Local Hessian estimate for second order parabolic equation in Hardy spaces - Trung, Nguyen Duc; Truong, Le Xuan; Do, Tan Duc; Trong, Nguyen Ngoc (Mathematica Scandinavica, 2025)
We derive an interior estimate up to second orders in Hardy spaces for solutions to the parabolic problem $$ \begin {cases} u_t - \sum _{i, j=1}^n a_{i j} \partial ^2_{ij} u = f & \text {in $\Omega _T$}, \\ u \in h^p(0,T; h^{1,p}(\Omega )) \cap h^{1,p}_{loc }(0,T ; h^{2,p}_{loc }(\Omega )) \end {cases} $$ 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.