16 problems

Rodino–Nicola's conjecture. If

Exact-cost computational conjecture. With a sharper choice of the enlarged regression supports, one can improve the computational-cost bound and achieve the exact optimal computati…

The paper considers more general Hörmander classes $S_0(M)$ and super operators acting on Hilbert-Schmidt operators. Super-operator extension conjecture. Extending these Hörmander…

Vladimirov sub-Laplacian conjecture. The Vladimirov sub-Laplacian of order $\alpha>0$ is a hypoelliptic operator on $G$ and is invertible on the space of mean-zero functions.

Hypoellipticity conjecture. The Vladimirov sub-Laplacian $\mathscr{L}^{\alpha}_{sub}$ is a hypoelliptic operator on $\mathbb{G}$.

The numerical tests consider eigenvalues $\lambda_j^{(\nu)}$ of the relevant zeroth-order pseudo-differential operators as the viscosity $\nu$ tends to $0^+$. In Tests 1–3, the com…

Let $G$ be the Grushin operator, let $\boldsymbol{L}$ and $\boldsymbol{U}$ denote the associated operator-valued spectral variables, and let…

Projection conjecture. The required extension of the Courrège theorem to symmetric spaces should be obtainable from the corresponding result for Lie groups by using projection tech…

Let $G$ be a homogeneous group, and let an operator on $G$ satisfy the positive maximum principle (PMP). Pseudo-differential representation conjecture. Such operators should be rep…

Let $K^{2,M}_{z_0}$ denote the limiting two-point correlation function of the rescaled eigenvalue process for the randomly perturbed operator, and let $K^{2,Gin}_{z_0}$ denote the…

Redundancy conjecture. The convergence of $a^m(x,D)u^m$ to $a(x,D)u$ in $\mathcal{S}'$ should be redundant; it should suffice to assume that $a(x,D)u\in\mathcal{S}'$.

Let $a(x,D)$ be a pseudo-differential operator with symbol $a$ in the type $1,1$ class $S^0_{1,1}$, and let $N_{\chi,\varepsilon,\alpha}(a)$ denote the corresponding frequency-modu…

Let $\psi$ be a non-constant complete Bernstein function such that $\psi(0)=0$, and let $\vartheta_\mu$ be the phase function considered in the preceding lemma, for $\mu>0$. Sharp…

Let $\psi$ be a non-constant complete Bernstein function such that $\psi(0)=0$, and let $\vartheta_\mu$ be the phase function considered in the preceding lemma. If $\psi$ is unboun…

A compact fractafold $X$ carries a Laplacian $\Delta$, and a pseudo-differential operator with variable coefficients is written as $p(x,-\Delta)$; its kernel is denoted by $K$. Cal…

Let $G$ be the underlying group and let $\bigcup_{\ell\text{?}}$...