Global \(L^{p}\) estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients (Q604219)

The authors consider a class of degenerate Ornstein-Uhlenbeck operators in \(\mathbb R^N\), of the kind \[ {\mathcal A}\equiv\sum_{i, j=1}^{p_0} a_{ij}\partial_{x_ix_j}^2+ \sum_{i, j=1}^N b_{ij}x_i \partial_{x_j}, \] where \((a _{ij}), (b _{ij })\) are constant matrices, \((a _{ij})\) is symmetric positive definite on \(\mathbb R^{p_0}\) \((p _0\leq N)\), and \((b _{ij})\) is such that \({\mathcal A}\) is hypoelliptic. For this class of operators it is proved a global \(L^p\) estimate \((1<p<\infty)\) of the kind \[ \left\|\partial_{x_ix_j}^2u\right\|_{L^p(\mathbb R^N)}\leq c\left\{\|{\mathcal A}u\|_{L^p(\mathbb R^N)}+ \|u\|_{L^p(\mathbb R^N)}\right\} \quad \text{for }i,j = 1, 2,\dots,p_0 \] and corresponding weak type \((1,1)\) estimates \[ \left|\left\{x\in\mathbb R^N: |\partial_{x_ix_j}^2u|> \alpha\right\}\right| \leq \frac{c_1}{\alpha} \left\{\| {\mathcal A}u\| _{L^1(\mathbb R^N)}+ \|u\|_{L^1(\mathbb R^N)}\right\}. \] This result seems to be the first case of global estimates, in Lebesgue \(L^{p}\) spaces, for complete Hörmander operators \[ \sum X_{i}^{2}+X_{0}, \] proved in absence of a structure of a homogeneous group. The previous estimates are obtained as a byproduct of the following one, which is of interest in its own: \[ \|\partial_{x_ix_j}^2u\|_{L^p(S)}\leq c\|Lu\|_{L^{p}(S)} \] for any \(u \in C_0^\infty(S)\), where \(S\) is the strip \(\mathbb R^N\times[-1,1]\) and \(L\) is the Kolmogorov-Fokker-Planck operator \({\mathcal A}-\partial_t\). To get this estimate it is used 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 [\textit{M. Bramanti}, Mat. Soc. Cult., Riv. Unione Mat. Ital. (1) 2, No.~3, 447--493 (2009; Zbl 1408.62180)].