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_αf 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 ∈ L^p(Rⁿ) 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_αf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)

where R f = D I 1 f is the vector-valued Riesz transform. 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_αμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rⁿ.

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,