Elementary Properties of Holomorphic Functions of Several Variables Harmonic Functions

Analysis ◽  
1980 ◽  
pp. 333-384
Author(s):  
Krzysztof Maurin
Keyword(s):  
Harmonic Functions ◽  
Holomorphic Functions ◽  
Several Variables

scholarly journals Foundations of Fatou Theory and a Tribute to the Work of E. M. Stein on Boundary Behavior of Holomorphic Functions

2021 ◽  
Author(s):  
Fausto Di Biase ◽  
Steven G. Krantz
Keyword(s):  
Harmonic Functions ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Several Variables ◽  
The Subject ◽  
Differentiation Of Integrals ◽  
Fatou Theory

AbstractWe lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.

Normal Functions: Lp Estimates

1997 ◽  
Vol 49 (1) ◽  
pp. 55-73 ◽  
Author(s):  
Huaihui Chen ◽  
Paul M. Gauthier
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Holomorphic Functions ◽  
Hyperbolic Metric ◽  
Lp Estimates ◽  
Normal Functions ◽  
The Hyperbolic Metric

AbstractFor ameromorphic (or harmonic) function ƒ, let us call the dilation of ƒ at z the ratio of the (spherical)metric at ƒ(z) and the (hyperbolic)metric at z. Inequalities are knownwhich estimate the sup norm of the dilation in terms of its Lp norm, for p > 2, while capitalizing on the symmetries of ƒ. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which ƒ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p = 2.

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