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

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 Modified Poisson Integral and Green Potential on a Half-Space

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Lei Qiao
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Poisson Integral ◽  
Dimensional Euclidean Space ◽  
Superharmonic Functions ◽  
Growth Properties ◽  
Green Potential

We discuss the behavior at infinity of modified Poisson integral and Green potential on a half-space of then-dimensional Euclidean space, which generalizes the growth properties of analytic functions, harmonic functions and superharmonic functions.

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