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In this paper, we first prove that, for a non-zero function f∈D(ℝ n ), its multi-Hilbert transform Hnf is bounded and does not have compact support. In addition, a new distribution space D' H (ℝ n ) is constructed and the definition of the multi-Hilbert transform is extended to it. It is shown that D' H (ℝ n ) is the biggest subspace of D'(ℝ n ) on which the extended multi-Hilbert transform is a homeomorphism.
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Apr 26, 2018
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