49 problems

Let $A_{\Omega}[1]$ denote the operator associated with a domain $\Omega\subset\mathbb R^2$ and the constant symbol $1$, and let $\Omega$ be a convex domain. Flandrin's conjecture.…

Let $\mathsf{M}$ be a smooth manifold and let $\mathsf{P}$ be a properly supported pseudodifferential operator on $\mathsf{M}$ with complex coefficients. Local solvability means th…

Let $M$ be a compact manifold and let $\Gamma\subset\mathbb{R}^p$ be a connected cone with nonempty interior and $H^1_{\rm dR}(\Gamma)=0$. Let…

Let $M$ be a compact oriented smooth manifold with smooth boundary $\partial M$, let $E_0$ and $E_1$ be smooth complex Hermitian vector bundles over $M$, and let…

Arnol'd's conjecture. Studying the asymptotic behaviour of $N_\chi(\lambda)$ should show that the multiplicity of each unitary irreducible representation in the decomposition of…

A locally compact abelian group is a locally compact topological group whose group operation is commutative. Let $L^2(\G)$ denote the associated Hilbert space of square-integrable…

Let $H$ be a manifold with corners and let $D_H$ be a cusp pseudodifferential operator of order $1$ that is of Dirac type near the boundary, as in the referenced theorem and remark…

Nonvanishing conjecture. The class $[c_k^\delta]$ is a nonzero multiple of the generator in degree $2k$.

Let $X$ be a compact Riemannian manifold, let $H:=\sqrt{-\Delta}$, and let $Q(f)$ be a pseudodifferential operator of order zero with principal symbol $f_P$ on $ST^*X$. Suppose the…

Let a massive quantum field theory be localized in a double cone and let its modular objects define the corresponding modular transformations. The source considers transformations…

Vanishing conjecture. All coefficients of $t^\ga\log^k t$ vanish for every $k\geq 1$.

Let $M$ be a filtered manifold and let $\Psi_H$DOs denote its Heisenberg pseudodifferential operators. For a negative-order operator in this calculus, consider its singular values…

Let $(M,H)$ be a compact Carnot manifold, and let $T\in \Psi^{-d_H}_H(M)$ be a pseudodifferential operator on $(M,H)$ in the van Erp-Yuncken sense. Let $\nu$ be a density on $M$, i…

Let $E_n$ be the Hilbert $\b7$-module introduced above, let $\a0\mathcal{L}_\mathcal{C}(E_n)$ denote its adjointable operators, and let $\mathcal{U}$ be the action of the Heisenber…

The Fourier multiplier $T_m$ and the anti-Wick operator $A^{g,g}_{1\otimes m}$ are considered in the setting where boundedness and smoothing effects are studied between Lebesgue sp…

Let $\mathcal{P}$ be the symbol of a semiclassical pseudodifferential operator with double characteristics, and suppose the operator has low-lying eigenfunctions. The symbol is ass…

Boutet de Monvel's conjecture. There is a maximal $\epsilon_0>0$ such that, for every $\epsilon\in(0,\epsilon_0)$, the extended flows define a real-analytic diffeomorphism

Let $P=P^*\in \Psi^{-m}(\mathbb{T}^n_\theta)$ with $m>0$ have principal symbol $\rho_{-m}(\xi)$, and set $p=nm^{-1}$. For a selfadjoint compact operator, write…

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…

Let $g,h>0$, let $p,q\geq1$, and let $a\in S^{2g,2h}_{p,q}=S(\langle x\rangle^{2g/p}\langle\xi\rangle^{2h/q};\langle\xi\rangle,\langle x\rangle)$. Denote by $a^w$ its Weyl quantiza…

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…

The classical and quantum notions involved are completeness of the Hamilton flow associated with a real principal symbol and essential self-adjointness of the corresponding pseudod…

Let $A^0$ be a scalar operator and let … be a strongly convex surface, meaning that … for all $\xi\in\Sigma^0_\lambda$ and all $\eta$ satisfying … where the sign depends on the con…

Characteristic-function conjecture. The function $W^\epsilon$ should characterize the full structure of $\mathcal{L}^\epsilon$ in the sense that

Let $M^{2l}$ be a closed compact quasi-conformal manifold, let $\Omega^{\ast}(M)$ be the algebra of differential forms on $M^{2l}$, and let $H$ be the Hilbert space of $L_{2}$-form…