Area integral estimates for caloric functions

We study the relationship between the area integral and the parabolic maximal function of solutions to the heat equation in domains whose boundary satisfies a (1/2, 1) mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in L^P(μ), 0 < p < ∞. The measure μ. must satisfy Muckenhoupt’s A∞-condition with respect to caloric measure. We also give a Fatou theorem which shows that the existence of parabolic limits is a.e. (with respect to caloric measure) equivalent to the finiteness of the area integral.