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
where the constant is given by
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)
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
where ''K''α is the locally integrable function:
and so, by the convolution theorem,
The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions
Furthermore, if 2 < Re α <''n'', then
One also has, for this class of functions,
* Bessel potential
* Fractional integration
* Sobolev space
* Fractional Schrödinger equation
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