Interpolation of Matrix Weighted Spaces and Commutator Estimates

TL;DR

This work extends interpolation theory to matrix-weighted $L^p$ spaces by establishing a general interpolation formula for vector-valued matrix weights via interpolation functors, applicable to both the complex and real methods. It shows that the complex-interpolation space satisfies $$(L^{p_0}(W_0), L^{p_1}(W_1))_{\theta} = L^{p_{\theta}}(|W_1W_0^{-1}|^{\theta}W_0)$$ and provides parallel real-interpolation characterizations, including Beurling/Lorentz-type constructions; the diagonalization reduction plays a key role. Through the differential/differentiation process, the paper derives commutator estimates for Calderón–Zygmund operators with matrix $BMO$ symbols, including higher-order iterates, and characterizes logarithms of matrix $A_2$ weights in terms of $BMO$. The results advance matrix-weighted harmonic analysis by linking interpolation, $A_p$ classes, and commutator theory, with implications for operator bounds in weighted, matrix-valued settings.

Abstract

We present a formula for the interpolation of matrix weighted spaces of vector valued functions via interpolation functors. We apply our formula to the particular case of interpolation of matrix weighted $L^p$ spaces by the real and complex methods, and present consequences regarding the matrix Muckenhoupt classes and commutator estimates of Calderón-Zygmund operators with matrix $BMO$ functions. In particular, we characterize the logarithms of Muckenhoupt weights through higher order commutators.

Theorems & Definitions (20)

• Theorem 2.1: FreitagGilbert
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• proof
• Remark 3.5
• Proposition 4.1
• Theorem 4.2
• proof