Soit une fonction harmonique dans le demi-plan , . Nous définissons une famille de fonctionnelles , qui sont les analogues géométriques de la famille des temps locaux associés au processus où est le mouvement brownien dans . Nous montrons que est borné dans si et seulement si la fonction appartien à , une équivalence qui a été déjà démontrée par Barlow et Yor pour le supremum des temps locaux. Signalons que notre démonstration tourne autour de la théorie des intégrales singulières de Caldéron-Zygmund plutôt que le calcul stochastique.
Let be a harmonic function in the half-plane , . We define a family of functionals , that are analogs of the family of local times associated to the process where is Brownian motion in . We show that is bounded in if and only if belongs to , an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.
@article{AIF_1985__35_1_215_0, author = {Gundy, Richard F. and Silverstein, Martin L.}, title = {The density of the area integral in ${\mathbb {R}}^{n+1}_+$}, journal = {Annales de l'Institut Fourier}, pages = {215--229}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {35}, number = {1}, year = {1985}, doi = {10.5802/aif.1006}, zbl = {0544.31012}, mrnumber = {86e:26012}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1006/} }
TY - JOUR AU - Gundy, Richard F. AU - Silverstein, Martin L. TI - The density of the area integral in ${\mathbb {R}}^{n+1}_+$ JO - Annales de l'Institut Fourier PY - 1985 SP - 215 EP - 229 VL - 35 IS - 1 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1006/ DO - 10.5802/aif.1006 LA - en ID - AIF_1985__35_1_215_0 ER -
%0 Journal Article %A Gundy, Richard F. %A Silverstein, Martin L. %T The density of the area integral in ${\mathbb {R}}^{n+1}_+$ %J Annales de l'Institut Fourier %D 1985 %P 215-229 %V 35 %N 1 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.1006/ %R 10.5802/aif.1006 %G en %F AIF_1985__35_1_215_0
Gundy, Richard F.; Silverstein, Martin L. The density of the area integral in ${\mathbb {R}}^{n+1}_+$. Annales de l'Institut Fourier, Tome 35 (1985) no. 1, pp. 215-229. doi : 10.5802/aif.1006. https://aif.centre-mersenne.org/articles/10.5802/aif.1006/
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