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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822702.png" />-potential''
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'' $  \alpha $-potential''
  
 
A potential of the form
 
A potential of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822703.png" /></td> </tr></table>
+
$$
 +
V _  \alpha  ( x)  = V( x; \alpha , \mu )  = \
 +
\int\limits
 +
\frac{d \mu ( y) }{| x- y |  ^  \alpha  }
 +
,\  \alpha > 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822704.png" /> is a positive [[Borel measure|Borel measure]] of compact support on the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822706.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822707.png" /> is the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822708.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r0822709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227010.png" />, the Riesz potential coincides with the classical [[Newton potential|Newton potential]]; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227012.png" />, the limit case of the Riesz potential is in some sense the [[Logarithmic potential|logarithmic potential]]. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227014.png" />, the Riesz potential is a [[Superharmonic function|superharmonic function]] on the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227015.png" />; moreover, in the classical case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227016.png" />, outside the support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227018.png" />, the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227019.png" /> is a [[Harmonic function|harmonic function]]. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227020.png" />, the Riesz potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227021.png" /> is a [[Subharmonic function|subharmonic function]] outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227022.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227023.png" /> the Riesz potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227024.png" /> is a lower semi-continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227025.png" />, continuous outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227026.png" />.
+
where $  \mu $
 +
is a positive [[Borel measure|Borel measure]] of compact support on the Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
and $  | x- y | $
 +
is the distance between the points $  x, y \in \mathbf R  ^ {n} $.  
 +
When $  n \geq  3 $
 +
and $  \alpha = n- 2 $,  
 +
the Riesz potential coincides with the classical [[Newton potential|Newton potential]]; when $  n= 2 $
 +
and $  \alpha \rightarrow 0 $,  
 +
the limit case of the Riesz potential is in some sense the [[Logarithmic potential|logarithmic potential]]. When $  n \geq  3 $
 +
and  $  0 < \alpha \leq  n- 2 $,  
 +
the Riesz potential is a [[Superharmonic function|superharmonic function]] on the entire space $  \mathbf R  ^ {n} $;  
 +
moreover, in the classical case $  \alpha = n- 2 $,  
 +
outside the support $  S( \mu ) $
 +
of $  \mu $,  
 +
the potential $  V( x) = V _ {n- 2} ( x) $
 +
is a [[Harmonic function|harmonic function]]. When $  \alpha > n- 2 $,  
 +
the Riesz potential $  V _  \alpha  ( x) $
 +
is a [[Subharmonic function|subharmonic function]] outside $  S( \mu ) $.  
 +
For all $  \alpha > 0 $
 +
the Riesz potential $  V _  \alpha  ( x) $
 +
is a lower semi-continuous function on $  \mathbf R  ^ {n} $,  
 +
continuous outside $  S( \mu ) $.
  
Among the general properties of Riesz potentials the following are the most important. The continuity principle: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227027.png" /> and if the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227028.png" /> is continuous at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227030.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227031.png" /> as a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227032.png" />. The restricted maximum principle: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227034.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227035.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227036.png" />, a more precise maximum principle is valid: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227038.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227039.png" /> (this statement remains valid also when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227041.png" />, that is, for the logarithmic potential).
+
Among the general properties of Riesz potentials the following are the most important. The continuity principle: If $  x _ {0} \in S( \mu ) $
 +
and if the restriction $  V _  \alpha  ( x) \mid  _ {S( \mu ) }  $
 +
is continuous at the point $  x _ {0} $,  
 +
then $  V _  \alpha  ( x) $
 +
is continuous at $  x _ {0} $
 +
as a function on $  \mathbf R  ^ {n} $.  
 +
The restricted maximum principle: If $  V _  \alpha  ( x) \mid  _ {S( \mu ) }  \leq  M $,  
 +
then $  V _  \alpha  ( x) \leq  2  ^  \alpha  M $
 +
everywhere on $  \mathbf R  ^ {n} $.  
 +
When $  n- 2 \leq  \alpha < n $,  
 +
a more precise maximum principle is valid: If $  V _  \alpha  ( x) \mid  _ {S( \mu ) }  \leq  M $,  
 +
then $  V _  \alpha  ( x) \leq  M $
 +
everywhere on $  \mathbf R  ^ {n} $ (this statement remains valid also when $  n= 2 $
 +
and $  \alpha \rightarrow 0 $,  
 +
that is, for the logarithmic potential).
  
