Abstract
In this paper, we first prove that, for a non-zero function f∈D(ℝn), its multi-Hilbert transform Hnf is bounded and does not have compact support. In addition, a new distribution space D' H (ℝn) is constructed and the definition of the multi-Hilbert transform is extended to it. It is shown that D' H (ℝn) is the biggest subspace of D'(ℝn) on which the extended multi-Hilbert transform is a homeomorphism.
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Supported by the National Natural Science Foundation of China (No: 11471309; 11271162; 11561062), the Nanhu Scholar Program for Young Scholars of XYNU and Doctoral Scientific Research Startup Fund of Xinyang Normal University (2016).
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Wei, Mq., Shen, F. & Yan, Dy. A Note about Multi-Hilbert Transform on D(ℝn). Acta Math. Appl. Sin. Engl. Ser. 34, 330–343 (2018). https://doi.org/10.1007/s10255-018-0761-y
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DOI: https://doi.org/10.1007/s10255-018-0761-y