Projections onto \(L^{p}\)-Bergman spaces of Reinhardt domains
The Bergman projection and Bergman kernel are crucial objects in several complex variables and complex geometry. However, in contrast to the \(L^2\) case, the Bergman projection operator from \(L^p(\Omega)\) to \(A^{p}(\Omega)\) may not be bounded for \(p\ne2\) and a domain \(\Omega\subseteq \mathbb{C}^n\), especially when the boundary of \(\Omega\) is not smooth.\N\NIn this paper, for Reinhardt domains \(\Omega\), using a Schauder basis in \(L^{p}(\Omega,\lambda)\) and Hahn-Banach extension, the authors define the monomial basis projection \(P^{\Omega}_{p,\lambda}:L^{p}(\Omega,\lambda)\to A^{p}(\Omega,\lambda)\), where \(\lambda\) is a weight function. When \(p=2\), the monomial basis projection \(P^{\Omega}_{2,\lambda}\) coincides with the Bergman projection with weight \(\lambda\). Furthermore, the authors prove that there is an integral expression for \(P^{\Omega}_{p,\lambda}\) and the integral kernel \(K^{\Omega}_{p,\lambda}(z,w)\) is the Bergman kernel when \(p=2\).\N\NThe authors also prove that for monomial polyhedrons (pseudoconvex Reinhardt domains) \(\mathscr{U}\) and any \(1<p<\infty\), the ``absolute'' operator \((P^{\mathscr{U}}_{p,1})^+ (f)(z):=\int_{\mathscr{U}}|K^{\mathscr{U}}_{p,1}(z,w)|f(w)dV(w)\) is bounded in \(L^p(\mathscr{U})\). This result indicates that comparing with the Bergman projection, the monomial basis projection holds better regularity for monomial polyhedrons. For proving the boundedness, the authors first compute the one dimensional case. Then they prove the equivalence of the boundedness of \((P^{\Omega_2}_{p,1})^+\) and the boundedness of the \(\Gamma\)-invariant projection \((P^{\Omega_1}_{p,\lambda_p,\Gamma})^+\), where \(\Gamma\subset \mathrm{Aut}(\Omega_1)\) is a finite group. Finally, they decompose the integral kernel of \((P^{\Omega_1}_{p,\lambda_p,\Gamma})^+\) into sums and products of integral kernels in the one dimensional case and finish the proof.\N\NThey also compute the dual of \(A^{p}(\mathscr{U})\) and get the identification \(A^{q}(\mathscr{U})'\simeq A^{p}(\mathscr{U},\eta_q)\) as an application.
scientific article; zbMATH DE number 7900904
scientific article