Sharp logarithmic inequalities for Riesz transforms

Abstract

Let d be a given positive integer and let {Rj}j=1d denote the collection of Riesz transforms on Rd. For any K>2/π we determine the optimal constant L such that the following holds. For any locally integrable Borel function f on Rd, any Borel subset A of Rd and any j=1,2,…,d we have∫A|Rjf(x)|dx⩽K∫RdΨ(|f(x)|)dx+|A|⋅L. Here Ψ(t)=(t+1)log(t+1)−t for t⩾0. The proof is based on probabilistic techniques and the existence of certain special harmonic functions. As a by-product, we obtain related sharp estimates for the so-called re-expansion operator, an important object in some problems of mathematical physics.