Non-symmetric differentially subordinate martingales and sharp weak-type bounds for Fourier multipliers

Let p> 2 be a given exponent. In this paper, we prove, with the best constant, the weak-type (p,p) inequality \(\Vert T_{m}f\Vert_{L^{p,\infty}(\mathbb{R}^{d})}\leqslant C_{p}\Vert f\Vert_{L^{p}(\mathbb{R}^{d})}\) $$\parallel T_{m}f{\parallel }_{L^{p,\mathrm{\infty }}({\mathbb{R}}^d)}⩽C_p\parallel f{\parallel }_{L^p({\mathbb{R}}^d)}$$

for a large class of non-symmetric Fourier multipliers T_m obtained via modulation of jumps of certain Lévy processes. In particular, the estimate holds for appropriate linear combinations of second-order Riesz transforms and skew versions of the Beurling-Ahlfors operator on the complex plane. The proof rests on a novel probabilistic bound for Hilbert-space-valued martingales satisfying a certain non-symmetric subordination principle. Further applications to harmonic functions and Riesz systems on Euclidean domains are indicated.