Schauder estimates for sub-elliptic equations (original) (raw)

Lp and Schauder estimates

2013

Let L = q i,j=1 aij (x)XiXj + a0 (x) X0, where X0, X1, ..., Xq are real smooth vector fields satisfying Hörmander's condition in some bounded domain Ω ⊂ R n (n > q + 1), the coefficients aij = aji, a0 are real valued, bounded measurable functions defined in Ω, satisfying the uniform positivity conditions: µ|ξ| 2 ≤ q i,j=1 aij (x)ξiξj ≤ µ −1 |ξ| 2 ; µ ≤ a0 (x) ≤ µ −1 for a.e. x ∈ Ω, every ξ ∈ R q , some constant µ > 0. We prove that if the coefficients aij , a0 belong to the Hölder space C α X (Ω) with respect to the distance induced by the vector fields, then local Schauder estimates of the following kind hold:

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).

L^p and Schauder estimates for nonvariational operators structured on H\

2011

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying H\"ormander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly positive. We prove that if the coefficients a_{ij} are H\"older

On limiting trace inequalities for vectorial differential operators

Indiana University Mathematics Journal, 2021

We establish that trace inequalities for vector fields u ∈ C ∞ c (R n , R N) D k−1 u L n−s n−1 (dµ) c µ n−1 n−s L 1,n−s A[D]u L 1 (dL n) (*) hold if and only if the k-th order homogeneous linear differential operator A[D] on R n is elliptic and cancelling, provided that s < 1, and give partial results for s = 1, where stronger conditions on A[D] are necessary. Here, µ L 1,λ denotes the Morrey norm of µ so that such traces can be taken, for example, with respect to H n−s-measure restricted to fractals of codimension s < 1. The class of inequalities (*) give a systematic generalisation of Adams' trace inequalities to the limit case p = 1 and can be used to prove trace embeddings for functions of bounded A-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We also prove a multiplicative version of (*), which implies strict continuity of the associated trace operators on BV A. the supremum ranging over all open balls B ⊂ R n ; r(B) denotes the radius of the ball B. Measures µ satisfying µ L 1,λ (R n) < ∞ are sometimes called λ-Ahlfors regular. Note that, even for s = 0, the inequality (1.2) does not extend to p = 1. This can Key words and phrases. Trace embeddings, overdetermined elliptic operators, elliptic and cancelling operators, C-elliptic operators, Triebel-Lizorkin spaces, functions of bounded variation, functions of bounded deformation, BV A-spaces, strict convergence, Sobolev spaces.

A product property of Sobolev spaces with application to elliptic estimates

Rendiconti del Seminario Matematico della Università di Padova, 2014

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for p P [1Y I), nY m P Z , and m ! 2 that satisfy m b nap, kfck mYpYV K sup Vs jfj 2 3 k ck m Y p Y V k ck m À 1 Y q Y V kck mÀ1YpYV kfk mYpYV 4 5 for all fY c P W mYp (V) that satisfy spt c & V s & V and domains V & R n that are nonempty, open, and satisfy the cone condition. Here q p if p b n, q P (naÇY pna(n À p)] if n b p, q P (naÇY I) if p n, K K(nYpYmYqYg), where g is the cone from the cone condition, and Ç X [[ nap ]], the largest integer less than or equal to nap.

A Kuran Type Regularity Criterion for Sub-Laplacians

Potential Analysis, 2010

The aim of this paper is to show the strong connection between regularity of bounded open set boundary points and quasi-boundedness, on the same set, of the fundamental solution of stratified Lie group sub-Laplacians. In the euclidean case the theorem was proved by Kuran (J Lond Math Soc 2(19): [301][302][303][304][305][306][307][308][309][310][311] 1979). We later give two examples using some direct consequences of main theorem.

Interior Schauder-Type Estimates for Higher-Order Elliptic Operators in Grand-Sobolev Spaces

In this paper an elliptic operator of the m-th order L with continuous coefficients in the n-dimensional domain Ω ⊂ R n in the non-standard Grand-Sobolev space W m q) (Ω) generated by the norm ∥ • ∥ q) of the Grand-Lebesgue space L q) (Ω) is considered. Interior Schauder-type estimates play a very important role in solving the Dirichlet problem for the equation Lu = f. The considered non-standard spaces are not separable, and therefore, to use classical methods for treating solvability problems in these spaces, one needs to modify these methods. To this aim, based on the shift operator, separable subspaces of these spaces are determined, in which finite infinitely differentiable functions are dense. Interior Schauder-type estimates are established with respect to these subspaces. It should be noted that Lebesgue spaces Lq (G) are strict parts of these subspaces. This work is a continuation of the authors of the work [28], which established the solvability in the small of higher order elliptic equations in grand-Sobolev spaces.