{
  "statement": "Prove 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\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(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.\n\nFor 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.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
  "starting_statement": null,
  "target_statement": "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\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(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.\n\nFor 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.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
  "case_id": "EP-1-H0-to-H1",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}