On a broad family of quasi-Banach modulation spaces

Let 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 , B ), iff V_φ f belongs to the Wiener amalgam space W ʳ(ω , B ), and \| f \| M(ω , B) \| V _φ f \, ω \| B \| V _φ f\| W ʳ(ω, B). We also show that the result can be useful in different situations. For example, we prove for f,g∈ Σ ₁' 0 that Vgf∈ Lr₀₍ᵥ₎, iff f,g∈ Mr₀₍ᵥ₎.