[Paper Review] On an extremal problem in analytic function spaces in tube domains over symmetric cones

Romi F. Shamoyan|arXiv (Cornell University)|May 15, 2012

Holomorphic and Operator Theory2 references3 citations

TL;DR

This paper establishes new sharp distance estimates in Bergman-type spaces of analytic functions on tube domains over symmetric cones, using the Bergman reproducing formula and generalized power functions. The key contribution is proving equivalence of distance norms in weighted Hardy and Bergman spaces, extending prior one-dimensional results to higher-rank symmetric cone settings with precise operator-theoretic bounds.

ABSTRACT

New sharp estimates concerning distance function in Bergman - type analytic function spaces on tube domains over symmetric cones are obtained. These are first results of this type in tube domains over symmetric cones.

Motivation & Objective

• To extend sharp distance estimates from one-dimensional tube domains to higher-rank symmetric cone settings.
• To establish new norm equivalence results for analytic function spaces on tube domains over symmetric cones.
• To generalize prior results on distance problems in Bergman and Hardy spaces to mixed-norm and weighted settings.
• To prove sharp estimates for distances from functions in $A^{ u}_{p,q}$ and $H^p_\alpha$ spaces to subspaces via integral norms involving the Bergman kernel and determinant functions.

Proposed method

• Utilizes the Bergman reproducing formula in tube domains over symmetric cones as the foundational analytical tool.
• Employs generalized power functions $\Delta^\nu(y)$ and determinant functions $\Delta(y)$ to define weighted norms in Bergman and Hardy spaces.
• Applies integral representations involving the Bergman kernel $B_\nu(z,w)$ to relate function norms to integral estimates over $T_\Omega$.
• Uses multi-index notation $t^\star$, $t < k$, and parameter conditions like $\nu > \frac{n}{r} - 1$ to control integrability and embeddings.
• Establishes norm equivalence via comparison of distance functionals $l_1(f)$ and $l_2(f)$ through integral convergence criteria.
• Relies on known embeddings between $A^{p,q}_\nu$, $B^{p,q}_\nu$, $H^p_\alpha$, and $A^\infty_\tau$ spaces to derive sharp estimates.

Experimental results

Research questions

• RQ1 What are the sharp distance estimates between analytic functions and their projections in Bergman-type spaces on tube domains over symmetric cones?
• RQ2 How do the norms of the distance functionals $l_1(f)$ and $l_2(f)$ relate in mixed-norm Bergman spaces $A^{p,q}_\nu$?
• RQ3 Can the distance from a function in $H^1_\nu$ to $A^1_\nu$ be characterized via an integral condition on level sets $L_{\epsilon,t}(f)$?
• RQ4 What conditions on parameters $\nu$, $p$, $q$, $\beta$, and $\alpha$ ensure the validity of the Bergman representation and embedding theorems in this setting?
• RQ5 How do the results generalize prior one-dimensional distance estimates in the upper half-plane to higher-rank symmetric cone domains?

Key findings

• For $\nu > \frac{n}{r} - 1$, the distance $l_1(f) = \text{dist}_{A^\infty_{\frac{n}{rp} + \frac{\nu}{p}}}(f, A^p_\nu)$ is bounded above by $C l_2(f)$, where $l_2(f)$ is defined via an integral involving the Bergman kernel and generalized power functions.
• The distance $l_1(f)$ in $H^1_\nu$ to $A^1_\nu$ is equivalent to $l_2(f)$, defined as the infimum of $\epsilon > 0$ such that $\int_\Omega \tau_{L_{\epsilon,t}(f)}(y) \Delta^{-n/r}(y) dy < \infty$, with $C$-dependence on parameters.
• For $0 < p < 1$, under the embedding $A^p_\nu \subset A^\infty_t$ and valid Bergman representation, $l_1(f) \leq C l_2(f)$ holds for $\beta > \beta_0$, with $\beta_0$ depending on $p$, $\nu$, and $n/r$.
• The embedding $A^{p,q}_\nu \subset H^s_\beta$ holds with $\beta = \frac{\nu}{q} + \frac{n}{rp} - \frac{n}{rs}$ for $q \leq s$, enabling distance problems in Hardy spaces.
• The results generalize one-dimensional distance estimates from [15] to higher-rank symmetric cone tube domains using the same core techniques based on the Bergman kernel and determinant function estimates.
• The sharpness of the estimates is confirmed by showing equivalence $l_1(f) \asymp l_2(f)$ in the $H^1_\nu$ case, extending known results in the unit ball and upper half-plane to this broader class of domains.