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

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

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where the constant is given by

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

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)

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

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

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

where ${\displaystyle Rf=DI_{1}f}$ 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

${\displaystyle I_{\alpha }f=f*K_{\alpha }}$

where K_α is the locally integrable function:

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

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.¹ In fact, one has

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

and so, by the convolution theorem,

${\displaystyle {\widehat {I_{\alpha }f}}(\xi )=|2\pi \xi |^{-\alpha }{\hat {f}}(\xi ).}$

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

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

provided

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

Furthermore, if 0 < Re α < n–2, then

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

One also has, for this class of functions,

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