Localization and interpolation of parabolic $L^p$ Neumann problems

We show a localization estimate for local solutions to the parabolic equation $-\partial_t u+\mbox{div} (A\nabla u)=0$ with zero Neumann data, assuming that the $L^p$ Neumann problem and $L^{p'}$ Dirichlet problem for the adjoint operator are solvable in a Lipschitz cylinder for some $p\in(1,\infty)$. Using this result, we establish the solvability of the Neumann problem in the atomic Hardy space for parabolic operators with bounded, measurable, time-dependent coefficients, and hence obtain the extrapolation of solvability of the $L^p$ Neumann problem.