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

Haoxian Liang, Haichou Li, Tao Qian

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.

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

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.
Paper Structure (17 sections, 27 theorems, 144 equations)

This paper contains 17 sections, 27 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardy Spaces on the Half-Plane
  4. Sobolev Spaces on the Real Line
  5. Weighted Lebesgue Spaces on $\mathbb{R}^+$
  6. Function Theory of $H_n^p(\mathbb{C}^+)$
  7. Boundary Characterization of $H_n^p(\mathbb{C}^+)$
  8. Sobolev-Type Embedding and Cauchy Representation
  9. Direct-Sum Decomposition of $W_n^p(\mathbb{R})$
  10. Generalized Banach Algebra Structure
  11. The Hilbert Case $p=2$
  12. A Paley--Wiener Theorem for $H_n^2(\mathbb{C}^+)$
  13. Reproducing Kernel of $H_n^2(\mathbb{C}^+)$
  14. Comparison with $\mathscr{H}_n^2(\mathbb{C}^+)$
  15. Operator Theory on $H_n^p(\mathbb{C}^+)$
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $n\in\mathbb N^+$. A function $F$ belongs to $W_n^0(\mathbb{R})$ if and only if there exists a finite sequence $\{F_k\}_{k=0}^n\subset L^1_{\mathrm{loc}}(\mathbb{R})$ such that $F=F_0$ almost everywhere and for all $-\infty<a<b<\infty$ and $k=0,\dots,n-1$.

Theorems & Definitions (48)

  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • ...and 38 more