#### Vol. 6, No. 8, 2013

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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
##### Abstract

Let

$\mathsc{ℒ}=\sum _{i,j=1}^{q}{a}_{ij}\left(x\right){X}_{i}{X}_{j}+{a}_{0}\left(x\right){X}_{0},$

where ${X}_{0},{X}_{1},\dots ,{X}_{q}$ are real smooth vector fields satisfying Hörmander’s condition in some bounded domain $\Omega \subset {ℝ}^{n}$ ($n>q+1$), and the coefficients ${a}_{ij}={a}_{ji}$, ${a}_{0}$ are real valued, bounded measurable functions defined in $\Omega$, satisfying the uniform positivity conditions

$\mu |\xi {|}^{2}\le \sum _{i,j=1}^{q}{a}_{ij}\left(x\right){\xi }_{i}{\xi }_{j}\le {\mu }^{-1}|\xi {|}^{2},\phantom{\rule{1em}{0ex}}\mu \le {a}_{0}\left(x\right)\le {\mu }^{-1},$

for a.e. $x\in \Omega$, every $\xi \in {ℝ}^{q}$, and some constant $\mu >0$.

We prove that if the coefficients ${a}_{ij}$, ${a}_{0}$ belong to the Hölder space ${C}_{X}^{\alpha }\left(\Omega \right)$ with respect to the distance induced by the vector fields, local Schauder estimates of the following kind hold:

$\parallel {X}_{i}{X}_{j}u{\parallel }_{{C}_{X}^{\alpha }\left({\Omega }^{\prime }\right)}+\parallel {X}_{0}u{\parallel }_{{C}_{X}^{\alpha }\left({\Omega }^{\prime }\right)}\le c\left\{\parallel Lu{\parallel }_{{C}_{X}^{\alpha }\left(\Omega \right)}+\parallel u{\parallel }_{{L}^{\infty }\left(\Omega \right)}\right\}$

for any ${\Omega }^{\prime }⋐\Omega$.

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

$\parallel {X}_{i}{X}_{j}u{\parallel }_{{L}^{p}\left({\Omega }^{\prime }\right)}+\parallel {X}_{0}u{\parallel }_{{L}^{p}\left({\Omega }^{\prime }\right)}\le c\left\{\parallel Lu{\parallel }_{{L}^{p}\left(\Omega \right)}+\parallel u{\parallel }_{{L}^{p}\left(\Omega \right)}\right\}.$

##### Keywords
Hörmander's vector fields, Schauder estimates, $L^p$ estimates, drift
##### Mathematical Subject Classification 2010
Primary: 35H20
Secondary: 42B20, 35B45, 53C17