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

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 $E_{p,q}^s$. By constructing the domain of the common coretraction, we are able to prove $E_{o,q}^s\hookrightarrow E_{p,q}^s$ for $o\le p$, i.e. the monotonicity in the spatial integrability parameter.