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A measure µ on R d is called reflectionless for the s-Riesz transform if the singular integral R s µ(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 s µ(f) = R s (fµ) is bounded in L 2 (µ). In this paper we show that the only reflectionless measure for the s-Riesz transform is the zero measure when 0 < s < 1.

Non-existence of reflectionless measures for the s-Riesz transform when 0 < s < 1