Sharp logarithmic inequalities for Riesz transforms

Let d   be a given positive integer and let {Rj}j=1d denote the collection of Riesz transforms on RdRd. For any K>2/πK>2/π we determine the optimal constant L such that the following holds. For any locally integrable Borel function f   on RdRd, any Borel subset A   of RdRd and any j=1,2,…,dj=1,2,…,d we have∫A|Rjf(x)|dx⩽K∫RdΨ(|f(x)|)dx+|A|⋅L. Here Ψ(t)=(t+1)log(t+1)−tΨ(t)=(t+1)log(t+1)−t for t⩾0t⩾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.