Sharp weak type estimates for Riesz transforms

Let d be a given positive integer and let { R j } j = 1 d denote the collection of Riesz transforms on R d . For 1 < p < ∞ , we determine the best constant C p such that the following holds. For any locally integrable function f on R d and any j ∈ { 1 , 2 , … , d } , | | ( R j f ) + | | L p , ∞ ( R d ) ≤ C p | | f | | L p , ∞ ( R d ) . A related statement for Riesz transforms on spheres is also established. The proofs exploit Gundy–Varopoulos representation of Riesz transforms and appropriate inequality for orthogonal martingales.