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

Comments

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