L²-bounded singular integrals on a purely unrectifiable set in\n \ℝ^d

2019/12/24 by Joan Mateu, Mateu, Joan, Laura Prat +2
Mathematics · #42B20 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1912.11257

openalex publication_date 2019/12/24 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We construct an example of a purely unrectifiable measure \μ in\n\ℝ^d for which the singular integrals associated to the kernels\n\K(x)= fracP_2k+1(x)|x|^2k+d, with k\≥ 1 and\nP_2k+1 a homogeneous harmonic polynomial of degree 2k+1, are bounded in\nL²(\μ). This contrasts starkly with the results concerning the Riesz kernel\n\ fracx|x|^d in \ℝ^d.\n

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