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 $mathbb {R}^d$ . For $1 , we determine the best constant $C_p$ such that the following holds. For any locally integrable function $f$ on $mathbb {R}^d$ and any $jin {1,,2,,ldots ,,d}$ , $$begin{aligned} ||(R_jf)_+||_{L^{p,infty }(mathbb {R}^d)}le C_p||f||_{L^{p,infty }(mathbb {R}^d)}. end{aligned}$$ 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.