Bessel potential

A potential of the form

where , are points in the Euclidean space , ; is a Borel measure on ;

and is the modified cylinder function (or Bessel function, cf. Cylinder functions) of the second kind of order or the Macdonald function of order ; is called a Bessel kernel.

The principal properties of the Bessel kernels are the same as those of the Riesz kernels (cf. Riesz potential), viz., they are positive, continuous for , can be composed

but, unlike the Riesz potentials, Bessel potentials are applicable for all , since

as .

If , where is a natural number, and the measure is absolutely continuous with square-integrable density , the Bessel potentials satisfy the identities:

and

where is the Laplace operator on . In other words, the function is a fundamental solution of the operator .

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

Comments

The function is usually called the modified Bessel function of the third kind.