$\alpha$-Modulation Spaces (I)

TL;DR

This paper explores fundamental properties, inclusion relations, and algebraic structures of $oldsymbol{ extalpha}$-modulation spaces, revealing their complex interpolation, scaling behavior, and algebraic criteria, with implications for their role in harmonic analysis.

Contribution

It provides new insights into the properties, inclusion relations, and algebraic structures of $oldsymbol{ extalpha}$-modulation spaces, including criteria for multiplication algebra and interpolation characteristics.

Findings

Derived dual spaces and complex interpolation results.

Established necessary and sufficient conditions for scaling and inclusion.

Identified an $oldsymbol{ extalpha}$-modulation space not interpolable between modulation and Besov spaces.

Abstract

First, we consider some fundamental properties including dual spaces, complex interpolations of $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$ with $0<p,q \le \infty$. Next, necessary and sufficient conditions for the scaling property and the inclusions between $\alpha_1$-modulation and $\alpha_2$-modulation spaces are obtained. Finally, we give some criteria for $\alpha$-modulation spaces constituting multiplication algebra. As a by-product, we show that there exists an $\alpha$-modulation space which is not an interpolation space between modulation and Besov spaces.