Global Stein Theorem on Hardy Spaces

A. Bonami; S. Grellier; B. F. Sehba

doi:10.1007/s10476-023-0223-5

Global Stein Theorem on Hardy Spaces

A. Bonami, S. Grellier, B. F. Sehba

Department of Mathematics

Université d’Orléans

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

1 Citation (Scopus)

Abstract

Let f be an integrable function which has integral 0 on ℝⁿ. What is the largest condition on ❘ f ❘ that guarantees that f is in the Hardy space ℋ¹ (ℝⁿ)? When f is compactly supported, it is well-known that the largest condition on ❘f❘ is the fact that ❘f❘ ∈ L log L(ℝⁿ). We consider the same kind of problem here, but without any condition on the support. We do so for ℋ¹ (ℝⁿ), as well as for the Hardy space ℋ_log (ℝⁿ) which appears in the study of pointwise products of functions in ℋ¹ (ℝⁿ) and in its dual BMO.

Original language	English
Journal	Analysis Mathematica
DOIs	https://doi.org/10.1007/s10476-023-0223-5
Publication status	Accepted/In press - 2023

Keywords

Hardy space
Hardy–Musielak space
Stein theorem

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T1 - Global Stein Theorem on Hardy Spaces

AU - Bonami, A.

AU - Grellier, S.

AU - Sehba, B. F.

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Y1 - 2023

N2 - Let f be an integrable function which has integral 0 on ℝ n. What is the largest condition on ❘ f ❘ that guarantees that f is in the Hardy space ℋ1 (ℝ n)? When f is compactly supported, it is well-known that the largest condition on ❘f❘ is the fact that ❘f❘ ∈ L log L(ℝ n). We consider the same kind of problem here, but without any condition on the support. We do so for ℋ1 (ℝ n), as well as for the Hardy space ℋlog (ℝ n) which appears in the study of pointwise products of functions in ℋ1 (ℝ n) and in its dual BMO.

AB - Let f be an integrable function which has integral 0 on ℝ n. What is the largest condition on ❘ f ❘ that guarantees that f is in the Hardy space ℋ1 (ℝ n)? When f is compactly supported, it is well-known that the largest condition on ❘f❘ is the fact that ❘f❘ ∈ L log L(ℝ n). We consider the same kind of problem here, but without any condition on the support. We do so for ℋ1 (ℝ n), as well as for the Hardy space ℋlog (ℝ n) which appears in the study of pointwise products of functions in ℋ1 (ℝ n) and in its dual BMO.

KW - Hardy space

KW - Hardy–Musielak space

KW - Stein theorem

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JF - Analysis Mathematica

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Global Stein Theorem on Hardy Spaces

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