On boundary behavior of harmonic functions in Hölder domains

1998 ◽  
Vol 4 (4-5) ◽  
pp. 447-461 ◽  
Author(s):  
Fausto Ferrari
Keyword(s):  
Harmonic Functions ◽  
Boundary Behavior

scholarly journals Equivalence between the boundary Harnack principle and the Carleson estimate

2008 ◽  
Vol 103 (1) ◽  
pp. 61 ◽  
Author(s):  
Hiroaki Aikawa
Keyword(s):  
Harmonic Functions ◽  
Boundary Behavior ◽  
Boundary Harnack Principle ◽  
Positive Harmonic Functions

Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this paper is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains.

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.

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