Reflectionless measures for Calderón-Zygmund operators

We study the properties of reflectionless measures for a Calderón-Zygmund operator T. Roughly speaking, these are measures $μ$ for which T(μ) vanishes (in a weak sense) on the support of the measure. We describe the relationship between certain well-known problems in harmonic analysis and geometric measure theory and the classification of reflectionless measures. As an application of our theory, we give a new proof of a recent theorem of Eiderman, Nazarov, and Volberg, which states that in $\mathbb{R}^d$, the s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is unbounded if $s\in (d-1,d)$.

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