Riesz potential

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.

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

(I_{\alpha}f) (x)= \frac{1}{c_\alpha} \int_{{\mathbb{R}}^n} \frac{f(y)}{| x - y |^{n-\alpha}} \, \mathrm{d}y

 

 

 

 

(1)

where the constant is given by

c_\alpha = \pi^{n/2}2^\alpha\frac{\Gamma(\alpha/2)}{\Gamma((n-\alpha)/2)}.

This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α. If p > 1, then the rate of decay of f and that of Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)

\|I_\alpha f\|_{p^*} \le C_p \|f\|_p,\quad p^*=\frac{np}{n-\alpha p}.

More generally, the operators Iα are well-defined for complex α such that 0 < Re α < n.

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

I_\alpha f = f*K_\alpha\,

where Kα is the locally integrable function:

K_\alpha(x) = \frac{1}{c_\alpha}\frac{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 Iαμ 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. In fact, one has

\widehat{K_\alpha}(\xi) = |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}\

provided

0 < \operatorname{Re\,} \alpha, \operatorname{Re\,} \beta < n,\quad 0 < \operatorname{Re\,} (\alpha+\beta) < n.

Furthermore, if 2 < Re α <n, then

\Delta I_{\alpha+2} = -I_\alpha.\

One also has, for this class of functions,

\lim_{\alpha\to 0^+} (I^\alpha f)(x) = f(x).

See also

References