A note on failure of energy reversal for classical fractional singular integrals

TL;DR

This paper shows that energy reversal, a property used in harmonic analysis, fails for certain fractional Riesz transforms and similar singular integrals with specific kernel properties.

Contribution

It provides a counterexample demonstrating the failure of energy reversal for a class of fractional singular integrals with kernels having zero integral on great circles.

Findings

Energy reversal fails for alpha-fractional Riesz transforms.

Failure extends to singular integrals with kernels vanishing on great circles.

Results impact the understanding of boundedness and regularity in harmonic analysis.

Abstract

For alpha in [0,n) we demonstrate the failure of energy reversal for the vector of alpha-fractional Riesz transforms, and more generally for any vector of alpha-fractional convolution singular integrals having a kernel with vanishing integral on every great circle of the sphere.