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Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees, that is,…
We establish compactness estimates for $\overline{\partial}_{b}$ on a compact pseudoconvex CR-submanifold of $\mathbb{C}^{n}$ of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved…
We use algebras of pseudodifferential operators on groupoids to study geometric operators on non-compact manifolds and singular spaces. The first step is to establish that the geometric operators are in our algebras. This then leads to…
The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P_q), a potential theoretic condition on $(0,q)$-forms that generalizes Catlin's property (P_q) to CR…
We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the $p$-Laplacian. The natural framework in which we consider such equations is that of Orlicz-Sobolev spaces. we exhibit two…
In this paper, we explore the positive solutions of the following nonlinear Choquard equation involving the green kernel of the fractional operator $(-\Delta_{\mathbb{B}^N})^{-\alpha/2}$ in the hyperbolic space \begin{equation}…
We prove the first positive results concerning boundary value problems in the upper half-space of second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so,…
Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over…
The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type…
In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are…
We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic…
We present a scheme of biquaternionic algebrodymamics based on a nonlinear generalization of the Cauchy-Riemann holomorphy conditions considered therein as fundamental field equations. The automorphism group SO(3,C) of the biquaternion…
The discrete Laplace operator on a triangulated polyhedral surface is related to geometric properties of the surface. This paper studies extremum problems for eigenvalues of the discrete Laplace operators. Among all triangles, an…
Non-local boundary conditions for Euclidean quantum gravity are proposed, consisting of an integro-differential boundary operator acting on metric perturbations. In this case, the operator P on metric perturbations is of Laplace type,…
The task to construct parametrices of elliptic differential operators on a manifold with edges requires a calculus of operators with a two-component principal symbolic hierarchy, consisting of (edge-degenerate) interior and…
This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional $p\& q$ Laplacian operator with indefinite weights $$\left(-\Delta_p\right)^{\alpha}u +…
Given i.i.d. observations uniformly distributed on a closed manifold $\mathcal{M}\subseteq \mathbb{R}^p$, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are…
Weighted $L^p-L^r$ inequalities with arbitrary measurable non-negative weights for positive quasilinear integral operators with Oinarov's kernel on the semiaxis are characterized. Application to the boundedness of maximal operator in the…
We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner…
The classical result of J.J. Kohn asserts that over a relatively compact subdomain $D$ with $C^\infty$ boundary of a Hermitian manifold whose Levi form has at least $n-q$ positive eigenvalues or at least $q+1$ negative eigenvalues at each…