Matrix $$A_p$$-weights relative to a pseudo-metric

Abstract Matrix weights satisfying a Muckenhoupt $$A_p$$ A p -condition relative to a family of anisotropic balls in $${{\mathbb {R}}}^d$$ R d 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 $$L^p(W)$$ L p ( W ) for Muckenhoupt $$A_p$$ A p weights W along with a corresponding multiplier result for $$L^p(W)$$ L p ( W ) . An application of the results to the study of anisotropic matrix-weighted Besov spaces is considered.