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.

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

where the constant is given by


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

This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f &isin; Lp(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 Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
 * $$\|I_\alpha f\|_{p^*} \le C_p \|Rf\|_p, \quad p^*=\frac{np}{n-\alpha p},$$

where $$Rf=DI_1f$$ is the vector-valued Riesz transform. 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
 * $$\widehat{K_\alpha}(\xi) = \int_{\R^n} K_{\alpha}(x) e^{-2\pi i x \xi }\, \mathrm{d}x = |2\pi\xi|^{-\alpha}$$

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
 * $$I_\alpha I_\beta = I_{\alpha+\beta} $$

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

Furthermore, if 0 < Re α < n–2, then
 * $$\Delta I_{\alpha+2} = I_{\alpha+2} \Delta=-I_\alpha. $$

One also has, for this class of functions,
 * $$\lim_{\alpha\to 0^+} (I_\alpha f)(x) = f(x).$$