Interpolation of Directional Banach Spaces and Commutator Estimates

Abstract

We generalize the complex interpolation formula $(X,X^{\prime }{)}_{\frac{1}{2}}=L^2$ from the context of Banach function spaces to that of directional Banach spaces. We also obtain a formula for the interpolation of matrix weighted spaces of vector valued functions via interpolation functors, and apply our formula to the particular case of interpolation of matrix weighted $L^p$ spaces by the real and complex methods. We present consequences regarding the matrix Muckenhoupt classes and commutator estimates of Calderón-Zygmund operators with matrix $BMO$ functions. We also characterize the logarithms of Muckenhoupt weights through higher order commutators.

New author and new results