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Let $u(x,y)$ be a harmonic function in the half-plane ${\mathbf{R}}_{+}^{n+1}$, $n\ge 2$. We define a family of functionals $D(u;r),-\infty \>r\>\infty$, that are analogs of the family of local times associated to the process $u(x_t,y_t)$ where $(x_t,y_t)$ is Brownian motion in ${\mathbf{R}}_{+}^{n+1}$. We show that $D\left(u\right)={sup}_{r}D(u;r)$ is bounded in $L^p$ if and only if $u(x,y)$ belongs to $H^p$, an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.

Let $u$ be harmonic in the half-space $R_{+}^{n+1}$, $n\ge 2$. We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^n=\partial R_{+}^{n+1}$ but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary. In $R_{+}^2$ it is known that the Hardy classes $H^p$, $0\<p\<\infty$, may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the...