mathcal{D}_{b_1} \phi(r).\] Applying the estimate in Lemma 6 to the right-hand side of the inequality ?? , we further reduce ?? to \[| \mathcal{D}_{b_{2m}} \cdots\mathcal{D}_{b_1} \phi(r)|\le \frac{h(r)}{r}\cdot C\sum_{j=0}^{2m-1}\frac{|\phi^{(j)}(r)|}{r^{2m-1 -j}} = Ch(r)\cdot\sum_{j=0}^{2m-1}\frac{|\phi^{(j)}(r)|}{r^{2m -j}}.\] It remains to show that ?? is equivalent to ?? , as the rest of the proof of Theorem 22 then follows directly from Theorem 13. The proof of Corollary 23 is similar to that of Theorem 2 and is therefore omitted.

For any \(j =1, \ldots, m,\) in the first case of 70 where \(b_{2j-1} = c_{2j-1}\) and \(b_{2j} = c_{2j}+n-1\), ?? implies \[\begin{align} &2j-\mathrm{Re}(c_{2j-1})-2m = 2j-\mathrm{Re}(b_{2j-1})-2m<2 \Rightarrow (2j-1)-\mathrm{Re}(b_{2j-1})-2m<1; \\ &2j-\mathrm{Re}(c_{2j})-2m = 2j-\left (\mathrm{Re}(b_{2j})-n+1 \right )-2m<n \Rightarrow 2j-\mathrm{Re}(b_{2j})-2m<1. \end{align}\] If \(b_{2j-1} = c_{2j}+n-2\) and \(b_{2j} =c_{2j-1}+1\) in 70 , then ?? implies \[\begin{align} & 2j-\mathrm{Re}(c_{2j-1})-2m = 2j-\left (\mathrm{Re}(b_{2j})-1 \right )-2m<2 \Rightarrow 2j-\mathrm{Re}(b_{2j})-2m<1; \\ & 2j-\mathrm{Re}(c_{2j})-2m = 2j-\left (\mathrm{Re}(b_{2j-1})-n+2 \right )-2m<n \Rightarrow (2j-1)-\mathrm{Re}(b_{2j-1})-2m<1. \end{align}\]

In either case ?? is satisfied with \(N=2m\). The proof is complete. ◻

Proof of Theorem 24: . Suppose \(u\) is radial about the origin. By Lemma 8, the odd-order derivatives must vanish: \[u^{(j)}(0) =0, j=1, 3,\ldots, 2m-1.\] Moreover, \(u\) satisfies \[\Delta_{(c_{2m-1}, c_{2m})}\cdots \Delta_{(c_1, c_2)} u = \mathcal{D}_{b_{2m}} \cdots\mathcal{D}_{b_1} u = f(r, u, \dots, u^{(n-1)}),\] where the coefficients \((b_{2j-1}, b_{2j})\) are related to \((c_{2j-1}, c_{2j})\) by 70 , or equivalently 71 .

Let \(\Lambda\) be the index set defined as in ?? with respect to \((b_1, \dots, b_{2m})\). We shall show that \[\Gamma = \Lambda\cap \{0, 2,\ldots,2m-2 \}.\] Indeed, by the definition of \(\Gamma\) in ?? , every \(\gamma\in \Gamma\) satisfies \[\gamma = -c_{2j-1}+2j-2 \quad \text{or} \quad -c_{2j}+2j -n\] for some \(j=1, \ldots, m\). Substituting these into 71 , we obtain \[\gamma +b_j-j+1 =0\] for some \(j=1, \ldots, 2m-1\).This establishes that \(\Gamma \subset \Lambda \cap \{0, 2, \dots, 2m-2\}\). The reverse inclusion is proved similarly.

Consequently, the proof reduces to establishing the existence of \(C^N([0, r_0))\) solutions to the following initial value problem: \[\begin{cases} &\mathcal{D}_{b_{2m}} \cdots\mathcal{D}_{b_1} u(r) = f(r, u, \dots, u^{(n-1)}),\;\;r\in (0, r_0),\\ &u^{(j)} (0) = \zeta_j, \quad j\in \Lambda\cap \{0, 2,\ldots,2m-2 \}, \\ &u^{(j)} (0) = 0, \quad j\in \{0, \ldots, 2m-1\}\setminus \left( \Lambda\cap \{0, 2,\ldots,2m-2 \}\right). \end{cases}\] The existence of such a solution is then guaranteed by Theorem 16. ◻