On the maximum principle for the Riesz transform

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: $$\underset{x\in {\mathbb{R}}^d}{\sup }|R^s\mu (x)|\le C\underset{x\in \text{supp}\mu }{\sup }|R^s\mu (x)|,C=C(d,s).$$ 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.