Continuous and localized Riesz bases for L2L2 spaces defined by Muckenhoupt weights

Let w   be an A∞A∞-Muckenhoupt weight in RR. Let L2(wdx)L2(wdx) denote the space of square integrable real functions with the measure w(x)dxw(x)dx and the weighted scalar product 〈f,g〉w=∫Rfgwdx. By regularization of an unbalanced Haar system in L2(wdx)L2(wdx) we construct absolutely continuous Riesz bases with supports as close to the dyadic intervals as desired. Also the Riesz bounds can be chosen as close to 1 as desired. The main tool used in the proof is Cotlar's Lemma.