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 .
Soit une fonction harmonique dans le demi-espace , . Nous montrons que peut avoir une limite fine en presque chaque point du cube unité dans 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 , il est connu que l’on peut caractériser les classes de Hardy , , par l’intégrabilité de la fonction maximale du mouvement brownien. Nous montrons que ce résultat est aussi valable dans , pour .
@article{AIF_1973__23_4_195_0, 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/} }
TY - JOUR TI - Boundary behaviour of harmonic functions in a half-space and brownian motion JO - Annales de l'Institut Fourier PY - 1973 DA - 1973/// SP - 195 EP - 212 VL - 23 IS - 4 PB - Imprimerie Durand PP - 28 - Luisant UR - https://aif.centre-mersenne.org/articles/10.5802/aif.487/ UR - https://www.ams.org/mathscinet-getitem?mr=51 #1943 UR - https://zbmath.org/?q=an%3A0253.31010 UR - https://doi.org/10.5802/aif.487 DO - 10.5802/aif.487 LA - en ID - AIF_1973__23_4_195_0 ER -
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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