Riesz potential explained

In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable.

Definition

If 0 < α < n, then the Riesz potential I_αf of a locally integrable function f on Rⁿ is the function defined by

where the constant is given by

c _{\alpha} = \pi ^{n/2} 2 ^{\alpha (\Gamma(\alpha/ 2 ))/(\Gamma((n - \alpha)/ 2 ))} .

This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ L^p(Rⁿ) with 1 ≤ p < n/α. In fact, for any 1 ≤ p (p>1 is classical, due to Sobolev, while for p=1 see, the rate of decay of f and that of I_αf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)

\|I _{\alpha} f\| _{p ^{*}} \le C _{p} \|Rf\| _{p,} p ^{* = (np)/(n - \alpha p)} , where Rf = DI _{1f} is the vector-valued Riesz transform. More generally, the operators I_α are well-defined for complex α such that .

The Riesz potential can be defined more generally in a weak sense as the convolution

I _{\alpha} f = f*K _{\alpha}

where K_α is the locally integrable function:

K _{\alpha(x)} = (1)/(c _{\alpha}) (1)/(|x| ^{n-\alpha}) . The Riesz potential can therefore be defined whenever f is a compactly supported distribution. In this connection, the Riesz potential of a positive Borel measure μ with compact support is chiefly of interest in potential theory because I_αμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rⁿ.

Consideration of the Fourier transform reveals that the Riesz potential is a Fourier multiplier.^[1] In fact, one has

\widehat{K _{\alpha}(\xi)} = \int _{\R ^{n}} K _{\alpha} (x) e ^{-2\pi} d x = | 2 \pi\xi| ^{-\alpha} and so, by the convolution theorem,\widehat{I _{\alpha} f}(\xi) = | 2 \pi\xi| ^{-\alpha} \hat{f}(\xi).

The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions

I _{\alpha} I _{\beta} = I _{\alpha+\beta} provided0 < \operatorname{Re} \alpha, \operatorname{Re} \beta < n, 0 < \operatorname{Re} (\alpha + \beta) < n. Furthermore, if, then\Delta I _{\alpha+2} = I _{\alpha+2} \Delta =- I _{\alpha.} One also has, for this class of functions,\lim _{\alpha\to 0 ^{+}} (I _{\alpha} f)(x) = f(x).

See also

• Bessel potential
• Fractional integration
• Sobolev space

References

• • .

Notes and References

1. .