The Phragemén‐Lindelöf principle of harmonic functions and conditions for nevanlinna class in the half space

2019 ◽  
Vol 42 (12) ◽  
pp. 4360-4364 ◽  
Author(s):  
Yan Hui Zhang
Keyword(s):  
Half Space ◽  
Harmonic Functions ◽  
Nevanlinna Class ◽  
Lindelöf Principle

Poisson’s integral formula via heat kernels

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 59-64
Author(s):  
Yoichi Miyazaki
Keyword(s):  
Fourier Transform ◽  
Heat Kernel ◽  
Half Space ◽  
Unbounded Domain ◽  
Harmonic Functions ◽  
Integral Formula ◽  
Heat Kernels ◽  
Transform Method ◽  
Fourier Transform Method ◽  
The Fourier Transform

Abstract We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.

scholarly journals Inferior Mean of Measures on Curves and Subspaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yevgenya Movshovich
Keyword(s):  
Half Space ◽  
Harmonic Functions ◽  
Upper Bounds ◽  
Finite Boundary ◽  
Positive Harmonic Functions

We obtain new sharp upper bounds of the inferior mean for positive harmonic functions defined by finite boundary measures that lie on curves or subspaces of the boundary of the half-space.

scholarly journals Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

2020 ◽  
Vol 30 (5) ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Terhi Kaarakka
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Laplace Operator ◽  
Harmonic Functions ◽  
Riemannian Metric ◽  
Hyperbolic Metric ◽  
Hyperbolic Functions ◽  
The Hyperbolic Metric ◽  
Hyperbolic Brownian Motion ◽  
Hyperbolic Harmonic Functions

Abstract We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ d s 2 = d x 1 2 + ⋯ + d x n 2 x n 2 α n - 2 in the upper half space $$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$ R + n = { x 1 , … , x n ∈ R n : x n > 0 } . They are called $$\alpha $$ α -hyperbolic harmonic. An important result is that a function f is $$\alpha $$ α -hyperbolic harmonic íf and only if the function $$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$ g x = x n - 2 - n + α 2 f x is the eigenfunction of the hyperbolic Laplace operator $$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$ △ h = x n 2 ▵ - n - 2 x n ∂ ∂ x n corresponding to the eigenvalue $$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$ 1 4 α + 1 2 - n - 1 2 = 0 . This means that in case $$\alpha =n-2$$ α = n - 2 , the $$n-2$$ n - 2 -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of $$\alpha $$ α -hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.

scholarly journals Some properties of conjugate harmonic functions in a half-space

2010 ◽  
Vol 369 (2) ◽  
pp. 686-693
Author(s):  
Anatoly Ryabogin ◽  
Dmitry Ryabogin
Keyword(s):  
Half Space ◽  
Harmonic Functions

scholarly journals From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class

2020 ◽  
Vol 7 (1) ◽  
pp. 91-115
Author(s):  
Xavier Massaneda ◽  
Pascal J. Thomas
Keyword(s):  
Algebraic Structure ◽  
Analytic Functions ◽  
Unit Disc ◽  
Harmonic Functions ◽  
General Rule ◽  
Corona Theorem ◽  
Uniform Bounds ◽  
Bounded Analytic Functions ◽  
Interpolating Sequences ◽  
Nevanlinna Class

AbstractThis survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.

scholarly journals Harmonic Functions in Upper Half Space

2012 ◽  
Vol 19 (4) ◽  
pp. 675-681
Author(s):  
Guo-Shuang Pan ◽  
Lei Qiao ◽  
Guan-Tie Deng
Keyword(s):  
Half Space ◽  
Harmonic Functions
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