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

Abstract

Let

$$\mathcal{ℒ}=\sum _{i,j=1}^q{a}_{ij}\left(x\right)X_i{X}_j+a_0\left(x\right)X_0,$$

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

$$\mu |\xi {|}^2\le \sum _{i,j=1}^q{a}_{ij}\left(x\right){\xi }_i{\xi }_j\le {\mu }^{-1}|\xi {|}^2,\mu \le a_0\left(x\right)\le {\mu }^{-1},$$

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 $p\in \left(1,\infty \right)$:

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