2017/06/07 by Chen, Zhen-Qing, Zhang, Xicheng
#42B37 #60H10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1706.02181
Consider the following Kolmogorov type hypoelliptic operator \mathscr Lt:=\mbox∑j=2nxj⋅∇_xj-1+\rm Tr (at ⋅∇2xn), where n≥ 2, x=(x1,⋯,xn)∈(\mathbb Rd)n =\mathbb Rnd and at is a time-dependent constant symmetric d× d-matrix that is uniformly elliptic and bounded.. Let \\mathcal Ts,t; t≥ s\ be the time-dependent semigroup associated with \mathscr Lt; that is, ∂s \mathcal Ts, t f = - \mathscr Ls \mathcal Ts, tf. For any p∈(1,∞), we show that there is a constant C=C(p,n,d)>0 such that for any f(t, x)∈ Lp(\mathbb R × \mathbb Rnd)=Lp(\mathbb R1+nd) and every λ≥ 0, ‖Δxj^1/(1+2(n-j))∫∞0 e-λt \mathcal Ts, s+t f(t+s, x)dt‖p≤ C‖f‖p, j=1,⋯, n, where ‖⋅‖p is the usual Lp-norm in Lp(\mathbb R1+nd; d s× d x). To show this type of estimates, we first study the propagation of regularity in L2-space from variable xn to x1 for the solution of the transport equation ∂t u+∑j=2nxj⋅∇_xj-1 u=f.