Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Schauder estimates for nondivergence stationary operators modeled on homogeneous Hörmander vector fields

Matteo Faini

Abstract

In this paper we prove global regularity results and Schauder estimates for non-divergence stationary operators of the form L=\sum_{i,j=1}^m a_{ij}(x) X_i X_j, where X_1, ..., X_m are homogeneous (but not necessarily left-invariant) Hörmander vector fields in R^n (n>m), and [a_{ij}(x)] is a symmetric uniformly positive-definite matrix with Hölder-continuous entries w.r.t. the control distance induced by the vector fields X_1, ..., X_m.

Global Schauder estimates for nondivergence stationary operators modeled on homogeneous Hörmander vector fields

Abstract

In this paper we prove global regularity results and Schauder estimates for non-divergence stationary operators of the form L=\sum_{i,j=1}^m a_{ij}(x) X_i X_j, where X_1, ..., X_m are homogeneous (but not necessarily left-invariant) Hörmander vector fields in R^n (n>m), and [a_{ij}(x)] is a symmetric uniformly positive-definite matrix with Hölder-continuous entries w.r.t. the control distance induced by the vector fields X_1, ..., X_m.
Paper Structure (7 sections, 22 theorems, 151 equations)

This paper contains 7 sections, 22 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Carnot groups
  3. Global Schauder estimates in Carnot groups
  4. Local estimates for second order derivatives
  5. From local to global
  6. Estimates for higher order derivatives
  7. Removal of the group structure

Key Result

Theorem 1.2

Under assumptions (H1)-(H2)-(H3) on $\mathcal{L}$, there exists a positive constant $c_0=c_0( \{X_i\}_{i=1}^m,\Lambda, \alpha, ||a||_{0,X})$ such that for every $u \in C^{2,\alpha}_X(\mathbb{R}^n)$ the following estimate holds: If moreover we assume that, for some $k \geq 1$, $a_{ij} \in C^{k,\alpha}_X(\mathbb{R}^n)$ for all $1 \leq i,j \leq m$, then there exists a constant $c_k=c(k,\{X_i\}_{i=1}

Theorems & Definitions (41)

  • Remark 1.1
  • Theorem 1.2: Global Schauder estimates
  • Theorem 1.3: Global lifting
  • Remark 1.4: Comparison with Rothschild-Stein's lifting
  • Example 1.5
  • Theorem 1.6: Global Schauder estimates in Carnot groups
  • Theorem 1.7: Higher order estimates in Carnot groups
  • Proposition 2.1
  • proof
  • Definition 2.2
  • ...and 31 more