Even singular integral operators that are well behaved on a purely unrectifiable set

TL;DR

This paper demonstrates the existence of a specific unrectifiable set where certain even singular integral operators remain bounded, highlighting unique geometric and analytical properties.

Contribution

It establishes the existence of a purely unrectifiable set supporting boundedness of even singular integral operators, a novel result in geometric measure theory.

Findings

Existence of a (d-2)-dimensional purely unrectifiable set

Boundedness of a family of even singular integral operators on this set

Insights into the behavior of singular integrals on irregular sets

Abstract

We prove the existence of a $(d-2)$-dimensional purely unrectifiable set upon which a family of \emph{even} singular integral operators is bounded.