Related papers: About the mass of certain second order elliptic op…

For any fixed $p>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transforms associated with second order elliptic operators with real, symmetric, bounded measurable coefficients.

The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using…

We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space…

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of m complex-valued half-densities over a connected compact n-dimensional manifold without boundary. The eigenvalues of the principal symbol are…

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex…

In [5] Herzlich proved a new positive mass theorem for Riemannian 3-manifolds $(N, g)$ whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the…

An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing…

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal…

We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times…

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding…

We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number…

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of…

The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian…

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…

We study second-order elliptic partial differential operators acting on sections of vector bundles over a compact manifold with boundary with a non-scalar positive definite leading symbol. Such operators, called non-Laplace type operators,…

The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. As a consequence,…

Answering a question by Chatterji--Dru\c{t}u--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected…