JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY, 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) (R-d) to be the subspace of tempered distributions f is an element of S' (R-d) such that the Gabor transform V-g (f) of f is in the Lorentz space L (p, q) (R-2d), 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) (R-d) becomes a Barlach space and is invariant under time-frequency shifts for 1 <= p, q <= infinity. We also discuss the dual space of M(p, q) (R-d) and the multipliers from L-1 (R-d) into M(p, q) (R-d). In the third section we intend to study the intersection space S (p, q) (R-d) = L-1 (R-d) boolean AND M (p, q) (R-d) for 1 < P < infinity, 1 <= q <= infinity. We endow it with the sum norm and show that S (p, q) (R-d) 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) (R-d) and M (p, q) (R-d).