Global L ^p estimates for degenerate Ornstein–Uhlenbeck operators

Abstract

We consider a class of degenerate Ornstein–Uhlenbeck operators in ${\mathbb{R}^{N}}$ , of the kind

$$\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}$$

where (a _ij ), (b _ij ) are constant matrices, (a _ij ) is symmetric positive definite on ${\mathbb{R} ^{p_{0}}}$ (p ₀ ≤ N), and (b _ij ) is such that ${\mathcal{A}}$ is hypoelliptic. For this class of operators we prove global L ^p estimates (1 < p < ∞) of the kind:

$$\left\Vert \partial_{x_{i}x_{j}}^{2}u\right\Vert _{L^{p}\left(\mathbb{R}^{N}\right)} \leq c \left\{\left\Vert \mathcal{A}u\right\Vert _{L^{p}\left(\mathbb{R}^{N}\right)} + \left\Vert u\right\Vert_{L^{p}\left(\mathbb{R}^{N}\right)}\right\} {\rm for} \, i, j = 1, 2, \ldots, p_{0}$$

and corresponding weak type (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue L ^p spaces, for complete Hörmander’s operators

$$\sum X_{i}^{2}+X_{0},$$

proved in absence of a structure of homogeneous group. We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:

$$\left\Vert \partial_{x_{i}x_{j}}^{2}u\right\Vert _{L^{p}\left( S\right)}\leq c\left\Vert Lu\right\Vert _{L^{p}\left( S\right)}$$

for any ${u \in C_{0}^{\infty} \left(S\right)}$ , where S is the strip ${\mathbb{R}^{N} \times \left[-1, 1\right]}$ and L is the Kolmogorov-Fokker-Planck operator ${\mathcal{A} - \partial_{t}}$ . To get this estimate we use in a crucial way the left invariance of L with respect to a Lie group structure in ${\mathbb{R}^{N+1}}$ and some results on singular integrals on nonhomogeneous spaces recently proved in Bramanti (Revista Matematica Iberoamericana, 2009, in press).