Motivated by studies of the nonlinear kinetic Landau equation in a bounded domain, we investigate the linearized equation with the specular reflection boundary condition. To study the regularity of the solution near the boundary, one needs to develop the $S_p$ theory (Calderon-Zygmund theory) for the degenerate Kolmogorov-Fokker-Planck equation with `irregular' coefficients in the whole space. This is joint work with Hongjie Dong and Yan Guo. In the second part of the talk, I explain how to derive such $S_p$ estimates using a `kernel-free' approach. This is joint work with Hongjie Dong.Motivated by studies of the nonlinear kinetic Landau equation in a bounded domain, we investigate the linearized equation with the specular reflection boundary condition. To study the regularity of the solution near the boundary, one needs to develop the $S_p$ theory (Calderon-Zygmund theory) for the degenerate Kolmogorov-Fokker-Planck equation with `irregular' coefficients in the whole space. This is joint work with Hongjie Dong and Yan Guo. In the second part of the talk, I explain how to derive such $S_p$ estimates using a `kernel-free' approach.

This is joint work with Hongjie Dong.