Boundary behaviour of harmonic functions in a half-space and brownian motion

D. L. Burkholder; Richard F. Gundy

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 4, page 195-212
  • ISSN: 0373-0956

Abstract

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Let be harmonic in the half-space , . We show that can have a fine limit at almost every point of the unit cubs in 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 it is known that the Hardy classes , , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability of a Brownian motion maximal function. This result is shown to hold in , for .

How to cite

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Burkholder, D. L., and Gundy, Richard F.. "Boundary behaviour of harmonic functions in a half-space and brownian motion." Annales de l'institut Fourier 23.4 (1973): 195-212. <http://eudml.org/doc/74146>.

@article{Burkholder1973,
abstract = {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&lt; p&lt; \infty $, may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability of a Brownian motion maximal function. This result is shown to hold in $R^\{n+1\}_+$, for $n\ge 2$.},
author = {Burkholder, D. L., Gundy, Richard F.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {195-212},
publisher = {Association des Annales de l'Institut Fourier},
title = {Boundary behaviour of harmonic functions in a half-space and brownian motion},
url = {http://eudml.org/doc/74146},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Burkholder, D. L.
AU - Gundy, Richard F.
TI - Boundary behaviour of harmonic functions in a half-space and brownian motion
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 4
SP - 195
EP - 212
AB - 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&lt; p&lt; \infty $, may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability of a Brownian motion maximal function. This result is shown to hold in $R^{n+1}_+$, for $n\ge 2$.
LA - eng
UR - http://eudml.org/doc/74146
ER -

References

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