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

Haichou Li, Tao Qian

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.

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

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 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 . 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.
Paper Structure (31 sections, 19 theorems, 203 equations)

This paper contains 31 sections, 19 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex cones and tube domains
  4. Hardy spaces on tube domains
  5. Fourier transform and Fourier--Laplace representations
  6. Sobolev spaces on Euclidean space
  7. Hilbertian Hardy--Sobolev Spaces on Tube
  8. A weighted Fourier-side definition
  9. Well-definedness and Hilbert structure
  10. Basic analytic properties
  11. Paley--Wiener Characterization
  12. Boundary translates
  13. Paley--Wiener characterization
  14. Analytic derivative estimates
  15. A Hardy--Sobolev decomposition
  16. ...and 16 more sections

Key Result

Theorem 3.4

For every $n\in\mathbb{N}$, the Fourier--Laplace transform is injective, and its range is precisely $H^n_{2,\rho}(T_\Omega)$. Consequently, the formula defines an inner product on $H^n_{2,\rho}(T_\Omega)$, with respect to which $H^n_{2,\rho}(T_\Omega)$ is a Hilbert space. In particular, $\mathcal{L}$ is an isometric isomorphism from $L^2(\Omega^\ast,w_n)$ onto $H^n_{2,\rho}(T_\Omega)$.

Theorems & Definitions (46)

  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Corollary 3.7
  • ...and 36 more