Real-variable characterizations and their applications of matrix-weighted Besov spaces on spaces of homogeneous type

In this article, the authors introduce matrix-weighted Besov spaces on a given space of homogeneous type, ( X , d , μ ) , in the sense of Coifman and Weiss and prove that matrix-weighted Besov spaces are independent of the choices of both approximations of the identity with exponential decay and spaces of distributions. Moreover, the authors establish the wavelet characterization of matrix-weighted Besov spaces, introduce almost diagonal operators on matrix-weighted Besov sequence spaces, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of matrix-weighted Besov spaces. As an application, the authors obtain the boundedness of Calderón–Zygmund operators on matrix-weighted Besov spaces. One novelty is that, by using the property of matrix weights and the boundedness of matrix-weighted Hardy–Littlewood maximal operators, all the proofs presented in this article are different from those on Euclidean spaces, the latter strongly rely on the fact that Schwartz functions on Euclidean spaces decay faster than any polynomial. Another novelty is that all the results of this article get rid of both the reverse doubling condition of the measure μ and the triangle inequality of the quasi-metric d under consideration, by fully using the geometrical properties of X expressed by dyadic reference points, dyadic cubes, and wavelets. Besides, all the results in this article are new even for Ahlfors regular spaces and RD-spaces.