Encyclopedia

Riesz potential

Potential in mathematics

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated July 3, 2026

In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines a fractional inverse power $(-\Delta )^{-\alpha /2}$ of the Laplace operator on Euclidean space. It generalizes to several variables the Riemann-Liouville integrals of one variable.

01Definition

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

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

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 ∈ L^p(Rⁿ) with 1 ≤ p < n/α. The classical result due to Sobolev states that the rate of decay of f and that of I_αf are related in the form of an inequality (the Hardy-Littlewood-Sobolev inequality)

$\|I_{\alpha }f\|_{p^{*}}\leq C_{p}\|f\|_{p},\quad p^{*}={\frac {np}{n-\alpha p}},\quad \forall 1<p<{\frac {n}{\alpha }}$

For p=1 the result was extended by (Schikorra, Spector & Van Schaftingen 2014),

$\|I_{\alpha }f\|_{1^{*}}\leq C_{p}\|Rf\|_{1}.$

where $Rf=DI_{1}f$ is the vector-valued Riesz transform. More generally, the operators I_α 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 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

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

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

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

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,

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

Watch videos about Riesz potentialExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Riesz potential”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Riesz potential

01Definition

Sources and credits

Related topics

Bessel potential

Differintegral

Sobolev space