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

We consider a broad class of modulation spaces M(ω, B),parameterized with weight function ω and a normal quasiBanach function space B of order r0 ∈ (0, 1]. Then we provethat f ∈ M(ω, B), if and only if Vϕf belongs to the Wieneramalgam space Wr(ω, B), and ∥f∥M(ω,B) ≍ ∥Vϕf · ω∥B ≍ ∥Vϕf∥Wr(ω,B), r ∈ [r0, ∞]. We use the results to extend and improve continuity andlifting properties for pseudo-differential and Toeplitz operatorswith symbols in weighted M∞,r0 -spaces, r0 ≤ 1, when actingon M(ω, B)-spaces.