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2013 $L^{p}$ and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift
Marco Bramanti, Maochun Zhu
Anal. PDE 6(8): 1793-1855 (2013). DOI: 10.2140/apde.2013.6.1793

Abstract

Let

= i , j = 1 q a i j ( x ) X i X j + a 0 ( x ) X 0 ,

where X0,X1,,Xq are real smooth vector fields satisfying Hörmander’s condition in some bounded domain Ωn (n>q+1), and the coefficients aij=aji, a0 are real valued, bounded measurable functions defined in Ω, satisfying the uniform positivity conditions

μ | ξ | 2 i , j = 1 q a i j ( x ) ξ i ξ j μ 1 | ξ | 2 , μ a 0 ( x ) μ 1 ,

for a.e. xΩ, every ξq, and some constant μ>0.

We prove that if the coefficients aij, a0 belong to the Hölder space CXα(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 ) c { L u C X α ( Ω ) + u L ( Ω ) }

for any ΩΩ.

If the coefficients aij, a0 belong to the space VMOX,loc(Ω) with respect to the distance induced by the vector fields, local Lp estimates of the following kind hold, for every p(1,):

X i X j u L p ( Ω ) + X 0 u L p ( Ω ) c { L u L p ( Ω ) + u L p ( Ω ) } .

Citation

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Marco Bramanti. Maochun Zhu. "$L^{p}$ and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift." Anal. PDE 6 (8) 1793 - 1855, 2013. https://doi.org/10.2140/apde.2013.6.1793

Information

Received: 24 December 2011; Revised: 5 April 2013; Accepted: 13 June 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 06322717
MathSciNet: MR3198585
Digital Object Identifier: 10.2140/apde.2013.6.1793

Subjects:
Primary: 35H20
Secondary: 35B45 , 42B20 , 53C17

Keywords: $L^P$ estimates , drift , Hörmander's vector fields , Schauder estimates

Rights: Copyright © 2013 Mathematical Sciences Publishers

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