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