no. 4
Boundary behaviour of harmonic functions in a half-space and brownian motion
Annales de l'Institut Fourier, Tome 23 (1973) no. 4, pp. 195-212.

Soit u(x,y) une fonction harmonique dans le demi-espace R + n+1 , n≥2. Nous montrons que u(x,y) peut avoir une limite fine en presque chaque point du cube unité dans R n =∂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<∞, par l’intégrabilité de la fonction maximale du mouvement brownien. Nous montrons que ce résultat est aussi valable dans R + n+1 , pour n≥2.

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.

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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 = {Institut Fourier},
     address = {Grenoble},
     volume = {23},
     number = {4},
     year = {1973},
     doi = {10.5802/aif.487},
     zbl = {0253.31010},
     mrnumber = {51 #1943},
     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, Tome 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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