Hilbertian Hardy--Sobolev Spaces on Tube Domains over Convex Cones

TL;DR

This paper introduces Hilbertian Hardy--Sobolev spaces on tube domains over convex cones, providing their Fourier-analytic structure, reproducing kernels, and applications to operators, advancing the understanding of these function spaces.

Contribution

It develops the structural theory of Hardy--Sobolev spaces on tube domains over convex cones using Fourier analysis, including representations, decompositions, kernels, and measure characterizations.

Findings

Established Paley--Wiener type representation for these spaces.

Derived explicit reproducing kernels and characterized Carleson measures.

Analyzed implications for multipliers and weighted composition operators.

Abstract

We introduce Hilbertian Hardy--Sobolev spaces on tube domains over convex cones and develop their structural theory from a Fourier-analytic point of view. We first establish a Paley--Wiener type representation, which identifies these spaces with weighted $L^2$ spaces on the dual cone and reveals their intrinsic Fourier structure. This representation leads naturally to a Hardy--Sobolev decomposition theorem for boundary Sobolev spaces on $\mathbb{R}^d$. Building on these structural results, we derive explicit reproducing kernels and characterize Carleson measures for the Hilbertian Hardy--Sobolev spaces. As a preliminary operator-theoretic application, we also derive basic consequences for multipliers and weighted composition operators on these spaces.