Riesz potential

 ( Fractional Calculus ) 

In mathematics, the Riesz potential is a potential theory 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.

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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

\, \mathrm{d}y|}}
     

where the constant is given by

$c\_\backslash alpha\; =\; \backslash pi^\{n/2\}2^\backslash alpha\backslash frac\{\backslash Gamma(\backslash alpha/2)\}\{\backslash Gamma((n-\backslash alpha)/2)\}.$

This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp space with 1 ≤  p <  n/ α. The classical result due to Sobolev states that 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)

For p=1 the result was extended by ,

$\backslash |I\_\backslash alpha\; f\backslash |\_\{1^*\}\; \backslash le\; C\_p\; \backslash |Rf\backslash |\_1.$

where $Rf=DI\_1f$ is the vector-valued Riesz transform. More generally, the operators I _α are well-defined for complex number α such that .

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

$I\_\backslash alpha\; f\; =\; f*K\_\backslash alpha$

where K_α is the locally integrable function:

$K\_\backslash alpha(x)\; =\; \backslash frac\{1\}\{c\_\backslash alpha\}\backslash frac\{1\}\{|x|^\{n-\backslash alpha\}\}.$

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 I _αμ 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

$\backslash widehat\{K\_\backslash alpha\}(\backslash xi)\; =\; \backslash int\_\{\backslash R^n\}\; K\_\{\backslash alpha\}(x)\; e^\{-2\backslash pi\; i\; x\; \backslash xi\; \}\backslash ,\; \backslash mathrm\{d\}x\; =\; |2\backslash pi\backslash xi|^\{-\backslash alpha\}$

and so, by the convolution theorem,

$\backslash widehat\{I\_\backslash alpha\; f\}(\backslash xi)\; =\; |2\backslash pi\backslash xi|^\{-\backslash alpha\}\; \backslash hat\{f\}(\backslash xi).$

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

$I\_\backslash alpha\; I\_\backslash beta\; =\; I\_\{\backslash alpha+\backslash beta\}$

provided

Furthermore, if , then

$\backslash Delta\; I\_\{\backslash alpha+2\}\; =\; I\_\{\backslash alpha+2\}\; \backslash Delta=-I\_\backslash alpha.$

One also has, for this class of functions,

$\backslash lim\_\{\backslash alpha\backslash to\; 0^+\}\; (I\_\backslash alpha\; f)(x)\; =\; f(x).$

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See also

• Bessel potential
• Fractional integration
• Sobolev space

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Notes

• .

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