scholarly journals Kelvin transform for α-harmonic functions in regular domains

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Krzysztof Michalik ◽  
Michał Ryznar
Keyword(s):  
Lipschitz Domain ◽  
Harmonic Functions ◽  
Stable Processes ◽  
Kelvin Transform

AbstractWe investigate conditional stable processes in a Lipschitz domain

scholarly journals Regularity of solutions to anisotropic nonlocal equations

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1135-1155 ◽  
Author(s):  
Jamil Chaker
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Harmonic Functions ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Stable Processes ◽  
Symmetric Stable Processes ◽  
One Dimensional ◽  
Nonlocal Equations ◽  
Bounded Harmonic Functions

Abstract We study harmonic functions associated to systems of stochastic differential equations of the form $$dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots +A_{id}(X_{t-})dZ_t^d$$ d X t i = A i 1 ( X t - ) d Z t 1 + ⋯ + A id ( X t - ) d Z t d , $$i\in \{1,\dots ,d\}$$ i ∈ { 1 , ⋯ , d } , where $$Z_t^j$$ Z t j are independent one-dimensional symmetric stable processes of order $$\alpha _j\in (0,2)$$ α j ∈ ( 0 , 2 ) , $$j\in \{1,\dots ,d\}$$ j ∈ { 1 , ⋯ , d } . In this article we prove Hölder regularity of bounded harmonic functions with respect to solutions to such systems.

On the Radial Symmetry Property for Harmonic Functions

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Symmetry Property ◽  
Radial Symmetry

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

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