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Riesz potential

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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 R is the function defined by

( I α f ) ( x ) = 1 c α ∫ R n f ( y ) | x − y | n − α d y {\displaystyle (I_{\alpha }f)(x)={\frac {1}{c_{\alpha }}}\int _{\mathbb {R} ^{n}}{\frac {f(y)}{|x-y|^{n-\alpha }}}\,\mathrm {d} y} 1

where the constant is given by

c α = π n / 2 2 α Γ ( α / 2 ) Γ ( ( n − 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 ∈ L(R) 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 & Van Schaftingen 2014), 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 f ‖ p ∗ ≤ C p ‖ R f ‖ p , p ∗ = n p n − α p , {\displaystyle \|I_{\alpha }f\|_{p^{*}}\leq C_{p}\|Rf\|_{p},\quad p^{*}={\frac {np}{n-\alpha p}},}

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

where Kα is the locally integrable function:

K α ( x ) = 1 c α 1 | x | n − 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 R.

Consideration of the Fourier transform reveals that the Riesz potential is a Fourier multiplier. In fact, one has

K α ^ ( ξ ) = ∫ R n K α ( x ) e − 2 π i x ξ d x = | 2 π ξ | − 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 f ^ ( ξ ) = | 2 π ξ | − α f ^ ( 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 I β = I α + β {\displaystyle I_{\alpha }I_{\beta }=I_{\alpha +\beta }}

provided

0 < Re ⁡ α , Re ⁡ β < n , 0 < Re ⁡ ( α + 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 I α + 2 = I α + 2 Δ = − I α . {\displaystyle \Delta I_{\alpha +2}=I_{\alpha +2}\Delta =-I_{\alpha }.}

One also has, for this class of functions,

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

See also

Notes

  1. Samko 1998, section II.

References

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