L Estimates for Nonvariational Hypoelliptic Operators with V Mo Coefficients (original) (raw)
$L^p$ estimates for nonvariational hypoelliptic operators with VMOVMOVMO coefficients
Transactions of the American Mathematical Society, 1999
Let X 1 , X 2 , … , X q X_1,X_2,\ldots ,X_q be a system of real smooth vector fields, satisfying Hörmander’s condition in some bounded domain Ω ⊂ R n \Omega \subset \mathbb {R}^n ( n > q n>q ). We consider the differential operator L = ∑ i = 1 q a i j ( x ) X i X j , \begin{equation*} \mathcal {L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*} where the coefficients a i j ( x ) a_{ij}(x) are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: μ | ξ | 2 ≤ ∑ i , j = 1 q a i j ( x ) ξ i ξ j ≤ μ − 1 | ξ | 2 \begin{equation*} \mu |\xi |^2\leq \sum _{i,j=1}^qa_{ij}(x)\xi _i\xi _j\leq \mu ^{-1}|\xi |^2 \end{equation*} for a.e. x ∈ Ω x\in \Omega , every ξ ∈ R q \xi \in \mathbb {R}^q , some constant μ \mu . Moreover, we assume that the coefficients a i j a_{ij} belong to the space VMO (“Vanishing Mean Oscillation”), defined with respect to the subelliptic metric induced by the vector fields X 1 , X 2 , … , X q X_1,X_2,\ldots ,X_q . We prove the follo...
On a class of hypoelliptic operators with unbounded coefficients in RNR^NRN
Communications on Pure and Applied Analysis, 2009
We consider a class of non-trivial perturbations A of the degenerate Ornstein-Uhlenbeck operator in R N . In fact we perturb both the diffusion and the drift part of the operator (say Q and B) allowing the diffusion part to be unbounded in R N . Assuming that the kernel of the matrix Q(x) is invariant with respect to x ∈ R N and the Kalman rank condition is satisfied at any x ∈ R N by the same m < N , and developing a revised version of Bernstein's method we prove that we can associate a semigroup {T (t)} of bounded operators (in the space of bounded and continuous functions) with the operator A . Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup {T (t)} both in isotropic and anisotropic spaces of (Hölder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator A .
Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators
Potential Analysis, 2006
In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: L = m i=1 X 2 i + ∆, in R n where ∆ is the Laplace operator, m < n, and the limit operator L = m i=1 X 2 i is hypoelliptic. It is well known that L admits a fundamental solution Γ . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in , for solutions of the approximated equation L u = f . These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.
On partially hypoelliptic operators. Part I: Differential operators
arXiv: Analysis of PDEs, 2015
This article gives a fundamental discussion on variable coefficients, self-adjoint, formally partially hypoelliptic differential operators. A generalization of the results to pseudo differential operators, is given in a following article in ArXiv. Close to N. Nilsson "Some estimates for spectral functions connected with formally hypoelliptic differential operators" in Arkiv f\"or matematik ,10, 1972, we give a construction and estimates for a fundamental solution to the operator in a suitable topology. We further give estimates of the corresponding spectral kernel.
arXiv (Cornell University), 2015
We consider a nonvariational degenerate elliptic operator of the kind Lu ≡ q i,j=1 aij(x)XiXj u where X1, ..., Xq are a system of left invariant, 1-homogeneous, Hörmander's vector fields on a Carnot group in R n , the matrix {aij } is symmetric, uniformly positive on a bounded domain Ω ⊂ R n and the coefficients satisfy aij ∈ V M O loc (Ω) ∩ L ∞ (Ω). We give a new proof of the interior W 2,p X estimates XiXj u L p (Ω ′) + Xiu L p (Ω ′) ≤ c Lu L p (Ω) + u L p (Ω) for i, j = 1, 2, ..., q, u ∈ W 2,p X (Ω) , Ω ′ ⋐ Ω and p ∈ (1, ∞), first proved by Bramanti-Brandolini in [3], extending to this context Krylov' technique, introduced in [15], consisting in estimating the sharp maximal function of XiXj u.
ω-hypoelliptic differential operators of constant strength
Journal of Mathematical Analysis and Applications, 2004
We study ω-hypoelliptic differential operators of constant strength. We show that any operator with constant strength and coefficients in E ω (Ω) which is homogeneous ω-hypoelliptic is also σ -hypoelliptic for any weight function σ = O(ω). We also present a sufficient condition in order to ensure that a differential operator admits a parametrix and, as a consequence, we obtain some conditions on the weights (ω, σ ) to conclude that, for any operator P (x, D) with constant strength, the σ -hypoellipticity of the frozen operator P (x 0 , D) implies the ω-hypoellipticity of P (x, D). This requires the use of pseudodifferential operators. 2004 Elsevier Inc. All rights reserved.
Annales de l’institut Fourier, 2016
The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators Tome 66, n o 2 (2016), p. 589-631. http://aif.cedram.org/item?id=AIF\_2016\_\_66\_2\_589\_0 © Association des Annales de l'institut Fourier, 2016, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION-PAS DE MODIFICATION 3.
Revista Matemática Complutense, 2016
We consider a nonvariational degenerate elliptic operator of the kind Lu ≡ q i,j=1 aij(x)XiXj u where X1, ..., Xq are a system of left invariant, 1-homogeneous, Hörmander's vector fields on a Carnot group in R n , the matrix {aij } is symmetric, uniformly positive on a bounded domain Ω ⊂ R n and the coefficients satisfy aij ∈ V M O loc (Ω) ∩ L ∞ (Ω). We give a new proof of the interior W 2,p X estimates XiXj u L p (Ω ′) + Xiu L p (Ω ′) ≤ c Lu L p (Ω) + u L p (Ω) for i, j = 1, 2, ..., q, u ∈ W 2,p X (Ω) , Ω ′ ⋐ Ω and p ∈ (1, ∞), first proved by Bramanti-Brandolini in [3], extending to this context Krylov' technique, introduced in [15], consisting in estimating the sharp maximal function of XiXj u.
Limiting Sobolev inequalities for vector fields and canceling linear differential operators
Journal of the European Mathematical Society, 2013
A . The estimate D k−1 u L n/(n−1) ≤ A(D)u L 1 is shown to hold if and only if A(D) is elliptic and canceling. Here A(D) is a homogeneous linear di erential operator A(D) of order k on R n from a vector space V to a vector space E. The operator A(D) is de ned to be canceling if ξ∈R n \{0} A(ξ)[V ] = {0}. Date: April 4, 2011. 2000 Mathematics Subject Classi cation. 46E35 (26D10 42B20).