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 RiemannâÂÂLiouville integrals of one variable.
If 0 < ñ < n, then the Riesz potential I<sub>ñ</sub>f of a locally integrable function f on R<sup>n</sup> 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<sup>p</sup>(R<sup>n</sup>) with 1 ⤠p < n/ñ. The classical result due to Sobolev states that the rate of decay of f and that of I<sub>ñ</sub>f are related in the form of an inequality (the HardyâÂÂLittlewoodâÂÂSobolev inequality)
For p=1 the result was extended by ,
where is the vector-valued Riesz transform. More generally, the operators I<sub>ñ</sub> are well-defined for complex ñ such that .
The Riesz potential can be defined more generally in a weak sense as the convolution
where K<sub>ñ</sub> 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 ü with compact support is chiefly of interest in potential theory because I<sub>ñ</sub>ü is then a (continuous) subharmonic function off the support of ü, and is lower semicontinuous on all of R<sup>n</sup>.
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 , then
One also has, for this class of functions,