# On the maximum principle for the Riesz transform

## TL;DR

This paper proves a maximum principle for the Riesz transform in certain cases and demonstrates the conjecture's failure for non-positive measures, advancing understanding of Riesz transforms in harmonic analysis.

## Contribution

The paper establishes the maximum principle for the Riesz transform when 0<s<1 and for radial measures when 0<s<d, and shows the conjecture fails for non-positive measures.

## Key findings

- Maximum principle proven for 0<s<1.
- Maximum principle proven for radial measures when 0<s<d.
- Conjecture invalid for non-positive measures.

## Abstract

Let $\mu$ be a measure in $\mathbb R^d$ with compact support and continuous density, and let $$ R^s\mu(x)=\int\frac{y-x}{|y-x|^{s+1}}\,d\mu(y),\ \ x,y\in\mathbb R^d,\ \ 0<s<d. $$ We consider the following conjecture: $$ \sup_{x\in\mathbb R^d}|R^s\mu(x)|\le C\sup_{x\in\text{supp}\,\mu}|R^s\mu(x)|,\quad C=C(d,s). $$ This relation was known for $d-1\le s<d$, and is still an open problem in the general case. We prove the maximum principle for $0< s<1$, and also for $0<s<d$ in the case of radial measure. Moreover, we show that this conjecture is incorrect for non-positive measures.