A note on operating functions of modulation spaces

Abstract

Let $A^1_\beta ({{\textbf{T}}})$ denote the set of all Lebesgue integrable functions F on the torus ${\textbf{T}}$ such that $\sum _{m \in {{\textbf{Z}}}} |{\widehat{F}}(m) |{ (1+ |m |) }^{\beta } < \infty$, where $\{ {\widehat{F}}(m) \}_{m \in {{\textbf{Z}}}}$ denote the Fourier coefficients of F. We consider necessary and sufficient conditions for all functions $F \in A^1_\beta ({{\textbf{T}}})$ to operate on all real-valued functions in the modulation spaces $M^{p,q}_s({{\textbf{R}}})$.

5. Appendix

We can easily obtain an estimate for $\Vert e^{inf } -1 \Vert _{M^{p,p}_s}$ instead of $\Vert e^{inf} -1 \Vert _{M^{p,2}_s}$ by applying a method similar to the proofs of Lemmas 17 through 19. This gives an estimate

$$\begin{aligned} \Vert e^{inf } -1 \Vert _{M^{p,q}_s} \lesssim C_f |n |^{s+ \frac{1}{q} - \frac{1}{p} } \end{aligned}$$

for $1<q<p<\infty$, $p \ge 2$ and $s>2$. The proof is left to the reader.