On the inclusion relations between Gelfand-Shilov spaces
Andreas Debrouwere, Lenny Neyt, Jasson Vindas
TL;DR
The paper develops a unified framework to compare Gelfand-Shilov type spaces defined by weight matrices $\mathfrak{M}$ and weight function systems $\mathcal{W}$ in translation-invariant Banach spaces $E$. It proves two main inclusion-characterization theorems: an inclusion $E^{[\mathfrak{M}]}_{[\mathcal{W}]} \subseteq E^{[\mathfrak{N}]}_{[\mathcal{V}]}$ holds if and only if the defining systems are dominated, i.e., $\mathfrak{M} \preceq \mathfrak{N}$ and $\mathcal{W} \subseteq \mathcal{V}$ (and a fixed-weight-system analogue under appropriate hypotheses). The proofs blend discrete sampling reductions, De Wilde’s closed graph theorem, Grothendieck factorization, and window/partition-of-unity techniques to connect space inclusions with growth relations of weights, covering both Fourier-invariant and anisotropic, non-Fourier-invariant settings. These results unify classical Gelfand-Shilov and Beurling-Björck spaces within a flexible operator-theoretic framework and clarify when one ultradifferentiable class contains another. The framework also clarifies how growth conditions on weight matrices and function systems govern inclusions, with implications for anisotropic and nonstandard norm choices in ultradifferentiable spaces.
Abstract
We study inclusion relations between Gelfand-Shilov type spaces defined via a weight (multi-)sequence system, a weight function system, and a translation-invariant Banach function space. We characterize when such spaces are included into one another in terms of growth relations for the defining weight sequence and function systems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\mathcal{S}^{[M]}_{[A]}$ (defined via weight sequences $M$ and $A$) and the Beurling-Björck spaces $\mathcal{S}^{[ω]}_{[η]}$ (defined via weight functions $ω$ and $η$).