The [[Capacity|capacity]] theory for Riesz potentials can be constructed, for example, on the basis of the concept of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227042.png" />-energy of a measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227043.png" />:
+
The [[Capacity|capacity]] theory for Riesz potentials can be constructed, for example, on the basis of the concept of the $  \alpha $-energy of a measures $  \mu $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227044.png" /></td> </tr></table>
+
$$
 +
E _  \alpha  ( \mu )  = \int\limits \int\limits
 +
\frac{d \mu ( x)  d \mu ( y) }{| x- y |  ^  \alpha  }
  
One may assume that for a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227045.png" />,
+
,\  \alpha > 0.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227046.png" /></td> </tr></table>
+
One may assume that for a compact set  $  K $,
  
where the infimum is taken over all measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227047.png" /> concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227048.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227049.png" />; then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227051.png" />-capacity is equal to
+
$$
 +
V _  \alpha  ( K)  = \inf \{ E _  \alpha  ( \mu ) \} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227052.png" /></td> </tr></table>
+
where the infimum is taken over all measures  $  \mu $
 +
concentrated on  $  K $
 +
and such that  $  \mu ( K) = 1 $;  
 +
then the  $  \alpha $-capacity is equal to
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227053.png" />, then the infimum is attained on the capacitary measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227054.png" /> (also called equilibrium measure), which is concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227056.png" />, generating the corresponding capacitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227058.png" />-potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227059.png" /> (cf. also [[Capacity potential|Capacity potential]]). The further construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227060.png" />-capacities of arbitrary sets is carried out in the same way as for the classical capacities.
+
$$
 +
C _  \alpha  ( K)  = [ V _  \alpha  ( K)] ^ {- 1/ \alpha } .
 +
$$
 +
 
 +
If  $  V _  \alpha  ( K) < + \infty $,  
 +
then the infimum is attained on the capacitary measure $  \lambda $ (also called equilibrium measure), which is concentrated on $  K $,  
 +
$  \lambda ( K) = 1 $,  
 +
generating the corresponding capacitary $  \alpha $-potential $  V( x;  \alpha , \lambda ) $ (cf. also [[Capacity potential|Capacity potential]]). The further construction of $  \alpha $-capacities of arbitrary sets is carried out in the same way as for the classical capacities.
  
 
The Riesz potential is called after M. Riesz (see [[#References|[2]]]), who obtained a number of important properties of Riesz potentials; for the first time such potentials were studied by O. Frostman (see [[#References|[1]]]).
 
The Riesz potential is called after M. Riesz (see [[#References|[2]]]), who obtained a number of important properties of Riesz potentials; for the first time such potentials were studied by O. Frostman (see [[#References|[1]]]).
Line 27: Line 95:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Frostman,  "Potentiel d'equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions"  ''Medd. Lunds Univ. Mat. Sem.'' , '''3'''  (1935)  pp. 1–118</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Riesz,  "Intégrales de Riemann–Liouville et potentiels"  ''Acata Sci. Math. Szeged'' , '''9'''  (1938)  pp. 1–42</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Frostman,  "Potentiel d'equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions"  ''Medd. Lunds Univ. Mat. Sem.'' , '''3'''  (1935)  pp. 1–118</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Riesz,  "Intégrales de Riemann–Liouville et potentiels"  ''Acata Sci. Math. Szeged'' , '''9'''  (1938)  pp. 1–42</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227061.png" /> even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227063.png" /> is a [[Fundamental solution|fundamental solution]] of the polyharmonic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227064.png" />, otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227065.png" /> is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227066.png" />, see [[#References|[a2]]]. A treatment of Riesz potentials in the framework of balayage spaces is given in [[#References|[a1]]].
+
For $  n $
 +
even and $  \alpha = n- 2 m \leq  0 $,  
 +
$  | x - y |  ^ {2m- n}  \mathop{\rm log}  | x- y | $
 +
is a [[Fundamental solution|fundamental solution]] of the polyharmonic equation $  \Delta  ^ {m} u = 0 $,  
 +
otherwise $  | x- y |  ^ {2m- n} $
 +
is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order > 2 $,  
 +
see [[#References|[a2]]]. A treatment of Riesz potentials in the framework of balayage spaces is given in [[#References|[a1]]].
  
The Riesz kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082270/r08227067.png" /> are the standard examples of convolution kernels. Thus, Riesz potentials may be regarded as special singular integrals. For more details on this interesting point of view see [[#References|[a3]]].
+
The Riesz kernels $  | x- y | ^ {- \alpha } $
 +
are the standard examples of convolution kernels. Thus, Riesz potentials may be regarded as special singular integrals. For more details on this interesting point of view see [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bliedtner,  W. Hansen,  "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.W. Schulze,  G. Wildenhain,  "Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Carleson,  "Selected problems on exceptional sets" , v. Nostrand  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bliedtner,  W. Hansen,  "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.W. Schulze,  G. Wildenhain,  "Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Carleson,  "Selected problems on exceptional sets" , v. Nostrand  (1967)</TD></TR></table>

