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.

Definition

If 0 < ''α'' < ''n'', then the Riesz potential ''I''_α''f'' of a locally integrable function ''f'' on R^''n'' is the function defined by

where the constant is given by

:$c_\alpha = \pi^2^\alpha\frac.$

This singular integral is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f'' ∈  L^''p''(R^''n'') with 1 ≤ ''p'' < ''n''/''α''. In fact, for any 1 ≤ ''p'' (''p''>1 is classical, due to Sobolev, while for ''p''=1 see ), the rate of decay of ''f'' and that of ''I''_''α''''f'' are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
:$\, I_\alpha f\, _ \le C_p \, Rf\, _p, \quad p^*=\frac,$
where $Rf=DI_1f$ is the vector-valued Riesz transform. More generally, the operators ''I''_''α'' are well-defined for complex α such that .

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:
:$K_\alpha(x) = \frac\frac.$
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''_''α''μ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of R^''n''.

Consideration of the Fourier transform reveals that the Riesz potential is a Fourier multiplier..
In fact, one has
:$\widehat(\xi) = \int_ K_(x) e^\, \mathrmx = , 2\pi\xi, ^$
and so, by the convolution theorem,
:$\widehat(\xi) = , 2\pi\xi, ^ \hat(\xi).$

The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions
:$I_\alpha I_\beta = I_$
provided
:$0 < \operatorname \alpha, \operatorname \beta < n,\quad 0 < \operatorname (\alpha+\beta) < n.$
Furthermore, if , then
:$\Delta I_ = I_ \Delta=-I_\alpha.$
One also has, for this class of functions,
:$\lim_ (I_\alpha f)(x) = f(x).$

See also

* Bessel potential
* Fractional integration
* Sobolev space

Notes

References

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* {{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

Fractional calculus
Partial differential equations
Potential theory
Singular integrals