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

Haoxian Liang; Haichou Li; Tao Qian

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arXiv · 2603.13983

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

Haoxian Liang·Haichou Li·Tao Qian

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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.

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BibTeX RIS

Haoxian Liang, Haichou Li, Tao Qian. 2026-03-14. Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory. https://arxiv.org/abs/2603.13983

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