NON-EXISTENCE OF REFLECTIONLESS MEASURES FOR THE s-RIESZ TRANSFORM WHEN 0 < s < 1

Laura Prat; Xavier Tolsa

doi:10.5186/aasfm.2015.4055

NON-EXISTENCE OF REFLECTIONLESS MEASURES FOR THE s-RIESZ TRANSFORM WHEN 0 < s < 1

Laura Prat, Xavier Tolsa

Department of Mathematics

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

2 Citations (Scopus)

Abstract

A measure μ on Rd is called reflectionless for the s-Riesz transform if the singular integral (Formula presented) is constant on the support of μ in some weak sense and, moreover, the operator defined by (Formula presented) is bounded in L2(μ). In this paper we show that the only reflectionless measure for the s-Riesz transform is the zero measure when 0 < s < 1.

Original language	English
Pages (from-to)	957-968
Number of pages	12
Journal	Annales Academiae Scientiarum Fennicae Mathematica
Volume	40
DOIs	https://doi.org/10.5186/aasfm.2015.4055
Publication status	Published - 1 Jan 2015

Keywords

rectifiability
reflectionless measure
Riesz transforms
Wolff potential

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title = "NON-EXISTENCE OF REFLECTIONLESS MEASURES FOR THE s-RIESZ TRANSFORM WHEN 0 < s < 1",

abstract = "A measure μ on Rd is called reflectionless for the s-Riesz transform if the singular integral (Formula presented) is constant on the support of μ in some weak sense and, moreover, the operator defined by (Formula presented) is bounded in L2(μ). In this paper we show that the only reflectionless measure for the s-Riesz transform is the zero measure when 0 < s < 1.",

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N2 - A measure μ on Rd is called reflectionless for the s-Riesz transform if the singular integral (Formula presented) is constant on the support of μ in some weak sense and, moreover, the operator defined by (Formula presented) is bounded in L2(μ). In this paper we show that the only reflectionless measure for the s-Riesz transform is the zero measure when 0 < s < 1.

AB - A measure μ on Rd is called reflectionless for the s-Riesz transform if the singular integral (Formula presented) is constant on the support of μ in some weak sense and, moreover, the operator defined by (Formula presented) is bounded in L2(μ). In this paper we show that the only reflectionless measure for the s-Riesz transform is the zero measure when 0 < s < 1.

KW - rectifiability

KW - reflectionless measure

KW - Riesz transforms

KW - Wolff potential

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VL - 40

SP - 957

EP - 968

ER -

NON-EXISTENCE OF REFLECTIONLESS MEASURES FOR THE s-RIESZ TRANSFORM WHEN 0 < s < 1

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Keywords

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