Banuelos, R.; Moore, C. N.
Distribution function inequalities for the density of the area integral
Annales de l'institut Fourier, Tome 41 (1991) no. 1 , p. 137-171
Zbl 0727.42016 | MR 92k:42025
doi : 10.5802/aif.1252
URL stable : http://www.numdam.org/item?id=AIF_1991__41_1_137_0

Nous démontrons des “inégalités des bons λ" pour l’intégale d’aire, la fonction maximale non-tangentielle, et la fonction maximale associée à la densité de l’intégrale d’aire. Nos résultats répondent à une question posée par R. F. Gundy. De plus nous démontrons un théorème du genre loi du logarithme itéré pour des fonctions harmoniques, semblable à celui de Kesten pour la suite des sommes partielles de variables indépendantes. Nos théorèmes 1 et 2 sont énoncés pour un domaine dont la frontière est lipschitzienne. Mais, ils sont tout aussi nouveaux pour R + 2 .
We prove good-λ inequalities for the area integral, the nontangential maximal function, and the maximal density of the area integral. This answers a question raised by R. F. Gundy. We also prove a Kesten type law of the iterated logarithm for harmonic functions. Our Theorems 1 and 2 are for Lipschitz domains. However, all our results are new even in the case of R + 2 .

Bibliographie

[1] R. Bañuelos, I. Klemes and C.N. Moore, An analogue of Kolmogorov's law of the iterated logarithm for harmonic functions, Duke Math. J., 57 (1988), 37-68. MR 89k:42019 | Zbl 0666.31002

[2] R. Bañuelos and C.N. Moore, Sharp estimates for the nontangential maximal function and the Lusin area function in Lipschitz domains, Trans. Amer. Math. Soc., 312 (1989) 641-662. MR 90i:42030 | Zbl 0675.42016

[3] R. Bañuelos and C.N. Moore, Laws of the iterated logarithm, sharp good-λ inequalities and Lp estimates for caloric and harmonic functions, Indiana Univ. Math. J., 38 (1989), 315-344. Zbl 0718.31002

[4] M. Barlow and M. Yor, (Semi) Martingale inequalities and local times, A. Wahrch. Verw. Gebiete, 55 (1981), 237-354. MR 82h:60092 | Zbl 0451.60050

[5] M. Barlow and M. Yor, Semi-martingale inequalities via the Garsia-Rodemich-Rumsey Lemma, and applications to local times, J. Funct. Anal., 49,2 (1989), 198-229. MR 84f:60073 | Zbl 0505.60054

[6] R. Bass, Séminaire de Probabilités XXI, Lecture Notes in Math., Springer-Verlag, New York, 1247 (1987). Numdam | Zbl 0616.60046

[7] J. Brossard, Densité de l'intégrale d'aire dans Rn+1+ et limites non tangentielles, Invent. Math., 93 (1988), 297-308. MR 89h:31008 | Zbl 0655.31004

[8] J. Brossard and L. Chevalier, Classe L log L et densité de l'intégrale d'aire dans Rn+1+, Ann. of Math., 128 (1988), 603-618. MR 90a:42013 | Zbl 0666.60071

[9] D.L. Burkholder and R.F. Gundy, Distribution function inequalities for the area integral, Studia Math., 44 (1972), 527-544. MR 49 #5309 | Zbl 0219.31009

[10] S.Y.A. Chang, J.M. Wilson and T.H. Wolff, Some weighted norm inequalities involving the Schrödinger operators, Comment. Math. Helv., 60 (1985), 217-246. MR 87d:42027 | Zbl 0575.42025

[11] B.E.J. Dahlberg, Weighted norm inequalities for the Lusin area integral and nontangential maximal functions for harmonic functions in Lipschitz domains, Studia Math., 47 (1980), 297-314. MR 82f:31003 | Zbl 0449.31002

[12] B. Davis, On the Barlow-Yor inequalities for local time, Séminaire de Probabilitiés XXI, Lecture Notes in Math., Springer-Verlag, New York, 1247 (1987). Numdam | MR 89h:60122 | Zbl 0617.60041

[13] J.L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, New York/Berlin, 1984. MR 85k:31001 | Zbl 0549.31001

[14] R.F. Gundy, The density of the area integral, Conference on Harmonic Analysis in Honor of A. Zygmund, Beckner, W., Calderón, A P., Fefferman, R., and Jones, P., editors, Wadsworth, Belmont, California, 1983. MR 85k:42048

[15] R.F. Gundy, Some topics in probability and analysis, BMS, #70 (1989). MR 91c:60059 | Zbl 0673.60050

[16] R.F. Gundy and M.L. Silverstein, The density of the area integral in Rn+1+, Ann. Inst. Fourier (Grenoble), 35,1 (1985), 215-224. Numdam | MR 86e:26012 | Zbl 0544.31012

[17] Ikeda and Watanabe, Stochastic differential equations and diffusion processes, North Holland Kodansha, 1981. MR 84b:60080 | Zbl 0495.60005

[18] D. Jerison and C. Kenig, Boundary value problems on Lipschitz domains, MAA Studies in Math., 23 (1982), 1-68. MR 85f:35057 | Zbl 0529.31007

[19] H. Kesten, An iterated law for local time, Duke Math. J., 32 (1965), 447-456. MR 31 #2751 | Zbl 0132.12701

[20] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970. MR 44 #7280 | Zbl 0207.13501

[21] J.O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J., 28 (1979), 511-544. MR 81f:42021 | Zbl 0429.46016