View source for Bessel potential

<!--
b0158701.png
$#A+1 = 30 n = 0
$#C+1 = 30 : ~/encyclopedia/old_files/data/B015/B.0105870 Bessel potential
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A potential of the form

$$ 
P _  \alpha  (x)  = \ 
\int\limits _ {\mathbf R  ^ {n} }
G _  \alpha  (x - y) \ 
d \mu (y),\ \ 
a > 0,
$$

where  $  x = (x _ {1} \dots x _ {n} ) $, 
$  y = (y _ {1} \dots y _ {n} ) $
are points in the Euclidean space  $  \mathbf R  ^ {n} $, 
$  n \geq  2 $; 
$  d \mu $
is a Borel measure on  $  \mathbf R  ^ {n} $;

$$ 
G _  \alpha  (x)  = \ 
2 ^ {(2 - n - \alpha ) / 2 }
\pi ^ {-n / 2 }
\left [ \Gamma \left (
{
\frac \alpha {2}
 } \right )  \right ]  ^ {-1}
K _ {(n - \alpha ) / 2 } 
(| x |)  | x | ^ {( \alpha - n) / 2 } ,
$$

$$ 
| x |  =  \left ( \sum _ {i = 1 } ^ { n }  | x _ {i}  ^ {2} | \right )  ^ {1/2} ,
$$

and  $  K _  \nu  (z) $
is the modified cylinder function (or Bessel function, cf. [[Cylinder functions|Cylinder functions]]) of the second kind of order  $  \nu $
or the Macdonald function of order  $  \nu $; 
$  G _  \alpha  (x) $
is called a Bessel kernel.

The principal properties of the Bessel kernels  $  G _  \alpha  (x) $
are the same as those of the Riesz kernels (cf. [[Riesz potential|Riesz potential]]), viz., they are positive, continuous for  $  x \neq 0 $, 
can be composed

$$ 
\int\limits _ {\mathbf R  ^ {n} }
G _  \alpha  (x - y)
G _  \beta  (y)  dy  = \ 
G _ {\alpha + \beta }  (x),
$$

but, unlike the Riesz potentials, Bessel potentials are applicable for all  $  \alpha > 0 $, 
since

$$ 
G _  \alpha  (x)  \sim \ 
2 ^ {(1 - n - \alpha ) / 2 }
\pi ^ {(1 - n) / 2 }
\left [ \Gamma \left (
{
\frac \alpha {2}
 } \right )  \right ]  ^ {-1}
| x | ^ {( \alpha - n - 1) / 2 }
e ^ {- | x | } ,
$$

as  $  | x | \rightarrow \infty $.

If  $  \alpha > 2m $, 
where  $  m $
is a natural number, and the measure  $  d \mu $
is absolutely continuous with square-integrable density  $  f(y) \in L _ {2} ( \mathbf R  ^ {2m} ) $, 
the Bessel potentials satisfy the identities:

$$ 
(1 - \Delta )  ^ {m}
P _  \alpha  (x)  = \ 
P _ {\alpha - 2m }  (x),
$$

and

$$ 
(1 - \Delta )  ^ {m}
P _ {2m} (x)  = \ 
f (x),
$$

where  $  \Delta $
is the [[Laplace operator|Laplace operator]] on  $  \mathbf R  ^ {2m} $. 
In other words, the function  $  G _ {2m} (x) $
is a fundamental solution of the operator  $  (1 - \Delta )  ^ {m} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Aronszajn,   K.T. Smith,   "Theory of Bessel potentials I"  ''Ann. Inst. Fourier (Grenoble)'' , '''11'''  (1961)  pp. 385–475</TD></TR></table>

====Comments====
The function  $  K _  \nu  (z) $
is usually called the modified Bessel function of the third kind.