On the Maximum Principle for the Riesz Transform

Abstract

Let μ be a measure in ℝd with compact support and continuous density, and let Rsμ(x) = | y – x |y – x|s+1 dμ(y), x,y∈ ℝd, 0 < s < d. We consider the following conjecture: sup x∈Rd |R^sμ(x)| ≤ C sup x∈supp μ |R^sμ(x)|, C= C(d, s). This relation was known for d – 1 ≤ 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.

Dedicated to the memory of Victor Petrovich Havin, a remarkable mathematician and personality