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

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

top

## Abstract

top
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 integrability of a Brownian motion maximal function. This result is shown to hold in ${R}_{+}^{n+1}$, for $n\ge 2$.

## How to cite

top

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

top
1. [1] J. M. BRELOT and L. DOOB, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble), 13, (1963), 395-415. Zbl0132.33902MR33 #4299
2. [2] D.L. BURKHOLDER and R.F. GUNDY, Distribution function inequalities for the area integral, Studia Math., 44, (1972), 527-544. Zbl0219.31009MR49 #5309
3. [3] D.L. BURKHOLDER, R.F. GUNDY and M. L. SILVERSTEIN, A maximal function characterization of the class Hp, Trans. Amer. Math. Soc., 157 (1971), 137-153. Zbl0223.30048MR43 #527
4. [4] A. P. CALDERÓN, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc., 68, (1950), 47-54. Zbl0035.18901MR11,357e
5. [5] A. P. CALDERÓN, On a theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc., 68, (1950), 55-61. Zbl0035.18903MR11,357f
6. [6] L. CARLESON, On the existence of boundary values for harmonic functions in several variables, Arkiv för Mathematik, 4, (1961), 393-399. Zbl0107.08402MR28 #2232
7. [7] C. CONSTANTINESCU and A. CORNEA, Über das Verhalten der analytischen Abildungen Riemannscher Flachen auf dem idealen Rand von Martin, Nagoya Math. J., 17, (1960), 1-87. Zbl0104.29901MR23 #A1025
8. [8] J. L. DOOB, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85, (1957), 431-458. Zbl0097.34004MR22 #844
9. [9] J. L. DOOB, Boundary limit theorems for a half-space, J. Math. Pures Appl., (9) 37, (1958), 385-392. Zbl0097.34101MR22 #845
10. [10] C. FEFFERMAN and E. M. STEIN, Hp-spaces in several variables, Acta Math., 129, (1972), 137-193. Zbl0257.46078MR56 #6263
11. [11] G. H. HARDY and J. E. LITTLEWOOD, Some properties of conjugate functions, J. fur Mat., 167, (1931), 405-423. Zbl0003.20203JFM58.0333.03
12. [12] J. LELONG, Review 4471, Math. Reviews, 40, (1970), 824-825.
13. [13] J. LELONG, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66, (1949), 125-159. Zbl0033.37301
14. [14] J. MARCINKIEWICZ and A. ZYGMUND, A theorem of Lusin, Duke Math. J., 4, (1938), 473-485. Zbl0019.42001JFM64.0268.01
15. [15] H. P. McKEAN Jr., Stochastic integrals, Academic Press, New York, 1969. Zbl0191.46603
16. [16] L. NAÏM, Sur le rôle de la frontière de R. S. Martin dans la Théorie du potential, Ann. Inst. Fourier (Grenoble), 7, (1957), 183-285. Zbl0086.30603MR20 #6608
17. [17] I. I. PRIVALOV, Integral Cauchy, Saratov, 1919.
18. [18] D. SPENCER, A function-theoretic identity, Amer. J. Math., 65, (1943), 147-160. Zbl0060.20603MR4,137f
19. [19] E. M. STEIN, On the theory of harmonic functions of several variables II. Behaviour near the boundary, Acta Math., 106, (1961), 137-174. Zbl0111.08001MR30 #3234
20. [20] E. M. STEIN and G. WEISS, On the theory of harmonic functions of several variables I. The theory of Hp-spaces, Acta Math., 103, (1960), 25-62. Zbl0097.28501MR22 #12315
21. [21] J. L. WALSH, The approximation of harmonic functions by harmonic polynomials and harmonic rational functions, Bull. Amer. Math. Soc., 35, (1929), 499-544. Zbl55.0889.05JFM55.0889.05

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.