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

TL;DR

The paper develops a unified framework based on common retraction and coretraction for families of Banach spaces to identify interpolation spaces, and applies it to exponentially weighted modulation spaces. It proves a monotonicity result $E^s_{p_0,q} \hookrightarrow E^s_{p_1,q}$ for $p_0 \le p_1$ and establishes complex interpolation rules for these spaces, namely $[E^{s_0}_{p_0,q_0}, E^{s_1}_{p_1,q_1}]_\theta = E^{s}_{p,q}$ with $s,p,q$ given by convex combinations. The method relies on recasting modulation spaces via a common coretraction, constructing suitable domain spaces $\mathcal{V}$ and $\mathcal{W}$ for the retraction pair, and leveraging Bernstein-type multiplier estimates. The results extend the interpolation theory for modulation spaces to the exponentially weighted setting and provide explicit norm-equivalences in the interpolated spaces, with potential applications to PDEs in contexts requiring 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.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2: Complex interpolation of $E^{s}_{p,q}$
• Remark 1.3
• Example 4.1: cf. triebel1978
• Theorem 4.2: Retract interpolation theorem (see triebel1978)
• proof
• Remark 4.3
• Example 4.4
• Remark 4.5
• Definition 4.6: Common retraction and coretraction