Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality

Abstract

In this paper we study some questions in connection with uniform rectifiability and the L2 boundedness of Calderón–Zygmund operators (CZOs). We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients that are related to the Jones’ β numbers. We also use these new coefficients to prove that n‐dimensional CZOs with odd kernel of type 𝒞2 are bounded in L2(μ), if μ is an n‐dimensional uniformly rectifiable measure.

Variation for the Riesz transform and uniform rectifiability

“…The following proposition is a direct consequence of the techniques used in the last section of [To4]. We give the proof for completeness.…”

“…D is also small enough, since it is bounded by R∈D: Q 0 ⊂R⊂C 2 D α µ (C 2 R) (see [To4,Lemma 5.2] for a related argument). It is not hard to show that then…”

“…The β ∞,µ coefficients were first introduced by P. Jones in his celebrated work on rectifiability [Jn], while the β p,µ 's for 1 ≤ p < ∞ were introduced by G. David and S. Semmes in their pioneering work on uniform rectifiability (see [DS1] for example). Other coefficients that have been proved useful in the study of uniform rectifiability and boundedness of Calderón-Zygmund operators are the α coefficients introduced in [To4]. Let F ⊂ R d be the closure of an open set.…”

“…The following result characterizes the uniform rectifiability of µ in terms of the α and β coefficients (see [DS1] for (a) ⇐⇒ (b) and [To4] for (a) ⇐⇒ (c)).…”

“…However, this is not enough to ensure the uniform n-rectifiability of µ. We will prove the uniform n-rectifiability by arguments partially inspired by some of the techniques in [To4].…”