Abstract
In this paper we present the following result on regularity of solutions of the second order parabolic equation ∂tu - div (A∇u) + B · ∇u = 0 on cylindrical domains of the form Ω = O × R where O ⊂ Rn is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is n - 1 -Ahlfors regular. Let u be a solution of such PDE in Ω and the non-tangential maximal function of its gradient in spatial directions N~(∇u) belongs to Lp(∂Ω) for some p > 1 . Furthermore, assume that for u|∂Ω = f we have that Dt1/2f ∈ Lp(∂Ω) . Then both N~(Dt1/2u) and N~(Dt1/2Htu) also belong to Lp(∂Ω) , where Dt1/2 and Ht are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the Lp parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.