The density of the area integral in ℝ + n+1
Gundy, Richard F. ; Silverstein, Martin L.
Annales de l'Institut Fourier, Tome 35 (1985), p. 215-229 / Harvested from Numdam
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Soit u(x,y) une fonction harmonique dans le demi-plan R + n+1 , n≥2. Nous définissons une famille de fonctionnelles D(u;r),-∞>r>∞, 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≥2. We define a family of functionals D(u;r),-∞>r>∞, 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.

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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},
     volume = {35},
     year = {1985},
     pages = {215-229},
     doi = {10.5802/aif.1006},
     mrnumber = {86e:26012},
     zbl = {0544.31012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1985__35_1_215_0}
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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) pp. 215-229. doi : 10.5802/aif.1006. http://gdmltest.u-ga.fr/item/AIF_1985__35_1_215_0/

[1] M. Barlow and M. Yor, (Semi) Martingale Iequalities and Local Times, A. Wahrsch. Verw. Gebiete, 55 (1981), 237-254. | MR 82h:60092 | Zbl 0451.60050

[2] M. Barlow and M. Yor, Semi-martingale Inequalities Via the Garsia-Rodemich-Rumsey Lemma, and Applications to Local Times, J. Funct. Anal., 49, 2 (1982), 198-229. | MR 84f:60073 | Zbl 0505.60054

[3] A. Benedek, A.P. Calderon and R. Panzone, Convolution Operators on Banach Space Valued Functions, Proc. Natl. Acad. Sci. U.S.A., 48, 3 (1962), 356-365. | MR 24 #A3479 | Zbl 0103.33402

[4] D. Burkholder and R. Gundy, Distribution Function Inequalities for the Area Integral, Studia Mathematica, 44 (1972), 527-544. | MR 49 #5309 | Zbl 0219.31009

[5] H. Federer, Geometric Measure Theory, Springer-Verlag, New York (1969). | MR 41 #1976 | Zbl 0176.00801

[6] R. Fefferman, R. Gundy, M. Silverstein and E.M. Stein, Inequalities for Ratios of Functionals of Harmonic Functions, Proc. Natl. Acad. Sci. U.S.A., 79 (1982), 7958-7960. | MR 85c:42024 | Zbl 0512.31008

[7] C. Fefferman and E.M. Stein, Hp Spaces of Several Variables, Acta Math, 129 (1972), 137-193. | MR 56 #6263 | Zbl 0257.46078

[8] A.M. Garsia, E. Rodemich and H. Rumsey Jr., A Real variable Lemma and the Continuity of Paths of Some Gaussian Processes, Indiana Univ. Math. J., 20 (1970-1971), 565-578. | MR 42 #2534 | Zbl 0252.60020

[9] R.F. Gundy, The Density of the Area Integral in Conference on Harmonic Analysis in Honor of A. Zygmund, Eds., Beckner, W. Calderón, A.P., Fefferman, R., and Jones, P. ; Wadsworth, Belmont, California (1983).

[10] E.M. Stein, Singular Integral a Differentiability Properties of Functions, Princeton, 1970. | MR 44 #7280 | Zbl 0207.13501