12 problems

Let $f$ and $g$ be functions or distributions for which the short-time Fourier transform $\operatorname{STFT}_g f$ is defined, and let $L^p_v$ and $M^p_v$ denote the weighted Lebes…

Let $\mathscr{G}$ be a locally compact abelian group and let $\widehat{\mathscr{G}}$ be its dual group. The modulation space $M^{\infty,1}(\mathscr{G}\times\widehat{\mathscr{G}})$…

Let $FIO(S,M^{\infty,q}_{1\otimes v_s})$ denote the class of continuous linear operators whose Wigner kernels have the localization and modulation-space structure specified above,…

Let $p_1,q_1,p_2,q_2$ be the Lebesgue parameters and let $w_1,w_2$ be the weight functions satisfying the assumptions of Theorem 4. For $F_1\in L_{w_1}^{p_1,q_1}(\mathbb{R}^{2d})$…

When a symbol $\sigma$ on $\mathbb{R}^{2d}$ satisfies the Gevrey-type estimate … for $s>0$ and a $v$-moderate weight $M$, the relevant weighted modulation spaces are those with wei…

Well-posedness conjecture. The well-posedness result proved in the paper for the stated modulation-space setting should hold as well for initial data in $M^{p,q}_s(\mathbb{R}^d)$ w…

Gabor-matrix characterization conjecture. Pseudodifferential operators in the Hörmander class $S^m_{0,0}$, for $m\in\mathbb{R}$, should be characterizable via this Gabor-matrix est…

Let $n$ be the dimension, let $0<s<\frac n2$, and let $\varepsilon>0$. Write $M^{2s-n/2-\varepsilon}_{1,2}$ for the weighted modulation space and $H^{2s-n/2}$ for the Sobolev space…

Orlik's conjecture. The Born-Jordan kernel does not belong to the modulation space $M^{1,\infty}(\mathbb{R}^{2d})$:

Optimal boundedness conjecture. The condition

Modulation-space boundedness conjecture. The operator $T_\sigma$ is bounded from $\mathcal{M}^{r_1,r_2}$ to $\mathcal{M}^{r_2,r_1}$. This conjecture proposes an analogue of the cor…

Let $n=2$, let $p=4/n$, and let … also holds in the case $p=4/n$ when $n=2$. This is an endpoint extension of the preceding estimate; the source gives no resolution of this conject…