Matrix Ap-weights relative to a pseudo-metric

Matrix weights satisfying a Muckenhoupt Ap-condition relative to a family of anisotropic balls in Rd defined by a pseudo-metric are studied. It is shown that such matrix weights satisfy a doubling condition and a reverse Hölder inequality. In the special case, where the pseudo-metric is homogeneous with respect to a one-parameter dilation group, the corresponding Muckenhoupt class is shown to satisfy an invariance property under composition with affine transformations generated by the dilation group. A general sampling theorem is derived for the matrix-weighted space Lp(W) for Muckenhoupt Ap weights W along with a corresponding multiplier result for Lp(W). An application of the results to the study of anisotropic matrix-weighted Besov spaces is considered.