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 $E^s_{p,q}$. 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.