# Berenstein rigidity EP-1-H1-to-H3

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

H1: Let \(m\ge2\) be an integer and let \(\rho>1\). Put \(D=\{z\in\mathbb C:|z|<1\}\). For \(\ell\in\mathbb Z\) and \(s\in\mathbb Z_{\ge0}\), set
\[
\Phi_{\ell,s}(r,\theta)=r^{m|\ell|}P_s^{(0,m|\ell|)}(2r^2-1)e^{im\ell\theta},
\]
where \(P_s^{(0,b)}\) is the Jacobi polynomial normalized by \(P_s^{(0,b)}(1)=1\).
Let \(W_{m,\rho}\) consist of complex coefficient series \(g=\sum_{\ell,s}g_{\ell,s}\Phi_{\ell,s}\) with
\[
\|g\|_\rho=\sum_{\ell,s}\rho^{|\ell|}|g_{\ell,s}|<\infty.
\]
Let \(Y_{m,\rho}\) be the real subspace of \(W_{m,\rho}\) consisting of real-valued functions invariant under \(\theta\mapsto-\theta\).
Let \(P_{m,\rho}\) consist of holomorphic series \(a(z)=\sum_{j\ge0}a_jz^{mj}\) with real coefficients and norm
\(\|a\|_P=\sum_{j\ge0}\rho^j|a_j|<\infty\).
Let \(B_{m,\rho}\) consist of real-valued, reflection-invariant boundary series
\(b(\theta)=\sum_{\ell\in\mathbb Z}b_\ell e^{im\ell\theta}\), with norm
\(\|b\|_B=\sum_\ell\rho^{|\ell|}|b_\ell|<\infty\).
For \(g\in Y_{m,\rho}\), let \(K_Dg\) denote the weak solution, when it exists, of \(\Delta U=g\) in \(D\) with zero Dirichlet trace; put \(Ng=\partial_r(K_Dg)|_{\partial D}\), with the derivative understood as its boundary trace.

For every integer \(m\ge2\) and every \(\rho>1\), the coefficient series define continuous functions on their respective closed domains. The spaces \(W_{m,\rho}\), \(Y_{m,\rho}\), \(P_{m,\rho}\), and \(B_{m,\rho}\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.
For every \(g\in Y_{m,\rho}\), the weak Dirichlet solution exists and belongs to
\(Y_{m,\rho}\cap W^{2,q}(D)\cap W_0^{1,q}(D)\) for every finite \(q>1\). The operators satisfy
\[
\|K_D\|_{Y_{m,\rho}\to Y_{m,\rho}}=\frac14,
\qquad
\|N\|_{Y_{m,\rho}\to B_{m,\rho}}=\frac12.
\]
With the operators extended complex linearly,
\[
N\Phi_{\ell,0}=\frac{e^{im\ell\theta}}{2(m|\ell|+1)},
\qquad N\Phi_{\ell,s}=0\quad(s\ge1).
\]

Prove H3: For each integer \(m\ge2\) and each \(\rho>1\), define the following spaces and operators. Put \(D=\{z\in\mathbb C:|z|<1\}\). For \(\ell\in\mathbb Z\) and \(s\in\mathbb Z_{\ge0}\), set
\[
\Phi_{\ell,s}(r,\theta)=r^{m|\ell|}P_s^{(0,m|\ell|)}(2r^2-1)e^{im\ell\theta},
\]
where \(P_s^{(0,b)}\) is the Jacobi polynomial normalized by \(P_s^{(0,b)}(1)=1\).
Let \(W_{m,\rho}\) consist of complex coefficient series \(g=\sum_{\ell,s}g_{\ell,s}\Phi_{\ell,s}\) with
\[
\|g\|_\rho=\sum_{\ell,s}\rho^{|\ell|}|g_{\ell,s}|<\infty.
\]
Let \(Y_{m,\rho}\) be the real subspace of \(W_{m,\rho}\) consisting of real-valued functions invariant under \(\theta\mapsto-\theta\).
Let \(P_{m,\rho}\) consist of holomorphic series \(a(z)=\sum_{j\ge0}a_jz^{mj}\) with real coefficients and norm
\(\|a\|_P=\sum_{j\ge0}\rho^j|a_j|<\infty\).
Let \(B_{m,\rho}\) consist of real-valued, reflection-invariant boundary series
\(b(\theta)=\sum_{\ell\in\mathbb Z}b_\ell e^{im\ell\theta}\), with norm
\(\|b\|_B=\sum_\ell\rho^{|\ell|}|b_\ell|<\infty\).
For \(g\in Y_{m,\rho}\), let \(K_Dg\) denote the weak solution of \(\Delta U=g\) in \(D\) with zero Dirichlet trace; put \(Ng=\partial_r(K_Dg)|_{\partial D}\), with the derivative understood as its boundary trace.

For \(g\in Y_{m,\rho}\), \(k\in\mathbb R\), and \(a\in P_{m,\rho}\) with \(a_0=1\), define
\[
F_{m,\rho}(g,k,a)=\bigl(g+k^2|a|^2K_Dg,\ (Ng)^2-|a|^2|_{\partial D}\bigr).
\]
Its affine domain is
\[
X_{m,\rho}=\{(g,k,a)\in Y_{m,\rho}\times\mathbb R\times P_{m,\rho}:a_0=1\},
\]
with tangent space \(X_{m,\rho}^{\rm tan}=Y_{m,\rho}\times\mathbb R\times\{\dot a\in P_{m,\rho}:\dot a_0=0\}\) and norm
\[
\|(\dot g,\dot k,\dot a)\|_X=\|\dot g\|_\rho+|\dot k|+\|\dot a\|_P.
\]
Its codomain is \(Z_{m,\rho}=Y_{m,\rho}\times B_{m,\rho}\), with norm
\(\|(y,b)\|_Z=\|y\|_\rho+\|b\|_B\).

There exist an integer \(m\ge2\), a real number \(\rho>1\), a point \(x_c=(g_c,k_c,a_c)\in X_{m,\rho}\) with \(g_c\) and \(a_c\) of finite coefficient support, a bounded injective real-linear operator
\[
A:Z_{m,\rho}\longrightarrow X_{m,\rho}^{\rm tan},
\]
and real numbers \(r>0\) and \(0\le q<1\) with the following properties.
Write
\[
B_r(x_c)=\{x\in X_{m,\rho}:\|x-x_c\|_X\le r\}.
\]
The map \(F_{m,\rho}\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \(B_r(x_c)\), and
\[
\|A F_{m,\rho}(x_c)\|_X<(1-q)r,
\qquad
\sup_{x\in B_r(x_c)}
\|I-A\,DF_{m,\rho}(x)\|_{X_{m,\rho}^{\rm tan}\to X_{m,\rho}^{\rm tan}}\le q,
\]
where \(I\) is the identity on \(X_{m,\rho}^{\rm tan}\).
For every \((g,k,a)\in B_r(x_c)\),
\[
k>0,\qquad
\operatorname{Re}a(z)>0\quad(z\in\overline D),\qquad
Ng(e^{i\theta})>0\quad(\theta\in\mathbb R),\qquad a_1\ne0.
\]

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.