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 I_αf of a locally integrable function f on Rⁿ is the function defined by

where the constant is given by

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

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 Template:Harv, the rate of decay of f and that of I_αf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)

$\Vert I_{\alpha }f{\Vert }_{p^{*}}\le C_p\Vert Rf{\Vert }_p,p^{*}=\frac{np}{n-\alpha p},$

where $Rf=D{I}_{1}f$ is the vector-valued Riesz transform. More generally, the operators I_α are well-defined for complex α such that Template:Nowrap.

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:

$K_{\alpha }(x)=\frac{1}{c_{\alpha }}\frac{1}{|x{|}^{n-\alpha }}.$

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 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. In fact, one has

$\widehat{K_{\alpha }}(\xi )=\underset{{ℝ}^n}{\int }{K}_{\alpha }(x)e^{-2\pi ix\xi }\mathrm{d}x=|2\pi \xi {|}^{-\alpha }$

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

$0<\mathrm{Re}\alpha ,\mathrm{Re}\beta <n,0<\mathrm{Re}(\alpha +\beta )<n.$

Furthermore, if Template:Nowrap, then

$\mathrm{\Delta }{I}_{\alpha +2}=I_{\alpha +2}\mathrm{\Delta }=-I_{\alpha }.$

One also has, for this class of functions,

$\underset{\alpha \to 0^{+}}{\lim }(I_{\alpha }f)(x)=f(x).$

See also

• Bessel potential
• Fractional integration
• Sobolev space

References

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