Area Integral Estimates for Caloric Functions
R. M. Brown · Transactions of the American Mathematical Society · 1989
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 $\left ({\frac {1}{2},1}\right )$ mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in ${L^p}(\mu )$, $0 < p < \infty$. The measure $\mu$ must satisfy Muckenhoupt’s ${A_\infty }$-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.