Matrix Ap Weights Via Maximal Functions

The matrix Ap condition extends several results in weighted norm theory to functions taking values in a finite-dimensional vector space. Here we show that the matrix Ap condition leads to Lp-boundedness of a Hardy-Littlewood maximal function, then use this estimate to establish a bound for the weighted Lp norm of singular integral operators. 1. Preliminaries. Weighted Norm theory forms a basic component of the study of singular integrals. Here one attempts to characterize those measure spaces over which a broad class of singular integral operators remain bounded. For the case of singular integral operators on C-valued functions in Euclidean space, the answer is given by the Hunt-Muckenhoupt-Wheeden theorem [10]. It states that the necessary and sufficient condition for boundedness in Lp(dµ) is that dµ = W (x) dx and the function W satisfies the Ap condition, namely: ( 1 |B