The density of the area integral in ${ℝ}_{+}^{n+1}$

Gundy, Richard F., and Silverstein, Martin L.. "The density of the area integral in ${\mathbb {R}}^{n+1}_+$." Annales de l'institut Fourier 35.1 (1985): 215-229. <http://eudml.org/doc/74666>.

@article{Gundy1985,
abstract = {Let $u(x,y)$ be a harmonic function in the half-plane $\{\bf R\}_+^\{n+1\}$, $n\ge 2$. We define a family of functionals $D(u;r), -\infty &gt;r&gt;\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 $\{\bf R\}_+^\{n+1\}$. We show that $D(u)=\sup _\{r\} D(u;r)$ 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.},
author = {Gundy, Richard F., Silverstein, Martin L.},
journal = {Annales de l'institut Fourier},
keywords = {Brownian motion; singular integrals},
language = {eng},
number = {1},
pages = {215-229},
publisher = {Association des Annales de l'Institut Fourier},
title = {The density of the area integral in $\{\mathbb \{R\}\}^\{n+1\}_+$},
url = {http://eudml.org/doc/74666},
volume = {35},
year = {1985},
}

TY - JOUR
AU - Gundy, Richard F.
AU - Silverstein, Martin L.
TI - The density of the area integral in ${\mathbb {R}}^{n+1}_+$
JO - Annales de l'institut Fourier
PY - 1985
PB - Association des Annales de l'Institut Fourier
VL - 35
IS - 1
SP - 215
EP - 229
AB - Let $u(x,y)$ be a harmonic function in the half-plane ${\bf R}_+^{n+1}$, $n\ge 2$. We define a family of functionals $D(u;r), -\infty &gt;r&gt;\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 ${\bf R}_+^{n+1}$. We show that $D(u)=\sup _{r} D(u;r)$ 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.
LA - eng
KW - Brownian motion; singular integrals
UR - http://eudml.org/doc/74666
ER -