Fundamental solution and Lp estimates for higher order subelliptic Schrodinger operators on stratified groups.

摘要

Let G be a nilpotent, stratified homogeneous group, and let X 1, ···, Xl be left invariant vector fields generating the Lie algebra G associated to G . In [13] G. Lu proved a Fefferman-Phong type inequality for degenerate vector fields and stated that various operators associated with the sub-Laplacian - i=1lX2 ix plus a nonnegative potential are Lp bounded on the homogeneous group G when V(x) is a nonnegative group polynomial on G or satisfies a certain Reverse Holder inequality in the metric space ( G , rho) defined by the homogeneous norms. In this paper, we provided details of proof of theorems stated in [13]. Furthermore, we extended the results to higher order subelliptic operators -i=1lX 2ix+V xm and -i=1lX 2ix 2+Vx2 when V is a nonnegative polynomial. We obtained the Lp boundedness for various operators related to the two operators above. We also gave the fundamental solution estimates for -i=1lX 2ix 2+Vx2 and proved that the fundamental solution to -i=1lX 2ix 2+Vx2 are differentiable away from the pole and behaves like that of -i=1lX 2ix 2 for rho(x, y) m( y, V)--1 while decays faster than any negative power of rho(x, y) for rho( x, y) > m(y, V)--1 . Finally, to get the fundamental solution estimates to -i=1lX 2ix 2+Vx2 , we proved Caccioppoli Inequality and Mean-Value Inequality for the equation -i=1l X2ix 2+Vx 2 u(x) = 0.