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 Boundary behavior of positive harmonic functions in balls of Rn

1981 ◽  
Vol 84 ◽  
pp. 1-8
Author(s):  
Michael Von Renteln
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Boundary Behavior ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Ball ◽  
The Real ◽  
Positive Harmonic Functions

Let Rn be the real n-dimensional euclidean space. Elements of Rn are denoted by x = (xl • • •, xn), and ‖ x ‖ denotes the euclidean norm of x.The open ball B(x, r) with center x and radius r is defined by

A note on positive harmonic functions

1948 ◽  
Vol 44 (2) ◽  
pp. 289-291 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Duke Math ◽  
Negative Number ◽  
Image Position ◽  
American Math ◽  
Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

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