Propagation of regularity in L^p -spaces for Kolmogorov-type hypoelliptic operators

Consider the following Kolmogorov-type hypoelliptic operator Lt:=∑j=2nxj·∇xj-1+tr(at·∇xn2)on R^nd, where n⩾ 2 , d⩾ 1 , x=(x1,…,xn)∈(Rd)n=Rnd and a_t is a time-dependent constant symmetric d× d-matrix that is uniformly elliptic and bounded. Let { T_{s , t}; t⩾ s} be the time-dependent semigroup associated with L_t; that is, ∂_sT_{s , t}f= - L_sT_{s , t}f. For any p∈ (1 , ∞) , we show that there is a constant C= C(p, n, d) > 0 such that for any f(t, x) ∈ L^p(R× R^nd) = L^p(R^{1 + n d}) and every λ⩾ 0 , ∥Δxj1/(1+2(n-j))∫0∞e-λtTs,t+sf(t+s,x)dt∥p⩽C‖f‖p,j=1,…,n,where ‖ · ‖ _p is the usual L^p-norm in L^p(R× R^nd; d s× d x). To show this type of estimates, we first study the propagation of regularity in L²-space from variable x_n to x_j, 1 ⩽ j⩽ n- 1 , for the solution of the transport equation ∂tu+∑j=2nxj·∇xj-1u=f.