Complex Interpolation and the Monotonicity in the Spatial Integrability Parameter of Exponentially Weighted Modulation Spaces

TL;DR

This paper develops a framework for complex interpolation of exponential weighted modulation spaces, establishing monotonicity in the spatial integrability parameter through the notion of common retraction and coretraction.

Contribution

It introduces the concepts of common retraction and coretraction for families of Banach spaces and applies them to modulation spaces to prove monotonicity in the integrability parameter.

Findings

Proved $E^s_{o, q} \hookrightarrow E^s_{p, q}$ for $o \leq p$ in exponential weighted modulation spaces.

Established a new interpolation framework using common retraction and coretraction.

Enhanced understanding of the structure of modulation spaces with exponential weights.

Abstract

We introduce the notion of common retraction and coretraction for families of Banach spaces, formulate a framework for identifying interpolation spaces, and apply it to modulation spaces with exponential weights $E^s_{p,q}$. By constructing the domain of the common coretraction, we are able to prove $E^s_{o, q} \hookrightarrow E^s_{p, q}$ for $o \leq p$, i.e. the monotonicity in the spatial integrability parameter.