Weighted inequalities for q-functions

Abstract

Let f be a martingale on an arbitrary atomic probability space equipped with a tree-like structure and let $S(f,q)$ denote the associated q-function. The paper is devoted to weighted $L^p$-estimates,

$$c_{p,q,w}^{-1}{\Vert S(f,q)\Vert }_{L^p(w)}\le \Vert f{\Vert }_{L^p(w)}\le C_{p,q,w}{\Vert S(f,q)\Vert }_{L^p(w)},1\le p<\mathrm{\infty },$$

for Muckenhoupt weights. Using the combination of the theory of sparse operators, extrapolation, and Bellman function method, we identify the optimal dependence of the constants $c_{p,q,w}$ and $C_{p,q,w}$ on the $A_p$ characteristics of the weights involved.