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

(1)
where the constant is given by
This singular integral is welldefined provided f decays sufficiently rapidly at infinity, specifically if f ∈ L^{p}(R^{n}) with 1 ≤ p < n/α. In fact, for any 1 ≤ p (p>1 is classical, due to Sobolev, while for p=1 see (Schikorra & Spector Van Schaftingen)), 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)
where is the vectorvalued Riesz transform. More generally, the operators I_{α} are welldefined 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:
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_{α}μ 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
and so, by the convolution theorem,
The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions
provided
Furthermore, if 2 < Re α <n, then
One also has, for this class of functions,
See also
References
 Landkof, N. S. (1972), Foundations of modern potential theory, Berlin, New York: SpringerVerlag, MR 0350027
 Riesz, Marcel (1949), "L'intégrale de RiemannLiouville et le problème de Cauchy", Acta Mathematica, 81: 1–223, doi:10.1007/BF02395016, ISSN 00015962, MR 0030102.
 Solomentsev, E.D. (2001), "Riesz potential", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Schikorra, Armin; Spector, Daniel; Van Schaftingen, Jean, An type estimate for Riesz potentials, arXiv:1411.2318
 Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0691080798