Local Hessian estimate for second order parabolic equation in Hardy spaces

Abstract

We derive an interior estimate up to second orders in Hardy spaces for solutions to the parabolic problem {■(u_t-∑_(i,j=1)^n▒a_ij ∂_ij^2 u=f&"in " Ω_T,@u∈h^p (0ⓜ,Tⓜ,h^(1,p) (Ω) )∩h_loc^(1,p) (0ⓜ,Tⓜ,h_loc^(2,p) (Ω) )&)┤ within an appropriate framework. In the course of proof, we also establish the boundedness results of parabolic singular integrals and their commutators on Hardy spaces which are of independent interest.