Some Calderón–Zygmund kernels and their relations to Wolff capacities and rectifiability

Vasilis Chousionis, Laura Prat

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

Abstract

© 2015, Springer-Verlag Berlin Heidelberg. We consider the Calderón–Zygmund kernels (Formula presentd.) in Rd for 0<α≤1 and n∈N. We show that, on the plane, for 0<α<1, the capacity associated to the kernels Kα,n is comparable to the Riesz capacity (Formula presented.) of non-linear potential theory. As consequences we deduce the semiadditivity and bilipschitz invariance of this capacity. Furthermore we show that for any Borel set E⊂Rd with finite length the L2(H1⌊E)-boundedness of the singular integral associated to K1,n implies the rectifiability of the set E. We thus extend to any ambient dimension, results previously known only in the plane.
Original languageEnglish
Pages (from-to)435-460
JournalMathematische Zeitschrift
Volume282
Issue number1-2
DOIs
Publication statusPublished - 1 Feb 2016

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