Matrix-weighted Besov spaces

Abstract

Nazarov, Treil and Volberg defined matrix Apweights and extended the theory of weighted norm inequalities on Lpto the case of vector-valued functions. We develop some aspects of Littlewood-Paley function space theory in the matrix-weight setting. In particular, we introduce matrixweighted homogeneous Besov spaces Ḃpαq(W) and matrix-weighted sequence Besov spaces ḃpαq(W), as well as ḃpαq({AQ}), where the AQare reducing operators for W. Under any of three different conditions on the weight W, we prove the norm equivalences ||f→||Ḃp αq (W)≈ ||{SQ→}Q||ḃ p αq (W)≈ || {SQ→}Q|| ḃp αq ({AQ})where {SQ→}Qis the vector-valued sequence of φ-transform coefficients of f→. In the process, we note and use an alternate, more explicit characterization of the matrix Apclass. Furthermore, we introduce a weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on ḃpαq(W) if W is doubling. We also obtain the boundedness of almost diagonal operators on Ḃpαq(W) under any of the three conditions on W. This leads to the boundedness of convolution and non-convolution type Calderón-Zygmund operators (CZOs) on Ḃpαq(W), in particular, the Hilbert transform. We apply these results to wavelets to show that the above norm equivalence holds if the φ-transform coefficients are replaced by the wavelet coefficients. Finally, we construct inhomogeneous matrix-weighted Besov spaces Bpαq(W) and show that results corresponding to those above are true also for the inhomogeneous case.