Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 957-968

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

Laura Prat and Xavier Tolsa

Universitat Autònoma de Barcelona, Departament de Matemàtiques
08193 Bellaterra (Barcelona), Catalonia; laurapb 'at' mat.uab.cat

Institució Catalana de Recerca i Estudis Avançats (ICREA) and
Universitat Autònoma de Barcelona, Departament de Matemàtiques
08193 Bellaterra (Barcelona), Catalonia; xtolsa 'at' mat.uab.cat

Abstract. A measure μ on R^d is called reflectionless for the s-Riesz transform if the singular integral R^sμ(x) = ∫ y - x / |y - x|^s+1 dμ(y) is constant on the support of μ in some weak sense and, moreover, the operator defined by R^sμ(f) = R^s(fμ) 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.

2010 Mathematics Subject Classification: Primary 42B20, 28A75, 31B15.

Key words: Riesz transforms, reflectionless measure, Wolff potential, rectifiability.

Reference to this article: L. Prat and X. Tolsa: Non-existence of reflectionless measures for the s-Riesz transform when 0 < s < 1. Ann. Acad. Sci. Fenn. Math. 40 (2015), 957-968.

Full document as PDF file

doi:10.5186/aasfm.2015.4055