Some remarks on linear closed subspaces in Bergman spaces

Abstract

Problems related to finite-dimensionality, compact embeddings, and quantitative estimates of dimensions of embedded subspaces in Bergman spaces play an important role in approximation theory, Banach space geometry, the theory of holomorphic function spaces, and the analysis of infinite-dimensional dynamical systems. Classical results of Grothendieck and Subramanian established fundamental principles for embeddings of subspaces in $L_p$ -spaces, while recent studies have extended these ideas to Bergman spaces of holomorphic functions on bounded complex domains. In this context, the investigation of quantitative dimension estimates for closed subspaces embedded into Bergman spaces with stronger integrability conditions becomes particularly relevant.   In this article the finite-dimensionality Grothendieck type problem for closed linear subspaces of a Bergman space $A_{p}(\Omega ,d\lambda )\ $of holomorphic and $p$ - integrable with respect to the Lebesgue measure $d\lambda $ on a bounded complex domain $\Omega \subset \mathbb{C}^{n},$ embedded into a Bergman space $A_{q}(\Omega, d\lambda )$ \ for $q>p\geq 1,$ is analyzed. {It is shown that if a closed linear subspace }$S_{p}^{(q)}\subset A_{p}(\Omega, d\lambda )$ ${\hookrightarrow A}_{q}(\Omega, d\lambda ),q>p\geq 1,\ $ its dimension $\dim S_{p}^{(q)}=N\in $ $\mathbb{N}$ \ proves to satisfy the numerical inequality $\frac{\omega _{N}^{2(q-1)/q}}{N}\frac{|\Omega |^{\frac{ 2-q}{q}}}{k_{N}(\xi _{0})^{2(q-1)/q}}\leq \tilde{K}_{p,q}^{2}$ \ for some bounded constant $\tilde{K}_{p,q}>0,$ where $\omega _{N}=|SU(N)|$ is the volume of the compact unitary group $SU(N),|\Omega |$ is the volume of the bounded domain $\Omega \subset \mathbb{C}^{n}$ and $k_{N}(\xi _{0})>0-$ is the corresponding homogeneity parameter at $\xi _{0}=(1,0,0,...,0)_{N}\in \partial \mathbb{D}^{N}$ $\subset \mathbb{C}^{N}$.