Related papers: Hypoellipticity and Higher Order Gaussian Bounds

Given a metric measure space $(\mathcal{X}, d, \mu)$ satisfying the volume doubling condition, we consider a semigroup $\{S_t\}$ and the associated heat operator. We propose general conditions on the heat kernel so that the solutions of the…

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le…

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…

An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated to a class of functional stochastic differential equations. Consequently, the semigroup $P_t$ converges exponentially to its unique…

We prove sufficient conditions for Topological Quantum Order at both zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries (that notably extend and differ from standard local…

This paper investigates the boundedness of bilinear pseudo-differential operators with symbols in the H\"{o}rmander class $BS_{\varrho,\delta}^m(\mathbb{R}^n)$ in the previously unexplored regime $0 \leq \varrho < \delta < 1$. We establish…

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal L\'evy processes. Our bounds are sharp under the…

Let $\mu$ and $\nu$ be fixed probability measures on a filtered space $(\Omega, {\cal F}, ({\cal F}_t)_{t\in {\bf R}^{+}})$. Denote by $\mu_T $ and $\nu_T $ (respectively, $\mu_{T-} $ and $\nu_{T-} $) the restrictions of the measures $\mu$…

We study the well-posedness of Cauchy problems on the upper half space $\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u =(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly elliptic coefficients.…

We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional…

Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian…

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega ) $ and let $B_x$ be the…

We propose a description of T-duality between general geometric and non-geometric backgrounds as higher groupoid bundles with connections. Our description extends the previous observation by Nikolaus and Waldorf that the topological aspects…

In this manuscript, we appeal to Potential Theory to provide a sufficient condition for existence of distributional solutions to fractional elliptic problems with non-linear first-order terms and measure data $\omega$: $$ \left\{…

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where \[…

We prove that under H\"ormander's type conditions on the coefficients of the unobservable component of a partially observable diffusion process the filtering density is infinitely differentiable and can be represented as the integral of an…

In this paper we establish a hypoellipticity result for second order linear operators comprised by a linear combination, with infinite vanishing coefficients, of subelliptic operators in separate spaces. This generalizes previous known…

We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.),…

We study representations of positive definite kernels $K$ in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for…

For a an arbitrary periodic Borel measure $\mu$, we prove order $O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb R}^d, d\mu^\varepsilon)$ with $\varepsilon$-periodic…