Non-tangential control of the solutions to deterministic and stochastic parabolic Cauchy problems

Abstract

We study a (stochastic) parabolic Cauchy problem. We first prove that we have a non-tangential control of the solution of the deterministic parabolic Cauchy problem by the tent space norm of the gradient of the solution. This is done by showing boundedness results for several maximal regularity operators and by developing energy solutions in L^2. Thereafter, we show that the stochastic problem satisfies a conical maximal L^p-regularity by using an extrapolation result based on L^2-L^2 and L^1-L^2 off-diagonal bounds in tent spaces. Afterwards, we try to combine these results. To this end, we first prove a necessary condition for the desired result. Then we introduce a vector-valued Banach \tilde{X}p space through a Rademacher maximal function and prove boundedness of a maximal regularity operator from L^p(\Omega; T^{p,2}_\beta) to \tilde{X}p. This provides a sufficient condition for our aim if we can develop a solution of the stochastic parabolic Cauchy problem.