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

Matrix weights satisfying a Muckenhoupt $A_p$-condition relative to a family of anisotropic balls in $\mathbb{R}^d$ defined by a pseudo-metric are studied. It is shown that such matrix weights satisfy a doubling condition and a reverse H\"older inequality. In the special case, where the pseudo-metric is homogeneous with respect to a one-parameter dilation group, the corresponding Muckenhoupt class is shows 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 $L^p(W)$ for Muckenhoupt $A_p$ weights $W$ along with a corresponding multiplier result for $L^p(W)$. An application of the results to the study of anisotropic matrix-weighed Besov spaces is considered.