9.9 KB · ⇊ 6 parts · ⤓ Micron 9.9 KB · ⤓ HTML 44.1 KB

top

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

• See also

• Notes

• References

If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by

where the constant is given by

c α α = π π n / 2 2 α α Γ Γ ( α α / 2 ) Γ Γ ( ( n − − α α ) / 2 ) . {\displaystyle c_{\alpha }=\pi ^{n/2}2^{\alpha }{\frac {\Gamma (\alpha /2)}{\Gamma ((n-\alpha )/2)}}.}

This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α. The classical result due to Sobolev states that the rate of decay of f and that of Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)

‖ ‖ I α α f ‖ ‖ p ∗ ∗ ≤ ≤ C p ‖ ‖ f ‖ ‖ p , p ∗ ∗ = n p n − − α α p , ∀ ∀ 1 < p < n α α {\displaystyle \|I_{\alpha }f\|_{p^{*}}\leq C_{p}\|f\|_{p},\quad p^{*}={\frac {np}{n-\alpha p}},\quad \forall 1<p<{\frac {n}{\alpha }}}

For p=1 the result was extended by (Schikorra, Spector & Van Schaftingen 2014),

‖ ‖ I α α f ‖ ‖ 1 ∗ ∗ ≤ ≤ C p ‖ ‖ R f ‖ ‖ 1 . {\displaystyle \|I_{\alpha }f\|_{1^{*}}\leq C_{p}\|Rf\|_{1}.}

where R f = D I 1 f {\displaystyle Rf=DI_{1}f} 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

I α α f = f ∗ ∗ K α α {\displaystyle I_{\alpha }f=f*K_{\alpha }}

where Kα is the locally integrable function:

K α α ( x ) = 1 c α α 1 | x | n − − α α . {\displaystyle K_{\alpha }(x)={\frac {1}{c_{\alpha }}}{\frac {1}{|x|^{n-\alpha }}}.}

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.cite-ref-1[1] In fact, one has

K α α ^ ^ ( ξ ξ ) = ∫ ∫ R n K α α ( x ) e − − 2 π π i x ξ ξ d x = | 2 π π ξ ξ | − − α α {\displaystyle {\widehat {K_{\alpha }}}(\xi )=\int _{\mathbb {R} ^{n}}K_{\alpha }(x)e^{-2\pi ix\xi }\,\mathrm {d} x=|2\pi \xi |^{-\alpha }}

and so, by the convolution theorem,

I α α f ^ ^ ( ξ ξ ) = | 2 π π ξ ξ | − − α α f ^ ^ ( ξ ξ ) . {\displaystyle {\widehat {I_{\alpha }f}}(\xi )=|2\pi \xi |^{-\alpha }{\hat {f}}(\xi ).}

The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions

I α α I β β = I α α + β β {\displaystyle I_{\alpha }I_{\beta }=I_{\alpha +\beta }}

provided

0 < Re ⁡ ⁡ α α , Re ⁡ ⁡ β β < n , 0 < Re ⁡ ⁡ ( α α + β β ) < n . {\displaystyle 0<\operatorname {Re} \alpha ,\operatorname {Re} \beta <n,\quad 0<\operatorname {Re} (\alpha +\beta )<n.}

Furthermore, if 0 < Re α < n–2, then

Δ Δ I α α + 2 = I α α + 2 Δ Δ = − − I α α . {\displaystyle \Delta I_{\alpha +2}=I_{\alpha +2}\Delta =-I_{\alpha }.}

One also has, for this class of functions,

lim α α → → 0 + ( I α α f ) ( x ) = f ( x ) . {\displaystyle \lim _{\alpha \to 0^{+}}(I_{\alpha }f)(x)=f(x).}

• Bessel potential

• Fractional integration

• Sobolev space

cite-note-11. ↑ Samko 1998, section II.

• citereflandkof1972Landkof, N. S. (1972), Foundations of modern potential theory, Berlin, New York: Springer-Verlag, MR 0350027

• citerefriesz1949Riesz, Marcel (1949), "L'intégrale de Riemann-Liouville et le problème de Cauchy", Acta Mathematica, 81: 1–223, doi:10.1007/BF02395016, ISSN 0001-5962, MR 0030102.

• citerefsolomentsev2001Solomentsev, E.D. (2001) [1994], "Riesz potential", Encyclopedia of Mathematics, EMS Press

• citerefschikorraspectorvan-schaftingen2014Schikorra, Armin; Spector, Daniel; Van Schaftingen, Jean (2014), An L 1 {\displaystyle L^{1}} -type estimate for Riesz potentials, arXiv:1411.2318, doi:10.4171/rmi/937, S2CID 55497245

• citerefstein1970Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8

• citerefsamko1998Samko, Stefan G. (1998), "A new approach to the inversion of the Riesz potential operator" (PDF), Fractional Calculus and Applied Analysis, 1 (3): 225–245