The hausdorff operator is bounded on the real hardy space H1(ℝ)

Elijah Liflyand; Ferenc Móricz

doi:10.1090/s0002-9939-99-05159-x

The hausdorff operator is bounded on the real hardy space H¹(ℝ)

Elijah Liflyand, Ferenc Móricz

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

143 Scopus citations

Abstract

We prove that the Hausdorff operator generated by a function ∈ L¹(ℝ) is bounded on the real Hardy space H¹(ℝ). The proof is based on the closed graph theorem and on the fact that if a function f in L¹ (ℝ) is such that its Fourier transform f̂(t) equals 0 for t < 0 (or for t > 0), then f ∈ H¹(ℝ).

Original language	English
Pages (from-to)	1391-1396
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	128
Issue number	5
DOIs	https://doi.org/10.1090/s0002-9939-99-05159-x
State	Published - 2000

Keywords

Cesàro operator
Closed graph theorem
Fourier transform
Hausdorff operator
Hubert transform
Real hardy space H(ℝ)

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AB - We prove that the Hausdorff operator generated by a function ∈ L1(ℝ) is bounded on the real Hardy space H1(ℝ). The proof is based on the closed graph theorem and on the fact that if a function f in L1 (ℝ) is such that its Fourier transform f̂(t) equals 0 for t < 0 (or for t > 0), then f ∈ H1(ℝ).

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KW - Closed graph theorem

KW - Fourier transform

KW - Hausdorff operator

KW - Hubert transform

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The hausdorff operator is bounded on the real hardy space H¹(ℝ)

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