On a class of hypoelliptic operators with unbounded coefficients in RNR^NRN (original) (raw)
Related papers
Schauder estimates for parabolic nondivergence operators of Hörmander type
Journal of Differential Equations, 2007
Series Information Technology (IT): Stefano Paraboschi Series Mathematics and Statistics (MS): Luca Brandolini, Alessandro Fassò § L'accesso alle Series è approvato dal Comitato di Redazione. I Working Papers ed i Technical Reports della Collana dei Quaderni del Dipartimento di Ingegneria Gestionale e dell'Informazione costituiscono un servizio atto a fornire la tempestiva divulgazione dei risultati dell'attività di ricerca, siano essi in forma provvisoria o definitiva. Abstract Let X1; X2; : : : ; Xq be a system of real smooth vector …elds satisfying Hörmander's rank condition in a bounded domain of R n . Let A = faij (t; x)g q i;j=1 be a symmetric, uniformly positive de…nite matrix of real functions de…ned in a domain U R . For operators of kind 1 A basic result proved by Macias-Segovia (see Theorem 2 in [25]) states that:
Regularity estimates for diffusion semigroups on weighted Sobolev spaces
arXiv (Cornell University), 2022
In this work, we consider a class of second order uniformly elliptic operators with smooth and bounded coefficients. We provide some estimates on the norm of the semigroup generated by these operators acting on weighted Sobolev spaces, where the weight satisfies some specific conditions. Our proof relies on a classical bound for the derivatives of fundamental solution to parabolic equations.
General Kernel estimates of Schr\"odinger type operators with unbounded diffusion terms
Cornell University - arXiv, 2022
We prove first that the realization A min of A := div(Q∇) − V in L 2 (R d) with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on L 2 (R d) which coincides on L 2 (R d) ∩ C b (R d) with the minimal semigroup generated by a realization of A on C b (R d). Moreover, using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of A and deduce some spectral properties of A min in the case of polynomially and exponentially diffusion and potential coefficients.
One-Dimensional Degenerate Diffusion Operators
Mediterranean Journal of Mathematics, 2013
The aim of this paper is to present some results about generation, sectoriality and gradient estimates both for the semigroup and for the resolvent of suitable realizations of the operators A γ,b u(x) = γxu ′′ (x) + bu ′ (x), with constants γ > 0 and b ≥ 0, in the space C([0, ∞]). A γ,b u(x) = γxu ′′ (x) + bu ′ (x), (0.3) with constants γ > 0 and b ≥ 0, in the space C([0, ∞]), because they model in 0 the behaviour of the operators (0.2) near the end points 0 and 1. To this end we review some results established mainly in [7, 22, 13, 14], and we propose them in a unified way. We point out that the works [7, 13, 14] are addressed mainly to the study of the operators (0.2), (0.3) in Hölder continuous function spaces, while we will
Gaussian estimates for elliptic operators with unbounded drift
Journal of Mathematical Analysis and Applications, 2008
We consider a strictly elliptic operatorAu=∑ijDi(aijDju)−b⋅∇u+div(c⋅u)−Vu, where 0⩽V∈Lloc∞, aij∈Cb1(RN), b,c∈C1(RN,RN). If divb⩽βV, divc⩽βV, 0β1, then a natural realization of A generates a positive C0-semigroup T in L2(RN). The semigroup satisfies pseudo-Gaussian estimates if|b|⩽k1Vα+k2,|c|⩽k1Vα+k2, where 12⩽α1. If α=12, then Gaussian estimates are valid. The constant α=12 is optimal with respect to this property.
On the Dirichlet semigroup for Ornstein–Uhlenbeck operators in subsets of Hilbert spaces
Journal of Functional Analysis, 2010
We consider a family of self-adjoint Ornstein-Uhlenbeck operators Lα in an infinite dimensional Hilbert space H having the same gaussian invariant measure µ for all ′ α ∈ [0, 1]. We study the Dirichlet problem for the equation λϕ − Lαϕ = f in a closed set K, with f ∈ L 2 (K, µ). We first prove that the variational solution, trivially provided by the Lax-Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to K. Then we address two problems: 1) the regularity of the solution ϕ (which is by definition in a Sobolev space W 1,2 α (K, µ)) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior W 2,2 α regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside K belongs to W 1,2 α (H, µ). In the second case we exploit the Malliavin's theory of surface integrals which is recalled in the Appendix of the paper, then we are able to give a meaning to the trace of ϕ at ∂K and to show that it vanishes, as it is natural.
arXiv (Cornell University), 2022
such that the corresponding model operator having constant aij is hypoelliptic, translation invariant w.r.t. a Lie group operation in R N+1 and 2-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients aij are bounded and Hölder continuous in space (w.r.t. some distance induced by L in R N) and only bounded measurable in time; the matrix {aij} q i,j=1 is symmetric and uniformly positive on R q. We prove "partial Schauder a priori estimates" of the kind q i,j=1 ∂ 2 x i x j u C α x (S T) + Y u C α x (S T) ≤ c Lu C α x (S T) + u C 0 (S T) for suitable functions u, where f C α x (S T) = sup t≤T sup x 1 ,x 2 ∈R N ,x 1 =x 2 |f (x1, t) − f (x2, t)| x1 − x2 α + f L ∞ (S T). We also prove that the derivatives ∂ 2 x i x j u are locally Hölder continuous in space and time while ∂x i u and u are globally Hölder continuous in space and time.