Area integral estimates for caloric functions

Abstract

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 ) \left ({\frac {1}{2},1}\right ) mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in L p ( μ ) {L^p}(\mu ) , 0 > p > ∞ 0 > p > \infty . The measure μ \mu must satisfy Muckenhoupt’s A ∞ {A_\infty } -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.