The density of the area integral in ${\mathbb {R}}^{n+1}_+$

Gundy, Richard F.; Silverstein, Martin L.

doi:10.5802/aif.1006

The density of the area integral in

ℝ_{+}^{n + 1}

Gundy, Richard F. ; Silverstein, Martin L.

Annales de l'Institut Fourier, Tome 35 (1985) no. 1, pp. 215-229.

Résumé
Abstract

Soit $u (x, y)$ une fonction harmonique dans le demi-plan $R_{+}^{n + 1}$ , $n \geq 2$ . Nous définissons une famille de fonctionnelles $D (u; r), - \infty > r > \infty$ , qui sont les analogues géométriques de la famille des temps locaux associés au processus $u (x_{t}, y_{t})$ où $(x_{t}, y_{t})$ est le mouvement brownien dans $R_{+}^{n + 1}$ . Nous montrons que $D (u) = {sup}_{r} D (u; r)$ est borné dans $L^{p}$ si et seulement si la fonction $u (x, y)$ appartien à $H^{p}$ , 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 $u (x, y)$ be a harmonic function in the half-plane $R_{+}^{n + 1}$ , $n \geq 2$ . We define a family of functionals $D (u; r), - \infty > r > \infty$ , that are analogs of the family of local times associated to the process $u (x_{t}, y_{t})$ where $(x_{t}, y_{t})$ is Brownian motion in $R_{+}^{n + 1}$ . We show that $D (u) = {sup}_{r} D (u; r)$ is bounded in $L^{p}$ if and only if $u (x, y)$ belongs to $H^{p}$ , 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.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1006

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     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},
     mrnumber = {86e:26012},
     zbl = {0544.31012},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1006/}
}

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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://www.numdam.org/articles/10.5802/aif.1006/

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