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In [[mathematics]], the '''Riesz potential''' is a [[potential theory|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 integral]]s of one variable.

== Definition ==
If 0&nbsp;<&nbsp;''α''&nbsp;<&nbsp;''n'', then the Riesz potential ''I''<sub>α</sub>''f'' of a [[locally integrable function]] ''f'' on '''R'''<sup>''n''</sup> is the function defined by

{{NumBlk|:|<math>(I_{\alpha}f) (x)= \frac{1}{c_\alpha} \int_{\R^n} \frac{f(y)}{| x - y |^{n-\alpha}} \, \mathrm{d}y</math>|{{EquationRef|1}}}}

where the constant is given by

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

This [[singular integral]] is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f''&nbsp;&isin;&nbsp;[[Lp space|L<sup>''p''</sup>('''R'''<sup>''n''</sup>)]] with 1&nbsp;≤&nbsp;''p''&nbsp;<&nbsp;''n''/''α''. The classical result due to Sobolev states that the rate of decay of ''f'' and that of ''I''<sub>''α''</sub>''f'' are related in the form of an inequality (the [[Hardy–Littlewood–Sobolev inequality]])

:<math>\|I_\alpha f\|_{p^*} \le C_p \|f\|_p, \quad p^*=\frac{np}{n-\alpha p}, \quad \forall 1 < p < \frac{n}{\alpha}</math> 

For ''p''=1 the result was extended by {{harv|Schikorra|Spector|Van Schaftingen|2014}}, 

:<math>\|I_\alpha f\|_{1^*} \le C_p \|Rf\|_1.</math>

where <math>Rf=DI_1f</math> is the vector-valued [[Riesz transform]].  More generally, the operators ''I''<sub>''α''</sub> are well-defined for [[complex number|complex]] α such that {{nowrap|0 < Re ''α'' < ''n''}}.

The Riesz potential can be defined more generally in a [[distribution (mathematics)|weak sense]] as the [[convolution]]

:<math>I_\alpha f = f*K_\alpha</math>

where ''K''<sub>α</sub> is the locally integrable function:
:<math>K_\alpha(x) = \frac{1}{c_\alpha}\frac{1}{|x|^{n-\alpha}}.</math>
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 [[support (measure theory)|compact support]] is chiefly of interest in [[potential theory]] because ''I''<sub>''α''</sub>μ is then a (continuous) [[subharmonic function]] off the support of μ, and is [[lower semicontinuous]] on all of '''R'''<sup>''n''</sup>.

Consideration of the [[Fourier transform]] reveals that the Riesz potential is a [[Fourier multiplier]].<ref>{{harvnb |Samko|1998|loc=section II}}.</ref>
In fact, one has
:<math>\widehat{K_\alpha}(\xi) = \int_{\R^n} K_{\alpha}(x) e^{-2\pi i x \xi }\, \mathrm{d}x = |2\pi\xi|^{-\alpha}</math>
and so, by the [[convolution theorem]],
:<math>\widehat{I_\alpha f}(\xi) = |2\pi\xi|^{-\alpha} \hat{f}(\xi).</math>

The Riesz potentials satisfy the following [[semigroup]] property on, for instance, [[rapidly decreasing]] [[continuous function]]s
:<math>I_\alpha I_\beta = I_{\alpha+\beta} </math>
provided
:<math>0 < \operatorname{Re} \alpha, \operatorname{Re} \beta < n,\quad 0 < \operatorname{Re} (\alpha+\beta) < n.</math>
Furthermore, if {{nowrap|0 < Re ''α'' < ''n''–2}}, then
:<math>\Delta I_{\alpha+2} = I_{\alpha+2} \Delta=-I_\alpha. </math>
One also has, for this class of functions,
:<math>\lim_{\alpha\to 0^+} (I_\alpha f)(x) = f(x).</math>

==See also==
* [[Bessel potential]]
* [[Fractional integration]]
* [[Sobolev space]]

== Notes ==
{{reflist|22em}}

==References==
*{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York |mr=0350027 | year=1972}}
*{{Citation | last1=Riesz | first1=Marcel | author1-link=Marcel Riesz | title=L'intégrale de Riemann-Liouville et le problème de Cauchy |mr=0030102 | year=1949 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=81 | pages=1–223 | doi=10.1007/BF02395016| doi-access=free }}.
* {{springer|last=Solomentsev|first=E.D.|id=R/r082270|title=Riesz potential}}
*{{Citation|last1=Schikorra| first1=Armin |last2=Spector| first2=Daniel |last3= Van Schaftingen| first3=Jean | title=An <math> L^1 </math>-type estimate for Riesz potentials| year=2014 |arxiv=1411.2318 | doi=10.4171/rmi/937| s2cid=55497245 }}
* {{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Singular integrals and differentiability properties of functions|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1970|isbn=0-691-08079-8|url-access=registration|url=https://archive.org/details/singularintegral0000stei}}
* {{Citation | last=Samko | first=Stefan G. | title=A new approach to the inversion of the Riesz potential operator | journal=[[Fractional Calculus and Applied Analysis]] | year=1998 | volume=1 | issue=3 | pages=225–245 | url=http://w3.ualg.pt/~ssamko/dpapers/files/New_Approach_FCAA.pdf }}

[[Category:Fractional calculus]]
[[Category:Partial differential equations]]
[[Category:Potential theory]]
[[Category:Singular integrals]]