scholarly journals Uniform rectifiability from Carleson measure estimates and $\mathbf{\varepsilon}$ -approximability of bounded harmonic functions

2018 ◽  
Vol 167 (8) ◽  
pp. 1473-1524 ◽  
Author(s):  
John Garnett ◽  
Mihalis Mourgoglou ◽  
Xavier Tolsa
Keyword(s):  
Harmonic Functions ◽  
Carleson Measure ◽  
Uniform Rectifiability ◽  
Bounded Harmonic Functions

scholarly journals Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions

2016 ◽  
Vol 165 (12) ◽  
pp. 2331-2389 ◽  
Author(s):  
Steve Hofmann ◽  
José María Martell ◽  
Svitlana Mayboroda
Keyword(s):  
Harmonic Functions ◽  
Carleson Measure ◽  
Uniform Rectifiability

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

2021 ◽  
Author(s):  
Steve Hofmann ◽  
José María Martell ◽  
Svitlana Mayboroda ◽  
Tatiana Toro ◽  
Zihui Zhao
Keyword(s):  
Carleson Measure ◽  
Elliptic Operators ◽  
Uniform Rectifiability

A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces

1970 ◽  
Vol 22 (4) ◽  
pp. 855-862
Author(s):  
Y. K. Kwon ◽  
L. Sario
Keyword(s):  
Maximum Principle ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Decomposition Theorem ◽  
Higher Dimensions ◽  
Integral Representations ◽  
Classification Theory ◽  
Topological Properties ◽  
Harmonic Decomposition ◽  
Bounded Harmonic Functions

Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).

scholarly journals P-harmonic dimensions on ends

1993 ◽  
Vol 132 ◽  
pp. 131-139
Author(s):  
Michihiko Kawamura ◽  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Boundary Component ◽  
Harmonic Functions ◽  
Ideal Boundary ◽  
Boundary Values ◽  
Hölder Continuous ◽  
Closed Curves ◽  
Open Riemann Surface ◽  
Relative Boundary ◽  
Bounded Harmonic Functions

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

scholarly journals On the Existence of Various Bounded Harmonic Functions with given Periods

1973 ◽  
Vol 49 ◽  
pp. 143-148
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Functions ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Consider a pair (R, Γ) of a Riemann surface R and a period Γ. By a period Γ we mean a real-valued function Γ(γ) on one-dimensional cycles {γ} of the Riemann surface R.

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