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Transactions of the American Mathematical Society

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

Authors: Rodrigo Bañuelos and Charles N. Moore
Journal: Trans. Amer. Math. Soc. 312 (1989), 641-662
MSC: Primary 42B25; Secondary 31A20
MathSciNet review: 957080
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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\Vert Au\right\Vert _p \leq C_p\left\Vert Nu\right\Vert _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.

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