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 Riesz potential ： ウィキペディア英語版
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.
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
\, \mathrmy|}}
where the constant is given by
:$c_\alpha = \pi^2^\alpha\frac.$
This singular integral is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f'' ∈ L''p''(R''n'') with 1 ≤ ''p'' < ''n''/α. If ''p'' > 1, then 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)
:$\|I_\alpha f\|_ \le C_p \|f\|_p,\quad p^$
*=\frac.
More generally, the operators ''I''α 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
:$I_\alpha f = f$
*K_\alpha\,
where ''K''α is the locally integrable function:
:$K_\alpha\left(x\right) = \frac\frac\left(\xi\right) = |2\pi\xi|^$
and so, by the convolution theorem,
:$\widehat\left(\xi\right) = |2\pi\xi|^ \hat\left(\xi\right).$
The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions
:$I_\alpha I_\beta = I_\$
provided
:$0 < \operatorname \alpha, \operatorname \beta < n,\quad 0 < \operatorname \left(\alpha+\beta\right) < n.$
Furthermore, if 2 < Re α <''n'', then
:$\Delta I_ = -I_\alpha.\$
One also has, for this class of functions,
:$\lim_ \left(I^\alpha f\right)\left(x\right) = f\left(x\right).$

* Bessel potential
* Fractional integration
* Sobolev space
* Fractional Schrödinger equation

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