Interpolation along manifolds in Hardy-Sobolev spaces
type: Item
instance of: scholarly article
title: Interpolation along manifolds in Hardy-Sobolev spaces
zbMATH Open document ID: 0918.32004
DOI: 10.1007/BF02921703
author: Joaquim Bruna
author: J. M. Ortega
published in: The Journal of Geometric Analysis
publication date: 1999-08-17T00:00:00Z
review text: Let \(B\) be the unit ball in \(\mathbb{C}^n\) and \(S\) be its boundary. For a holomorphic function \(f\) on \(B\) with a homogeneous expansion \(\Sigma_k f_k\), the fractional derivative of order \(s>0\) is defined by \(R^s(f)= \Sigma_k (k+1)^sf_k\). For \(p\geq 0\) and \(s>0\) the Hardy-Sobolev space \(H^p_s\) consists of those holomorphic functions on \(B\) for which \(\| f\|^p_{p,s} =\sup_{r<1}\int_S | R^sf(r) |^pdm< +\infty\) where \(dm\) denotes the normalised Lebesgue measure on \(S\). The aim of this paper is to study the trace of \(H^p_s\) on closed complex tangential submanifolds \(\Gamma\) of \(S\), that is to study the restriction on \(\Gamma\) not only of the function \(f\in H^p_s\) itself but of its fractional derivatives in direction to given vector field that may be or not be tangential to \(S\) as well. The main result states that each \(f\in H^p_s\) induces on \(\Gamma\) a ``jet'' in Besov type class \(B^p_\beta (\Gamma)\), where the smoothness index \(\beta\) depends on \(p,s\) and the dimension \(d\) of \(\Gamma\).
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 32A40
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1192502
zbMATH Keywords: interpolation
zbMATH Keywords: Hardy-Sobolev space
MaRDI profile type: Publication
cites work: Exceptional sets for holomorphic Sobolev functions
cites work: Maximal and area integral characterizations of Hardy-Sobolev spaces in the unit ball of \({\mathbb{C}}^ n\)
cites work: Q3032805
cites work: Strong \(L^ p\) estimates for maximal functions with respect to singular measures; with applications to exceptional sets
cites work: Estimates for derivatives of holomorphic functions in pseudoconvex domains
cites work: Boundary Regularity for Harmonic Hardy-Sobolev Spaces
cites work: Henkin-Ramirez kernels with weight factors
cites work: Holomorphic approximation and estimates for the \({\bar \partial}\)-equation on strictly pseudoconvex nonsmooth domains
cites work: Interpolation by holomorphic functions smooth to the boundary in the unit ball of \({\mathbb{C}}^ n\)
cites work: Traces on curves of Sobolev spaces of holomorphic functions
cites work: A Whitney extension theorem in \(L^p\) and Besov spaces
cites work: Smooth zero sets and interpolation sets for some algebras of holomorphic functions on strictly pseudoconvex domains
description: scientific article; zbMATH DE number 1192502