Global L^p estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients

2012/09/03 by Marco Bramanti, Giovanni Cupini, Bramanti, Marco +5
Mathematics · #35B45 #35H10 #35K70 #42B20 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B45 #msc:35H10 #msc:35K70 #msc:42B20

paper · pdf · doi:10.48550/arxiv.1209.0387

arxiv created 2012/09/03 · arxiv updated 2012/09/04

Abstract

We consider a class of degenerate Ornstein-Uhlenbeck operators in ℝ^N, of the kind [A≡∑_i,j=1^p₀a_ij(x) ∂_x_ix_j²+∑_i,j=1^Nb_ijx_i∂_x_j%] where (a_ij) is symmetric uniformly positive definite on ℝ^p₀ (p₀≤ N), with uniformly continuous and bounded entries, and (b_ij) is a constant matrix such that the frozen operator A_x₀ corresponding to a_ij(x₀) is hypoelliptic. For this class of operators we prove global L^p estimates (1<p<∞) of the kind:% [|∂_x_ix_j²u|_{L^p(ℝ% ^N)}≤ c|Au|_L^p(ℝ^N)+|u|_{L^p(ℝ% ^N)} for i,j=1,2,...,p₀.] We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% [|∂_x_ix_j²u|_L^p(S_T)≤ c|Lu|_L^p(S_T)+|u|_L^p(S_T)] for any u∈ C₀^∞(S_T), where S_T is the strip ℝ^N×[-T,T], T small, and L is the Kolmogorov-Fokker-Planck operator% [L≡∑_i,j=1^p₀a_ij(x,t) ∂_x_ix_j% ²+∑_i,j=1^Nb_ijx_i∂_x_j-∂_t%] with uniformly continuous and bounded a_ij's.

Related

• Global L^p estimates for degenerate Ornstein-Uhlenbeck operators
• On a class of hypoelliptic operators with unbounded coefficients in \matbb R^N
• L^p-parabolic regularity and non-degenerate Ornstein-Uhlenbeck type operators
• On weak uniqueness for some degenerate SDEs by global L^p estimates
• L^p Estimates For Degenerate Non-Local Kolmogorov Operators