Related papers: Local Solvability on H_1: Non-homogeneous Operator…

Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older…

We prove interior boundedness and H\"{o}lder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et. al. in 2022 and 2023 for…

We consider a class of nonvariational linear operators formed by homogeneous left invariant Hormander's vector fields with respect to a structure of Carnot group. The bounded coefficients of the operators belong to "vanishing logarithmic…

We consider a class of multiparameter singular Radon integral operators on the Heisenberg group ${\mathbb H}^1$ where the underlying variety is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg…

We establish the Krylov Safonov Harnack inequalities and Holder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal…

We prove boundary H\"older and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the…

The purpose of this paper is to study microlocal conditions for inclusion relations between the ranges of square systems of pseudodifferential operators which fail to be locally solvable. The work is an extension of earlier results for the…

We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these…

The paper deals with homogenization problem for a non-local linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behaviour of…

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…

The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by…

M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum $(sl_n,\land V_n)$ link invariant, where…

We give a necessary and sufficient criterion for the solvability of $\operatorname{HH}^1(kP)$ as a Lie algebra, where $P$ is a $p$-group with $p$ odd, in terms of a directed graph constructed from the group $P$. This gives non-trivial…

The purpose of this paper is to obtain microlocal analogues of results by L. H \"ormander about inclusion relations between the ranges of first order differential operators with coefficients in $C^\infty$ which fail to be locally solvable.…

In the context of a connected, simply connected, nilpotent Lie group, whose representations are square-integrable modulo the center, we find characterization results of extra-invariant spaces under the left translations associated with the…

A nonpolynomial one-dimensional quantum potential representing an oscillator, that can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is…

Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$…

Let $G$ be the semidirect product $N \rtimes \mathbb{R}$, where $N$ is a stratified Lie group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$,…

We present a construction of a large class of Laplace invariants for linear hyperbolic partial differential operators of fairly general form and arbitrary order.

In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics $(M,g,X)$ with $X$ a left-invariant vector field, which we call totally left-invariant quasi-Einstein metrics. We give a complete…