Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators

2024/07/15 by Toft, Joachim, Pfeuffer, Christine, Teofanov, Nenad
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2407.10503

Abstract

Let \mathscr B be a normal quasi-Banach function space with respect to r₀ ∈ (0,1] and v₀, ω be v-moderate, and let r∈ [r₀,∞ ]. Then we prove that f belongs to the modulation space M(ω,\mathscr B ), iff V_ϕf belongs to the Wiener amalgam space W ^r(ω,\mathscr B ), and ‖ f ‖ _{M(ω, \mathscr B)} \asymp ‖ V _ϕf ω‖ _{\mathscr B} \asymp ‖ V _ϕf‖ _{W ^r(ω, \mathscr B)}. We also use the results to deduce continuity for pseudo-differential operators with symbols in weighted M^∞,r₀-spaces, with r₀≤ 1, when acting on M(ω,\mathscr B )-spaces.

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