Global hypoelliptic estimates for Landau-type operators with external potential (Q371220)

The authors study Landau-type equations with an external force, that is, NEWLINE\[NEWLINEP= i(yD_x- \nabla_x V(x) D_y)- (B(y) D_y)^* B(y)D_y+ F(y),NEWLINE\]NEWLINE where \(x\in\mathbb{R}^3\) is the space variable, \(y\in\mathbb{R}^3\) is the velocity variable, and \(B(y)\) is a \(3\times 3\) matrix with elements satisfying NEWLINE\[NEWLINE|D^\alpha_y B_{jk}(y)|\leq C_\alpha\langle y\rangle^{1-|\alpha|+\lambda/2},\quad 1\leq j,\;k\leq 3,\;0\leq\lambda\leq 1,\;y\in\mathbb{R}^3.NEWLINE\]NEWLINE Here, the real-valued function \(V(x)\) denotes the potential of the macroscopic external force and \(F(y)\) is a smooth function that satisfies suitable global estimates.NEWLINENEWLINE The main result of the paper is an a priori estimate for a weighted norm of the solutions \(u\) of the equation \(Pu= f\in L^2(\mathbb{R}^6)\). The present paper is the natural continuation of a previous work of the first author and \textit{K. Pravda-Starov} [J. Math. Pures Appl. (9) 95, No. 5, 513--552 (2011; Zbl 1221.35107)] concerning the case when there is no potential.