Sharp estimates for martingale transforms with unbounded transforming sequences

Abstract

Suppose that $p,q,r\ge 1$ satisfy the condition $\frac{1}{p}=\frac{1}{q}+\frac{1}{r}$. The paper contains the identification of the best constants $c_{p,q,r}$ and $C_{p,q,r}$ in the estimates

$$\Vert g{\Vert }_{p,\mathrm{\infty }}\le c_{p,q,r}\Vert f{\Vert }_q\Vert v^{\ast }{\Vert }_r,\Vert g{\Vert }_p\le C_{p,q,r}\Vert f{\Vert }_q\Vert v^{\ast }{\Vert }_r$$

where f is an arbitrary Hilbert-space valued martingale, g is its transform by a predictable sequence v, and $v^{\ast }$ is the maximal function of v. This is extended to the more general context of differential subordination for continuous-time processes.