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Sharp stability of convex functionals on weighted Bergman spaces with applications

Petar Melentijević

TL;DR

The paper proves a sharp quantitative stability for convex functionals on weighted Bergman spaces, identifying optimal exponents and constants and providing asymptotically sharp behavior as α approaches −1 and +∞. The authors leverage concentration inequalities, majorization, and localization to show that extremizers are reproducing kernels and extend the results to higher dimensions and to Hardy spaces. They also present two proofs for the operator-induced functional case, connect the theory to Cauchy wavelets, and derive analogous stability results in the unit ball of C^n. Applications include recovering Fock-space results and interpreting the Cauchy wavelet transform within this convex-functional stability framework.

Abstract

Recently, Kulikov (\cite{Ku}) has shown that certain convex functionals on weighted Bergman spaces are maximized by reproducing kernels. We show a sharp quantitative stability of these estimates with the optimal norm and the exponent and an explicit constant asymptotically sharp in both directions ($α\rightarrow -1$ and $α\rightarrow +\infty$). Several applications of this result include recovering the appropriate result for Fock spaces, interpretation to Cauchy wavelets, and the Hardy space counterpart for functionals induced by increasing function. In addition, we prove a higher-dimensional analog of the main result assuming that all convex functionals on the weighted Bergman space $\mathcal{A}^2_α(\mathbb{B}_n)$ attain their extrema in reproducing kernels.

Sharp stability of convex functionals on weighted Bergman spaces with applications

TL;DR

The paper proves a sharp quantitative stability for convex functionals on weighted Bergman spaces, identifying optimal exponents and constants and providing asymptotically sharp behavior as α approaches −1 and +∞. The authors leverage concentration inequalities, majorization, and localization to show that extremizers are reproducing kernels and extend the results to higher dimensions and to Hardy spaces. They also present two proofs for the operator-induced functional case, connect the theory to Cauchy wavelets, and derive analogous stability results in the unit ball of C^n. Applications include recovering Fock-space results and interpreting the Cauchy wavelet transform within this convex-functional stability framework.

Abstract

Recently, Kulikov (\cite{Ku}) has shown that certain convex functionals on weighted Bergman spaces are maximized by reproducing kernels. We show a sharp quantitative stability of these estimates with the optimal norm and the exponent and an explicit constant asymptotically sharp in both directions ( and ). Several applications of this result include recovering the appropriate result for Fock spaces, interpretation to Cauchy wavelets, and the Hardy space counterpart for functionals induced by increasing function. In addition, we prove a higher-dimensional analog of the main result assuming that all convex functionals on the weighted Bergman space attain their extrema in reproducing kernels.

Paper Structure

This paper contains 14 sections, 9 theorems, 163 equations.

Table of Contents

  1. Introduction and main result
  2. Convex functionals on weighted Bergman spaces
  3. Optimal stability for $\mathcal{A}^2_{\alpha}$- proof via concentration estimates and majorization theory
  4. Case $(1+\frac{s}{\pi})^{\alpha+2}\leqslant \frac{2}{t}$
  5. Case $(1+\frac{s}{\pi})^{\alpha+2}\geqslant \frac{2}{t}$
  6. Asymptotics $\alpha\rightarrow +\infty$ and stability of convex functionals acting on Fock space
  7. Case $\alpha=-1$ and stability of functionals on Hardy space
  8. Sharpness of the stability exponent
  9. Stability for convex functionals associated to a semi-definite positive operator on the weighted Bergman space. The application to the Cauchy wavelets
  10. First proof.
  11. Second proof.
  12. Stability of the estimate for the weighted Bergman spaces in the unit ball in $\mathbb{C}^n$
  13. Hardy space counterpart
  14. Acknowledgements

Key Result

Theorem 1

Let $G:[0,1]\rightarrow\mathbb{R}$ be a continuous and convex non-linear function such that $G(0)=0.$ Then, for every $f\in\mathcal{A}^2_{\alpha}(\mathbb{D})$ there exists a constant $c_{G}$ depending on $G$ and $\alpha>-1$ for which

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Remark 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 5 more