Besov--Triebel--Lizorkin-type spaces with matrix \bm{<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="TIMEQ1"><mml:msub><mml:mi>A</mml:mi><mml:mi>∞</mml:mi></mml:msub></mml:math></inline-formula>}weights

Introduced by A. Volberg (1997), matrix $A_{p,\infty}$ weights provide a suitable generalization of Muckenhoupt $A_\infty$ weights from the classical theory. In our previous work, we established new characterizations of these weights.Here, we use these results to study inhomogeneousBesov-type and Triebel–Lizorkin-type spaces with such weights.In particular, we characterize these spaces, in terms of the $\varphi$-transform, molecules, and wavelets,and obtain the boundedness of almost diagonal operators, pseudo-differential operators, trace operators, pointwise multipliers, and Calderón–Zygmund operators on these spaces.This is the first systematic study of inhomogeneous Besov–Triebel–Lizorkin-typespaces with $A_{p,\infty}$-matrix weights, but some of the results are new even when specialized to the scalar unweighted case.