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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451101.png" /> on which harmonic and superharmonic functions (cf. [[Harmonic function|Harmonic function]]; [[Subharmonic function|Subharmonic function]]) are defined and for which a [[Green function|Green function]] exists (for the [[Dirichlet problem|Dirichlet problem]] in the class of harmonic functions) or, which amounts to the same thing, for which there exists a non-constant superharmonic function. More exactly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451102.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451104.png" />-space, i.e. a connected separable topological space in which: 1) each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451105.png" /> has an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451106.png" /> homeomorphic to some open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451107.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451108.png" /> (or of its [[Aleksandrov compactification|Aleksandrov compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g0451109.png" />); and 2) the images of any non-empty intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511010.png" /> (under the two homeomorphisms to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511012.png" />) of two neighbourhoods in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511014.png" /> are isometric, and are conformally equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511015.png" />. Harmonic and superharmonic functions on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511016.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511017.png" /> are locally defined by passing to the images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511018.png" />. If, in addition, there exists a non-constant positive superharmonic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511019.png" /> or, which amounts to the same thing, a positive potential, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511020.png" /> is known as a Green space. Thus, the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511021.png" />, its compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511022.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511023.png" />) and Riemann surfaces are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511024.png" />-spaces. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511025.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511026.png" /> and Riemann surfaces of hyperbolic type (cf. [[Riemann surfaces, classification of|Riemann surfaces, classification of]]) are Green spaces, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511027.png" /> and Riemann surfaces of parabolic type are not. Any domain in a Green space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511028.png" /> is again a Green space.
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A topological space $X$ on which harmonic and superharmonic functions (cf. [[Harmonic function|Harmonic function]]; [[Subharmonic function|Subharmonic function]]) are defined and for which a [[Green function|Green function]] exists (for the [[Dirichlet problem|Dirichlet problem]] in the class of harmonic functions) or, which amounts to the same thing, for which there exists a non-constant superharmonic function. More exactly, let $X$ be an $E$-space, i.e. a connected separable topological space in which: 1) each point $x\in X$ has an open neighbourhood $V_x$ homeomorphic to some open set $V_x'$ of a Euclidean space $\mathbf R^n$ (or of its [[Aleksandrov compactification|Aleksandrov compactification]] $\overline{\mathbf R^n}$); and 2) the images of any non-empty intersection $V_x\cap V_y$ (under the two homeomorphisms to $\mathbf R^n$ or $\overline{\mathbf R^n}$) of two neighbourhoods in $V_x'$ and $V_y'$ are isometric, and are conformally equivalent if $n=2$. Harmonic and superharmonic functions on an $E$-space $X$ are locally defined by passing to the images $V_x'$. If, in addition, there exists a non-constant positive superharmonic function on $X$ or, which amounts to the same thing, a positive potential, $X$ is known as a Green space. Thus, the Euclidean space $\mathbf R^n$, its compactification $\overline{\mathbf R^n}$ ($n\geq2$) and Riemann surfaces are all $E$-spaces. Here, $\mathbf R^n$ $(n\geq3)$ and Riemann surfaces of hyperbolic type (cf. [[Riemann surfaces, classification of|Riemann surfaces, classification of]]) are Green spaces, while $\mathbf R^2$ and Riemann surfaces of parabolic type are not. Any domain in a Green space $X$ is again a Green space.
  
A [[Harmonic space|harmonic space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045110/g04511029.png" /> with a positive potential on it can also be regarded as a generalization of a Green space in the framework of axiomatic potential theory (cf. [[Potential theory, abstract|Potential theory, abstract]]).
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A [[Harmonic space|harmonic space]] $X$ with a positive potential on it can also be regarded as a generalization of a Green space in the framework of axiomatic potential theory (cf. [[Potential theory, abstract|Potential theory, abstract]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brelot,  "On topologies and boundaries in potential theory" , Springer  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Brélot,  G. Choquet,  "Espaces et lignes de Green"  ''Ann. Inst. Fourier'' , '''3'''  (1952)  pp. 199–263</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brelot,  "On topologies and boundaries in potential theory" , Springer  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Brélot,  G. Choquet,  "Espaces et lignes de Green"  ''Ann. Inst. Fourier'' , '''3'''  (1952)  pp. 199–263</TD></TR></table>

Latest revision as of 11:31, 27 October 2014

A topological space $X$ on which harmonic and superharmonic functions (cf. Harmonic function; Subharmonic function) are defined and for which a Green function exists (for the Dirichlet problem in the class of harmonic functions) or, which amounts to the same thing, for which there exists a non-constant superharmonic function. More exactly, let $X$ be an $E$-space, i.e. a connected separable topological space in which: 1) each point $x\in X$ has an open neighbourhood $V_x$ homeomorphic to some open set $V_x'$ of a Euclidean space $\mathbf R^n$ (or of its Aleksandrov compactification $\overline{\mathbf R^n}$); and 2) the images of any non-empty intersection $V_x\cap V_y$ (under the two homeomorphisms to $\mathbf R^n$ or $\overline{\mathbf R^n}$) of two neighbourhoods in $V_x'$ and $V_y'$ are isometric, and are conformally equivalent if $n=2$. Harmonic and superharmonic functions on an $E$-space $X$ are locally defined by passing to the images $V_x'$. If, in addition, there exists a non-constant positive superharmonic function on $X$ or, which amounts to the same thing, a positive potential, $X$ is known as a Green space. Thus, the Euclidean space $\mathbf R^n$, its compactification $\overline{\mathbf R^n}$ ($n\geq2$) and Riemann surfaces are all $E$-spaces. Here, $\mathbf R^n$ $(n\geq3)$ and Riemann surfaces of hyperbolic type (cf. Riemann surfaces, classification of) are Green spaces, while $\mathbf R^2$ and Riemann surfaces of parabolic type are not. Any domain in a Green space $X$ is again a Green space.

A harmonic space $X$ with a positive potential on it can also be regarded as a generalization of a Green space in the framework of axiomatic potential theory (cf. Potential theory, abstract).

References

[1] M. Brelot, "On topologies and boundaries in potential theory" , Springer (1971)
[2] M. Brélot, G. Choquet, "Espaces et lignes de Green" Ann. Inst. Fourier , 3 (1952) pp. 199–263
How to Cite This Entry:
Green space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Green_space&oldid=12148
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article