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$L^{p}$ and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift

Marco Bramanti and Maochun Zhu

Vol. 6 (2013), No. 8, 1793–1855


= i,j=1qa ij(x)XiXj + a0(x)X0,

where X0,X1,,Xq are real smooth vector fields satisfying Hörmander’s condition in some bounded domain Ω n (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=1qa ij(x)ξiξj μ1|ξ|2,μ a 0(x) μ1,

for a.e. x Ω, every ξ q, 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:

XiXjuCXα(Ω) + X0uCXα(Ω) c{LuCXα(Ω) + uL(Ω)}

for any Ω Ω.

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,):

XiXjuLp(Ω) + X0uLp(Ω) c{LuLp(Ω) + uLp(Ω)}.

Hörmander's vector fields, Schauder estimates, $L^p$ estimates, drift
Mathematical Subject Classification 2010
Primary: 35H20
Secondary: 42B20, 35B45, 53C17
Received: 24 December 2011
Revised: 5 April 2013
Accepted: 13 June 2013
Published: 20 April 2014
Marco Bramanti
Dipartimento di Matematica
Politecnico di Milano
Via Bonardi 9
I-20133 Milano
Maochun Zhu
Department of Applied Mathematics
Northwestern Polytechnical University
127 West Youyi Road
Xi’an, 710072