Constructive description of Hardy-Sobolev spaces in strictly pseudoconvex domains with minimal smoothness

TL;DR

This paper characterizes Hardy-Sobolev spaces in strictly pseudoconvex domains with minimal smoothness by linking derivatives of holomorphic functions to polynomial approximation sequences in boundary integrability conditions.

Contribution

It provides a constructive description of Hardy-Sobolev spaces in domains with only $C^2$ smoothness, extending classical results to less smooth settings.

Findings

Characterization of derivatives in Hardy-Sobolev spaces via polynomial approximation.

Equivalence between derivative existence and polynomial approximation sequences.

Extension of classical theory to domains with minimal smoothness.

Abstract

Let $\Omega\subset\mathbb{C}^n$ be a strictly pseudoconvex Runge domain with $C^2$-smooth defining function, $l\in\mathbb{N},$ $p\in(1,\infty).$ We prove that the holomorphic function $f$ has derivatives of order $l$ in $H^p(\Omega)$ if and only if there exists a sequence on polynomials $P_n$ of degree $n$ such that $\sum\limits_{k=1}^{\infty}2^{2lk}\left\lvert f(z)-P_{2^k}(z) \right\rvert^2\in L^p(\partial\Omega).$