Sharp estimates for the nontangential maximal function and the Lusin area function in Lipschitz domains

Abstract:

Let $u$ be a harmonic function on a domain of the form $D = \{ (x,y):x \in {{\mathbf {R}}^n},y \in {\mathbf {R}},y > \phi (x)\}$ where $\phi :{{\mathbf {R}}^n} \to {\mathbf {R}}$ is a Lipschitz function. The authors show a good-$\lambda$ inequality between $Au$, the Lusin area function of $u$, and $Nu$, the nontangential maximal function of $u$. This leads to an ${L^p}$ inequality of the form $\left \|Au\right \|_p \leq C_p\left \|Nu\right \|_p$ which is sharp in the sense that ${C_p}$ is of the smallest possible order in $p$ as $p \to \infty$. For $P \in \partial D$ and $t > 0$ we also consider the functions $Au(P + (0,t))$ and $Nu(P + (0,t))$ and show that a corollary of the good-$\lambda$ inequality is a law of the iterated logarithm involving these two functions as $t \to 0$. If $n = 1$ and $\phi$ has a small Lipschitz constant the above results are shown valid with the roles of $Nu$ and $Au$ interchanged.