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Kyoto Journal of Mathematics, vol.46, no.3, pp.595-616, 2006 (SCI-Expanded)
In the second section of this paper, in analogy to modulation spaces, we define the space M(p, q) (Rd) to be the subspace of tempered distributions f ∈ S′ (Rd) such that the Gabor transform Vg (f) of f is in the Lorentz space L (p, q) (R2d), where the window function g is a rapidly decreasing function. We endow this space with a suitable norm and show that the M(p, q) (Rd) becomes a Banach space and is invariant under time-frequency shifts for 1 ≤ p, q ≤ ∞. We also discuss the dual space of M(p, q) (Rd) and the multipliers from L1 (Rd) into M(p, q) (Rd). In the third section we intend to study the intersection space S (p, q) (Rd) = L1 (Rd) ∩ M (p, q) (Rd) for 1 < p < ∞, 1 ≤ q ≤ ∞. We endow it with the sum norm and show that S (p, q) (Rd) becomes a Banach convolution algebra. Further we prove that it is also a Segal algebra. In the last section we discuss the multipliers of S (p, q) (Rd) and M (p, q) (Rd).