| Given the class of Ap weights, 1 < p < ∞, we are able to define a metric d ∗ on this set such that the operator norm of any Calderón-Zygmund operator T on Lp( w), w ∈ Ap, is a continuous function with respect to w. Moreover, we find the “rate” of this continuity with respect to the weight and prove that it is sharp. This is done by finding the exact “rate” for the Hilbert transform H on the unit disk. We also study many properties of this new metric space (Ap, d∗) and identify its completion as a subset of BMO( Rd ). In addition, we extend the continuity result to the case of matrix-valued A2 weights W, for the Martingale transform MWs and we show that it does not hold for the classical Martingale transform. The problem of continuity of weighted estimates with respect to the weight appears naturally in problems of PDE (Partial Differential Equations) with random coeffcients, and can also be important to multivariate stationary processes. |
