Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory

TL;DR

This paper develops a comprehensive theory of Hardy--Sobolev spaces on the upper half-plane, including boundary characterizations, embeddings, Fourier analysis, and operator boundedness criteria, extending beyond the Hilbert space case.

Contribution

It introduces new boundary and Fourier-analytic characterizations of Hardy--Sobolev spaces and establishes operator boundedness conditions, broadening the understanding of these spaces.

Findings

Isometric boundary characterization via nontangential limits

Sobolev-type embedding theorem for Hardy--Sobolev spaces

Spectral formula for multiplication operators

Abstract

We study Hardy--Sobolev spaces H_n^p(C^+) on the upper half-plane for 1<=p<=infty and n is a nonnegative integer, from both function-theoretic and operator-theoretic viewpoints. We establish an isometric boundary characterization of H_n^p(C^+) via nontangential limits, together with a Sobolev-type embedding theorem, a Cauchy integral representation, a direct-sum decomposition of W_n^p(R) for 1<p<infty, and a generalized Banach algebra structure under pointwise multiplication. We also obtain a finer Fourier-analytic description in the Hilbert case p=2 by proving a Paley--Wiener theorem and deriving the reproducing kernel of H_n^2(C^+).On the operator-theoretic side, we prove the spectral formula for multiplication operators and establish two verifiable sufficient conditions for the boundedness of weighted composition operators. These results provide a systematic theory of Hardy--Sobolev spaces on the upper half-plane beyond the Hilbert setting.