Lp and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift

where $X_{0}, X_{1}, \dots, X_{q}$ are real smooth vector fields satisfying Hörmander’s condition in some bounded domain $Ω \subset ℝ^{n}$ ( $n > q + 1$ ), and the coefficients $a_{i j} = a_{j i}$ , $a_{0}$ are real valued, bounded measurable functions defined in $Ω$ , satisfying the uniform positivity conditions

μ | ξ |^{2} \leq \sum_{i, j = 1}^{q} a_{i j} (x) ξ_{i} ξ_{j} \leq μ^{- 1} | ξ |^{2}, μ \leq a_{0} (x) \leq μ^{- 1},

for a.e. $x \in Ω$ , every $ξ \in ℝ^{q}$ , and some constant $μ > 0$ .

We prove that if the coefficients $a_{i j}$ , $a_{0}$ belong to the Hölder space $C_{X}^{α} (Ω)$ with respect to the distance induced by the vector fields, local Schauder estimates of the following kind hold:

∥ X_{i} X_{j} u ∥_{C_{X}^{α} (Ω^{'})} + ∥ X_{0} u ∥_{C_{X}^{α} (Ω^{'})} \leq c {∥ L u ∥_{C_{X}^{α} (Ω)} + ∥ u ∥_{L^{\infty} (Ω)}}

for any $Ω^{'} ⋐ Ω$ .

If the coefficients $a_{i j}$ , $a_{0}$ belong to the space ${VMO}_{X, loc} (Ω)$ with respect to the distance induced by the vector fields, local $L^{p}$ estimates of the following kind hold, for every $p \in (1, \infty)$ :

∥ X_{i} X_{j} u ∥_{L^{p} (Ω^{'})} + ∥ X_{0} u ∥_{L^{p} (Ω^{'})} \leq c {∥ L u ∥_{L^{p} (Ω)} + ∥ u ∥_{L^{p} (Ω)}} .

Keywords

Hörmander's vector fields, Schauder estimates, $L^p$ estimates, drift

Mathematical Subject Classification 2010

Primary: 35H20

Secondary: 42B20, 35B45, 53C17

Milestones

Received: 24 December 2011

Revised: 5 April 2013

Accepted: 13 June 2013

Published: 20 April 2014

Authors

Marco Bramanti
Dipartimento di Matematica Politecnico di Milano Via Bonardi 9 I-20133 Milano Italy

Maochun Zhu
Department of Applied Mathematics Northwestern Polytechnical University 127 West Youyi Road Xi’an, 710072 China

Vol. 6, No. 8, 2013

Marco Bramanti and Maochun Zhu

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors