Boundedness of the square function and rectifiability

TL;DR

This paper demonstrates that if the square function related to Riesz transforms is finite on a set with respect to Hausdorff measure, then the set must be rectifiable, linking harmonic analysis to geometric measure theory.

Contribution

It establishes a new criterion connecting the finiteness of a square function to the rectifiability of sets, extending previous work by X. Tolsa.

Findings

Finiteness of the square function implies rectifiability of the set.

Extends the understanding of the relationship between harmonic analysis and geometric measure theory.

Provides a new characterization of rectifiable sets via Riesz transforms.

Abstract

Following a recent paper by X. Tolsa [JFA, 2008] we show that the finiteness of square function associated with the Riesz transforms with respect to Hausdorff measure $H^n$ ($n$ is interger) on a set $E$ implies that $E$ is rectifiable.