On the Operator ℒ(f) = f log |f|

Tadeusz Iwaniec; Anne Verde

doi:10.1006/jfan.1999.3443

On the Operator ℒ(f) = f log |f|

Tadeusz Iwaniec, Anne Verde

Department of Mathematics

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

58 Scopus citations

Abstract

The present note deals with the operator ℒ: ℋ¹(ℝⁿ) → script D sign¹(ℝⁿ) which takes a given element f of the Hardy space to the function f log |f|. In general, this function need not be locally integrable. Nevertheless, due to peculiar cancellations of large positive and negative terms in the integral (Latin small letter esh) φ f log \f\ with φ ∈ C^∞₀ (ℝⁿ), we are able to give meaning to f log |f| as a Schwartz distribution. We find several alternatives for this interpretation of f log |f|.

Original language	English (US)
Pages (from-to)	391-420
Number of pages	30
Journal	Journal of Functional Analysis
Volume	169
Issue number	2
DOIs	https://doi.org/10.1006/jfan.1999.3443
State	Published - Dec 20 1999

Keywords

Distributions
Hardy space
Jacobians

ASJC Scopus subject areas

Analysis

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abstract = "The present note deals with the operator ℒ: ℋ1(ℝn) → script D sign1(ℝn) which takes a given element f of the Hardy space to the function f log |f|. In general, this function need not be locally integrable. Nevertheless, due to peculiar cancellations of large positive and negative terms in the integral (Latin small letter esh) φ f log \f\ with φ ∈ C∞0 (ℝn), we are able to give meaning to f log |f| as a Schwartz distribution. We find several alternatives for this interpretation of f log |f|.",

keywords = "Distributions, Hardy space, Jacobians",

author = "Tadeusz Iwaniec and Anne Verde",

note = "Funding Information: 1Research supported by NSF Grant DMS 97-06611, and in part by GNAFA-CNR, Firenze.",

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AU - Iwaniec, Tadeusz

AU - Verde, Anne

N1 - Funding Information: 1Research supported by NSF Grant DMS 97-06611, and in part by GNAFA-CNR, Firenze.

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N2 - The present note deals with the operator ℒ: ℋ1(ℝn) → script D sign1(ℝn) which takes a given element f of the Hardy space to the function f log |f|. In general, this function need not be locally integrable. Nevertheless, due to peculiar cancellations of large positive and negative terms in the integral (Latin small letter esh) φ f log \f\ with φ ∈ C∞0 (ℝn), we are able to give meaning to f log |f| as a Schwartz distribution. We find several alternatives for this interpretation of f log |f|.

AB - The present note deals with the operator ℒ: ℋ1(ℝn) → script D sign1(ℝn) which takes a given element f of the Hardy space to the function f log |f|. In general, this function need not be locally integrable. Nevertheless, due to peculiar cancellations of large positive and negative terms in the integral (Latin small letter esh) φ f log \f\ with φ ∈ C∞0 (ℝn), we are able to give meaning to f log |f| as a Schwartz distribution. We find several alternatives for this interpretation of f log |f|.

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KW - Hardy space

KW - Jacobians

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On the Operator ℒ(f) = f log |f|

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Keywords

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