# Riesz potential

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{{Short description|Potential in mathematics}}
In [mathematics](/source/mathematics), the '''Riesz potential''' is a [potential](/source/potential_theory) named after its discoverer, the Hungarian mathematician [Marcel Riesz](/source/Marcel_Riesz).  In a sense, the Riesz potential defines an inverse for a power of the [Laplace operator](/source/Laplace_operator) on Euclidean space.  They generalize to several variables the [Riemann–Liouville integral](/source/Riemann%E2%80%93Liouville_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](/source/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](/source/singular_integral) is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f''&nbsp;&isin;&nbsp;[L<sup>''p''</sup>('''R'''<sup>''n''</sup>)](/source/Lp_space) 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](/source/Hardy%E2%80%93Littlewood%E2%80%93Sobolev_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](/source/Riesz_transform).  More generally, the operators ''I''<sub>''α''</sub> are well-defined for [complex](/source/complex_number) α such that {{nowrap|0 < Re ''α'' < ''n''}}.

The Riesz potential can be defined more generally in a [weak sense](/source/distribution_(mathematics)) as the [convolution](/source/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](/source/Borel_measure) μ with [compact support](/source/support_(measure_theory)) is chiefly of interest in [potential theory](/source/potential_theory) because ''I''<sub>''α''</sub>μ is then a (continuous) [subharmonic function](/source/subharmonic_function) off the support of μ, and is [lower semicontinuous](/source/lower_semicontinuous) on all of '''R'''<sup>''n''</sup>.

Consideration of the [Fourier transform](/source/Fourier_transform) reveals that the Riesz potential is a [Fourier multiplier](/source/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](/source/convolution_theorem),
:<math>\widehat{I_\alpha f}(\xi) = |2\pi\xi|^{-\alpha} \hat{f}(\xi).</math>

The Riesz potentials satisfy the following [semigroup](/source/semigroup) property on, for instance, [rapidly decreasing](/source/rapidly_decreasing) [continuous function](/source/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](/source/Bessel_potential)
* [Fractional integration](/source/Fractional_integration)
* [Sobolev space](/source/Sobolev_space)

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

==References==
*{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[Springer-Verlag](/source/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](/source/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](/source/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](/source/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 | access-date=2018-03-22 | archive-date=2016-02-22 | archive-url=https://web.archive.org/web/20160222024834/http://w3.ualg.pt/~ssamko/dpapers/files/New_Approach_FCAA.pdf | url-status=dead }}

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

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Adapted from the Wikipedia article [Riesz potential](https://en.wikipedia.org/wiki/Riesz_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Riesz_potential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.