Weighted inequalities for q-functions

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 Lp-estimates c−1 p,q,w‖S(f, q)‖Lp(w) ≤ ‖f‖Lp(w) ≤ Cp,q,w‖S(f, q)‖Lp(w), 1 ≤ p <∞, 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 cp,q,w and Cp,q,w on the Ap characteristics of the weights involved.