# Berenstein rigidity — 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,
\[
N\Phi_{\ell,0}=\frac{e^{im\ell\theta}}{2(m|\ell|+1)},
\qquad N\Phi_{\ell,s}=0\quad(s\ge1).
\]