New matrix-weighted norm inequalities and extrapolation

Abstract

In this paper, we propose a new type of matrix-weighted theory and show the corresponding Rubio de Francia extrapolation theorems by means of noncommutative maximal inequalities. We introduce a new matrix $A_p$ condition following Volberg's geometric formulation, and characterize it via a noncommutative Hardy--Littlewood maximal inequality. We then establish a noncommutative analogue of Rubio de Francia's abstract extrapolation theorem for trace-finite von Neumann algebras. The argument extends to $L_{\infty }(\mathbb{R})\bar{\otimes }{M}_d(\mathbb{C})$ through pointwise matrix operations. This yields the desired extrapolation results for individual operators and maximal families, with applications to matrix-weighted boundedness of Calderón--Zygmund operators.

arXiv comment27 pages