# 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

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:

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

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