Time frequency analysis and multipliers of the spaces $M(p,q) (R^d)$ and $S(p,q) (R^d)$

Abstract

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 $f \in S'(R^{d})$ 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 Banach space and is invariant under time-frequency shifts for $1 \leq p,q \leq \infty$. 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 $S (p, q)(R^{d}) = L^{1}(R^{d})\cap M (p, q)(R^{d})$ for $1 < p < \infty$, $1 \leq q \leq \infty$. 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 $M (p, q)(R^{d})$.