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

Burkholder, D. L.; Gundy, Richard F.

doi:10.5802/aif.487

Burkholder, D. L.; Gundy, Richard F.

Annales de l'Institut Fourier, Volume 23 (1973) no. 4, pp. 195-212.

Abstract
Résumé

Let $u$ be harmonic in the half-space $R_{+}^{n + 1}$ , $n \geq 2$ . 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 $n \geq 2$ .

Soit $u (x, y)$ une fonction harmonique dans le demi-espace $R_{+}^{n + 1}$ , $n \geq 2$ . Nous montrons que $u (x, y)$ peut avoir une limite fine en presque chaque point du cube unité dans $R^{n} = \partial R_{+}^{n + 1}$ sans avoir pourtant de limite non tangentielle en aucun point du cube. La méthode est probabiliste et utilise l’équivalence entre limites conditionnelles du mouvement brownien et limites fines à la frontière.

Dans $R_{+}^{2}$ , il est connu que l’on peut caractériser les classes de Hardy $H^{p}$ , $0 < p < \infty$ , par l’intégrabilité de la fonction maximale du mouvement brownien. Nous montrons que ce résultat est aussi valable dans $R_{+}^{n + 1}$ , pour $n \geq 2$ .

MR: 51 #1943 | Zbl: 0253.31010

DOI: 10.5802/aif.487

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     author = {Burkholder, D. L. and Gundy, Richard F.},
     title = {Boundary behaviour of harmonic functions in a half-space and brownian motion},
     journal = {Annales de l'Institut Fourier},
     pages = {195--212},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {23},
     number = {4},
     year = {1973},
     doi = {10.5802/aif.487},
     mrnumber = {51 #1943},
     zbl = {0253.31010},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.487/}
}

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Burkholder, D. L.; Gundy, Richard F. Boundary behaviour of harmonic functions in a half-space and brownian motion. Annales de l'Institut Fourier, Volume 23 (1973) no. 4, pp. 195-212. doi : 10.5802/aif.487. https://aif.centre-mersenne.org/articles/10.5802/aif.487/

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