We identify the generalised modulation spaces associated with tensor products of amalgam having a large class Banach as their local component. As consequences main results, we describe various $$L^p$$ spaces.

Gabor frames {e2πinβ·xg(x− kα)}n,k∈Zd provide series representations not only of functions in L(R) but of the entire range of spaces M ν known as the modulation spaces. Membership of a function or distribution f in the modulation space is characterized by a sequence-space norm of the Gabor coefficients of f depending only on the magnitudes of those coefficients, and the Gabor series representat...

Since the theory of pseudo-differential operators was established in 1970’s, the L-boundedness of them with symbols in the Hörmander class S ρ,δ has been well investigated by many authors. Among them, Calderón-Vaillancourt [5] first treated the boundedness for the class S 0,0, which means that the boundedness of all the derivatives of symbols assures the L-boundedness of the corresponding opera...

We introduce new classes of Banach algebras of pseudodifferential operators with symbols in certain modulation spaces and investigate their composition and the functional calculus. Operators in these algebras possess the spectral invariance property on the associated family of modulation spaces. These results extend and contain Sjöstrand’s theory, and they are obtained with new phase space meth...

We prove a Beurling-Helson type theorem on modulation spaces. More precisely, we show that the only C changes of variables that leave invariant the modulation spaces M(R) are affine functions on R. A special case of our result involving the Sjöstrand algebra was considered earlier by A. Boulkhemair.

The fact that symbols in the modulation space M1,1 generate pseudo-differential operators of the trace class was first mentioned by Feichtinger and the proof was given by Gröchenig [12]. In this paper, we show that the same is true if we replace M1,1 by more general α-modulation spaces which include modulation spaces (α = 0) and Besov spaces (α = 1) as special cases. The result with α = 0 corre...