Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory
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| Format: | Preprint |
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2026
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| _version_ | 1866918388162363392 |
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| author | Liang, Haoxian Li, Haichou Qian, Tao |
| author_facet | Liang, Haoxian Li, Haichou Qian, Tao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_13983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory Liang, Haoxian Li, Haichou Qian, Tao Functional Analysis 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. |
| title | Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2603.13983 |