Boundary behaviour of harmonic functions in a half-space and brownian motion

Burkholder, D. L.; Gundy, Richard F.

Annales de l'Institut Fourier, Tome 23 (1973) no. 4, pp. 195-212.

Soit $u(x,y)$ une fonction harmonique dans le demi-espace $R_{+}^{n+1}$, $n\ge 2$. Nous montrons que $u(x,y)$ peut avoir une limite fine en presque chaque point du cube unité dans $R^n=\partial R_{+}^{n+1}$ 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 $R_{+}^2$, il est connu que l’on peut caractériser les classes de Hardy $H^p$, $0<p<\infty$, par l’intégrabilité de la fonction maximale du mouvement brownien. Nous montrons que ce résultat est aussi valable dans $R_{+}^{n+1}$, pour $n\ge 2$.

Let $u$ be harmonic in the half-space $R_{+}^{n+1}$, $n\ge 2$. We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^n=\partial R_{+}^{n+1}$ 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 $R_{+}^2$ it is known that the Hardy classes $H^p$, $0<p<\infty$, 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 $R_{+}^{n+1}$, for $n\ge 2$.

[1] J. M. Brelot and L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble), 13, (1963), 395-415. | Numdam | MR | Zbl

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

[3] D.L. Burkholder, R.F. Gundy and M. L. Silverstein, A maximal function characterization of the class Hp, Trans. Amer. Math. Soc., 157 (1971), 137-153. | MR | Zbl

[4] A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc., 68, (1950), 47-54. | MR | Zbl

[5] A. P. Calderón, On a theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc., 68, (1950), 55-61. | MR | Zbl

[6] L. Carleson, On the existence of boundary values for harmonic functions in several variables, Arkiv för Mathematik, 4, (1961), 393-399. | MR | Zbl

[7] C. Constantinescu and A. Cornea, Über das Verhalten der analytischen Abildungen Riemannscher Flachen auf dem idealen Rand von Martin, Nagoya Math. J., 17, (1960), 1-87. | MR | Zbl

[8] J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85, (1957), 431-458. | Numdam | MR | Zbl

[9] J. L. Doob, Boundary limit theorems for a half-space, J. Math. Pures Appl., (9) 37, (1958), 385-392. | MR | Zbl

[10] C. Fefferman and E. M. Stein, Hp-spaces in several variables, Acta Math., 129, (1972), 137-193. | MR | Zbl

[11] G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. fur Mat., 167, (1931), 405-423. | JFM | Zbl

[12] J. Lelong, Review 4471, Math. Reviews, 40, (1970), 824-825.

[13] J. Lelong, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66, (1949), 125-159. | Numdam | Zbl

[14] J. Marcinkiewicz and A. Zygmund, A theorem of Lusin, Duke Math. J., 4, (1938), 473-485. | JFM | Zbl

[15] H. P. Mckean Jr., Stochastic integrals, Academic Press, New York, 1969. | Zbl

[16] L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la Théorie du potential, Ann. Inst. Fourier (Grenoble), 7, (1957), 183-285. | Numdam | MR | Zbl

[17] I. I. Privalov, Integral Cauchy, Saratov, 1919.

[18] D. Spencer, A function-theoretic identity, Amer. J. Math., 65, (1943), 147-160. | MR | Zbl

[19] E. M. Stein, On the theory of harmonic functions of several variables II. Behaviour near the boundary, Acta Math., 106, (1961), 137-174. | MR | Zbl

[20] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables I. The theory of Hp-spaces, Acta Math., 103, (1960), 25-62. | MR | Zbl

[21] J. L. Walsh, The approximation of harmonic functions by harmonic polynomials and harmonic rational functions, Bull. Amer. Math. Soc., 35, (1929), 499-544. | JFM