Singular integrals on Sierpinski gaskets

TL;DR

This paper constructs singular integral operators on Sierpinski gaskets that are bounded in L^2 but have divergent principal values almost everywhere, revealing complex behaviors of such operators on fractal sets.

Contribution

It introduces a new class of Calderón-Zygmund singular integrals on fractal gaskets and analyzes their boundedness and divergence properties.

Findings

Operators are bounded in L^2(rac{d}{ ext{measure}})

Principal values diverge rac{d}{ ext{measure}} almost everywhere

Extends understanding of singular integrals on fractal geometries

Abstract

We construct a class of singular integral operators associated with homogeneous Calder\'{o}n-Zygmund standard kernels on $d$-dimensional, $d <1$, Sierpinski gaskets $E_d$. These operators are bounded in $L^2(\mu_d)$ and their principal values diverge $\mu_d$ almost everywhere, where $\mu_d$ is the natural (d-dimensional) measure on $E_d$.