Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

We study the connection between STFT multipliers $A_{1\otimes m}^{g_1,g_2}$ having windows $g_1,g_2$, symbols $a(x,\omega)=(1\otimes m)(x,\omega)=m(\omega)$,$(x,\omega)\in$\mathbb{R}$^{2d}$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on ${\mathbb{R}}^d$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $g_1,g_2$ for the equality $T_{m_2}=A_{1\otimes m}^{g_1,g_2}$. For $m=m_2$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A_{1\otimes m}^{g_1,g_2}$, also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of $A_{1\otimes m}^{g_1,g_2}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A_{1\otimes m}^{g,g}:L^p\to L^q$ having multiplier $m$ in weak $L^r$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.