Reflectionless measures for Calderón-Zygmund operators

2013/09/25 by Benjamin Jaye, Fëdor Nazarov, Jaye, Benjamin +1
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1309.6661

openalex publication_date 2013/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ℝ^d, the s-dimensional Riesz transform of a non-trivial s-dimensional measure is unbounded if s∈ (d-1,d).

Citations

• On the uniform rectifiability of AD regular measures with bounded Riesz\n transform operator: the case of codimension 1

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