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| In [[mathematics]], the '''Riesz potential''' is a [[potential theory|potential]] named after its discoverer, the [[Hungary|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.
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| If 0 < α < ''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
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| {{NumBlk|:|<math>(I_{\alpha}f) (x)= \frac{1}{c_\alpha} \int_{{\mathbb{R}}^n} \frac{f(y)}{| x - y |^{n-\alpha}} \, \mathrm{d}y</math>|{{EquationRef|1}}}}
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| where the constant is given by
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| :<math>c_\alpha = \pi^{n/2}2^\alpha\frac{\Gamma(\alpha/2)}{\Gamma((n-\alpha)/2)}.</math>
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| This [[singular integral]] is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f'' ∈ [[Lp space|L<sup>''p''</sup>('''R'''<sup>''n''</sup>)]] with 1 ≤ ''p'' < ''n''/α. If ''p'' > 1, then 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]])
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| :<math>\|I_\alpha f\|_{p^*} \le C_p \|f\|_p,\quad p^*=\frac{np}{n-\alpha p}.</math>
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| More generally, the operators ''I''<sub>α</sub> are well-defined for [[complex number|complex]] α such that 0 < Re α < ''n''.
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| The Riesz potential can be defined more generally in a [[distribution (mathematics)|weak sense]] as the [[convolution]]
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| :<math>I_\alpha f = f*K_\alpha\,</math> | |
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| where ''K''<sub>α</sub> is the locally integrable function:
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| :<math>K_\alpha(x) = \frac{1}{c_\alpha}\frac{1}{|x|^{n-\alpha}}.</math>
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| 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>.
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| Consideration of the [[Fourier transform]] reveals that the Riesz potential is a [[Fourier multiplier]]. In fact, one has
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| :<math>\widehat{K_\alpha}(\xi) = |2\pi\xi|^{-\alpha}</math>
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| and so, by the [[convolution theorem]],
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| :<math>\widehat{I_\alpha f}(\xi) = |2\pi\xi|^{-\alpha} \hat{f}(\xi).</math>
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| The Riesz potentials satisfy the following [[semigroup]] property on, for instance, rapidly decreasing continuous functions
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| :<math>I_\alpha I_\beta = I_{\alpha+\beta}\ </math>
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| provided
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| :<math>0 < \operatorname{Re\,} \alpha, \operatorname{Re\,} \beta < n,\quad 0 < \operatorname{Re\,} (\alpha+\beta) < n.</math>
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| Furthermore, if 2 < Re α <''n'', then
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| :<math>\Delta I_{\alpha+2} = -I_\alpha.\ </math>
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| One also has, for this class of functions,
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| :<math>\lim_{\alpha\to 0^+} (I^\alpha f)(x) = f(x).</math>
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| ==See also==
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| * [[Bessel potential]]
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| * [[Fractional integration]]
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| * [[Sobolev space]]
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| * [[Fractional Schrödinger equation]]
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| ==References==
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| *{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | id={{MathSciNet | id = 0350027}} | year=1972}}
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| *{{Citation | last1=Riesz | first1=Marcel | author1-link=Marcel Riesz | title=L'intégrale de Riemann-Liouville et le problème de Cauchy | id={{MathSciNet | id = 0030102}} | year=1949 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=81 | pages=1–223 | doi=10.1007/BF02395016}}.
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| * {{springer|last=Solomentsev|first=E.D.|id=R/r082270|title=Riesz potential}}
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| * {{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}}
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| [[Category:Fractional calculus]]
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| [[Category:Partial differential equations]]
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| [[Category:Potential theory]]
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| [[Category:Singular integrals]]
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