# Bessel potential

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

#### References

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

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