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 Iαf of a locally integrable function f on Rn is the function defined by
-
[math]\displaystyle{ (I_{\alpha}f) (x)= \frac{1}{c_\alpha} \int_{\R^n} \frac{f(y)}{| x - y |^{n-\alpha}} \, \mathrm{d}y }[/math]
()
where the constant is given by
- [math]\displaystyle{ 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 ∈ Lp(Rn) with 1 ≤ p < n/α. In fact, for any 1 ≤ p (p>1 is classical, due to Sobolev, while for p=1 see (Schikorra Spector), the rate of decay of f and that of Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
- [math]\displaystyle{ \|I_\alpha f\|_{p^*} \le C_p \|Rf\|_p, \quad p^*=\frac{np}{n-\alpha p}, }[/math]
where [math]\displaystyle{ Rf=DI_1f }[/math] is the vector-valued Riesz transform. More generally, the operators Iα 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
- [math]\displaystyle{ I_\alpha f = f*K_\alpha }[/math]
where Kα is the locally integrable function:
- [math]\displaystyle{ 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 compact support is chiefly of interest in potential theory because Iαμ 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
- [math]\displaystyle{ \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]\displaystyle{ \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 functions
- [math]\displaystyle{ I_\alpha I_\beta = I_{\alpha+\beta} }[/math]
provided
- [math]\displaystyle{ 0 \lt \operatorname{Re} \alpha, \operatorname{Re} \beta \lt n,\quad 0 \lt \operatorname{Re} (\alpha+\beta) \lt n. }[/math]
Furthermore, if 0 < Re α < n–2, then
- [math]\displaystyle{ \Delta I_{\alpha+2} = I_{\alpha+2} \Delta=-I_\alpha. }[/math]
One also has, for this class of functions,
- [math]\displaystyle{ \lim_{\alpha\to 0^+} (I_\alpha f)(x) = f(x). }[/math]
See also
- Bessel potential
- Fractional integration
- Sobolev space
Notes
- ↑ Samko 1998, section II.
References
- Landkof, N. S. (1972), Foundations of modern potential theory, Berlin, New York: Springer-Verlag
- Riesz, Marcel (1949), "L'intégrale de Riemann-Liouville et le problème de Cauchy", Acta Mathematica 81: 1–223, doi:10.1007/BF02395016, ISSN 0001-5962.
- Hazewinkel, Michiel, ed. (2001), "Riesz potential", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=R/r082270
- Schikorra, Armin; Spector, Daniel; Van Schaftingen, Jean (2014), An [math]\displaystyle{ L^1 }[/math]-type estimate for Riesz potentials, doi:10.4171/rmi/937
- Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8, https://archive.org/details/singularintegral0000stei
- Samko, Stefan G. (1998), "A new approach to the inversion of the Riesz potential operator", Fractional Calculus and Applied Analysis 1 (3): 225–245, http://w3.ualg.pt/~ssamko/dpapers/files/New_Approach_FCAA.pdf
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