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A note about multi-Hilbert transform on \(\mathcal D(\mathbb R^n)\). (English) Zbl 1391.42019

Summary: In this paper, we first prove that, for a non-zero function \(f\in\mathcal D(\mathbb R^n)\), its multi-Hilbert transform \(H_nf\) is bounded and does not have compact support. In addition, a new distribution space \(\mathcal D'_H(\mathbb R^n)\) is constructed and the definition of the multi-Hilbert transform is extended to it. It is shown that \(\mathcal D'_H(\mathbb R^n)\) is the biggest subspace of \(\mathcal D'(\mathbb R^n)\) on which the extended multi-Hilbert transform is a homeomorphism.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis
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