Latest revision as of 06:02, 12 July 2022


$ \alpha $-potential

A potential of the form

$$ V _ \alpha ( x) = V( x; \alpha , \mu ) = \ \int\limits \frac{d \mu ( y) }{| x- y | ^ \alpha } ,\ \alpha > 0, $$

where $ \mu $ is a positive Borel measure of compact support on the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, and $ | x- y | $ is the distance between the points $ x, y \in \mathbf R ^ {n} $. When $ n \geq 3 $ and $ \alpha = n- 2 $, the Riesz potential coincides with the classical Newton potential; when $ n= 2 $ and $ \alpha \rightarrow 0 $, the limit case of the Riesz potential is in some sense the logarithmic potential. When $ n \geq 3 $ and $ 0 < \alpha \leq n- 2 $, the Riesz potential is a superharmonic function on the entire space $ \mathbf R ^ {n} $; moreover, in the classical case $ \alpha = n- 2 $, outside the support $ S( \mu ) $ of $ \mu $, the potential $ V( x) = V _ {n- 2} ( x) $ is a harmonic function. When $ \alpha > n- 2 $, the Riesz potential $ V _ \alpha ( x) $ is a subharmonic function outside $ S( \mu ) $. For all $ \alpha > 0 $ the Riesz potential $ V _ \alpha ( x) $ is a lower semi-continuous function on $ \mathbf R ^ {n} $, continuous outside $ S( \mu ) $.

Among the general properties of Riesz potentials the following are the most important. The continuity principle: If $ x _ {0} \in S( \mu ) $ and if the restriction $ V _ \alpha ( x) \mid _ {S( \mu ) } $ is continuous at the point $ x _ {0} $, then $ V _ \alpha ( x) $ is continuous at $ x _ {0} $ as a function on $ \mathbf R ^ {n} $. The restricted maximum principle: If $ V _ \alpha ( x) \mid _ {S( \mu ) } \leq M $, then $ V _ \alpha ( x) \leq 2 ^ \alpha M $ everywhere on $ \mathbf R ^ {n} $. When $ n- 2 \leq \alpha < n $, a more precise maximum principle is valid: If $ V _ \alpha ( x) \mid _ {S( \mu ) } \leq M $, then $ V _ \alpha ( x) \leq M $ everywhere on $ \mathbf R ^ {n} $ (this statement remains valid also when $ n= 2 $ and $ \alpha \rightarrow 0 $, that is, for the logarithmic potential).

The capacity theory for Riesz potentials can be constructed, for example, on the basis of the concept of the $ \alpha $-energy of a measures $ \mu $:

$$ E _ \alpha ( \mu ) = \int\limits \int\limits \frac{d \mu ( x) d \mu ( y) }{| x- y | ^ \alpha } ,\ \alpha > 0. $$

One may assume that for a compact set $ K $,

$$ V _ \alpha ( K) = \inf \{ E _ \alpha ( \mu ) \} , $$

where the infimum is taken over all measures $ \mu $ concentrated on $ K $ and such that $ \mu ( K) = 1 $; then the $ \alpha $-capacity is equal to

$$ C _ \alpha ( K) = [ V _ \alpha ( K)] ^ {- 1/ \alpha } . $$

If $ V _ \alpha ( K) < + \infty $, then the infimum is attained on the capacitary measure $ \lambda $ (also called equilibrium measure), which is concentrated on $ K $, $ \lambda ( K) = 1 $, generating the corresponding capacitary $ \alpha $-potential $ V( x; \alpha , \lambda ) $ (cf. also Capacity potential). The further construction of $ \alpha $-capacities of arbitrary sets is carried out in the same way as for the classical capacities.

The Riesz potential is called after M. Riesz (see [2]), who obtained a number of important properties of Riesz potentials; for the first time such potentials were studied by O. Frostman (see [1]).

References

[1] O. Frostman, "Potentiel d'equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions" Medd. Lunds Univ. Mat. Sem. , 3 (1935) pp. 1–118
[2] M. Riesz, "Intégrales de Riemann–Liouville et potentiels" Acata Sci. Math. Szeged , 9 (1938) pp. 1–42
[3] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[4] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)

Comments

For $ n $ even and $ \alpha = n- 2 m \leq 0 $, $ | x - y | ^ {2m- n} \mathop{\rm log} | x- y | $ is a fundamental solution of the polyharmonic equation $ \Delta ^ {m} u = 0 $, otherwise $ | x- y | ^ {2m- n} $ is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order $ > 2 $, see [a2]. A treatment of Riesz potentials in the framework of balayage spaces is given in [a1].

The Riesz kernels $ | x- y | ^ {- \alpha } $ are the standard examples of convolution kernels. Thus, Riesz potentials may be regarded as special singular integrals. For more details on this interesting point of view see [a3].

References

[a1] J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2] B.W. Schulze, G. Wildenhain, "Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung" , Birkhäuser (1977)
[a3] E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970)
[a4] L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967)
How to Cite This Entry:
Riesz potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_potential&oldid=14705
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article