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 Iαf of a locally integrable function f on Rn 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 ∈ Lp(Rn) with 1 ≤ p < n/α. If p > 1, then the rate of decay of f and that of Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
More generally, the operators Iα are well-defined for complex α such that 0 < Re α < n.
where Kα is the locally integrable function:
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 Rn.
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,
- Landkof, N. S. (1972), Foundations of modern potential theory, Berlin, New York: Springer-Verlag, MR 0350027
- Riesz, Marcel (1949), "L'intégrale de Riemann-Liouville et le problème de Cauchy", Acta Mathematica 81: 1–223, doi:10.1007/BF02395016, ISSN 0001-5962, MR 0030102.
- Solomentsev, E.D. (2001), "Riesz potential", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8