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 u be harmonic in the half-space R + n + 1 , n ≥ 2 . We show that u can have a fine limit at almost every point of the unit cubs in R n = ∂ 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 < ∞ , 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 ≥ 2 .

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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