FATOU THEOREM AND EMBEDDING THEOREMS FOR THE MEAN LIPSCHITZ FUNCTIONS ON THE UNIT BALL

Title & Authors
FATOU THEOREM AND EMBEDDING THEOREMS FOR THE MEAN LIPSCHITZ FUNCTIONS ON THE UNIT BALL
Cho, Hong-Rae; Lee, Jin-Kee;

Abstract
We investigate the boundary values of the holomorphic mean Lipschitz function. In fact, we prove that the admissible limit exists at every boundary point of the unit ball for the holomorphic mean Lipschitz functions under some assumptions on the Lipschitz order. Moreover, we get embedding theorems of holomorphic mean Lipschitz spaces into Hardy spaces or into the Bloch space on the unit ball in $\small{\mathbb{C}_n}$.
Keywords
Fatou theorem;mean Lipschitz function;embedding theorems;admissible limit;
Language
English
Cited by
References
1.
P. Ahern and W. Cohn, Exceptional Sets for Hardy Sobolev Functions, p>1, Indiana Univ. Math. J. 38 (1989), no. 2, 417–453

2.
O. Blasco, D. Girela, and M. A. M´arquez, Mean growth of the derivative of analytic functions, bounded mean oscillation, and normal functions, Indiana Univ. Math. J. 47 (1998), no. 3, 893–912

3.
H. R. Cho, H. W. Koo, and E. G. Kwon, Holomorphic functions satisfying mean Lipschitz condition in the ball, J. Korean Math. Soc. 44 (2007), no. 4, 931–940

4.
P. L. Duren, Theory of Hp Spaces, Academic Press, New York, 1970

5.
M. Nozomu, Inequalities of Fej'er-Riesz and Hardy-Littlewood, Tohoku Math. J. (2) 40 (1988), no. 1, 77–86

6.
R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, (GTM 108) Springer-Verlag, New York Inc., 1986