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Riesz potential

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Potential in mathematics

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

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

( I a f ) ( x ) = 1 c a R n f ( y ) | x - y | n - a 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 a = p n / 2 2 a G ( a / 2 ) G ( ( n - a ) / 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/a. The classical result due to Sobolev states that the rate of decay of f and that of Iaf are related in the form of an inequality (the Hardy-Littlewood-Sobolev inequality)

|| I a f || p * <= C p || f || p , p * = n p n - a p , 1 < p < n a {\displaystyle \|I_{\alpha }f\|_{p^{*}}\leq C_{p}\|f\|_{p},\quad p^{*}={\frac {np}{n-\alpha p}},\quad \forall 1

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

|| I a 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 Ia are well-defined for complex a such that 0 < Re a < n.

The Riesz potential can be defined more generally in a weak sense as the convolution

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

where Ka is the locally integrable function:

K a ( x ) = 1 c a 1 | x | n - a . {\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 m with compact support is chiefly of interest in potential theory because Iam is then a (continuous) subharmonic function off the support of m, 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

K a ^ ( x ) = R n K a ( x ) e - 2 p i x x d x = | 2 p x | - a {\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 a f ^ ( x ) = | 2 p x | - a f ^ ( x ) . {\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 a I b = I a + b {\displaystyle I_{\alpha }I_{\beta }=I_{\alpha +\beta }}

provided

0 < Re a , Re b < n , 0 < Re ( a + b ) < n . {\displaystyle 0<\operatorname {Re} \alpha ,\operatorname {Re} \beta

Furthermore, if 0 < Re a < n-2, then

D I a + 2 = I a + 2 D = - I a . {\displaystyle \Delta I_{\alpha +2}=I_{\alpha +2}\Delta =-I_{\alpha }.}

One also has, for this class of functions,

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

See also

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Notes

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  1. ^ Samko 1998, section II.

References

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