In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the

**Riesz potential** is a

potentialIn mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

named after its discoverer, the

HungarianHungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Marcel RieszMarcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...

. In a sense, the Riesz potential defines an inverse for a power of the

Laplace operatorIn mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

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}*ƒ* of a

locally integrable functionIn mathematics, a locally integrable function is a function which is integrable on any compact set of its domain of definition. Their importance lies on the fact that we do not care about their behavior at infinity.- Formal definition :...

*ƒ* on

**R**^{n} is the function defined by

where the constant is given by

This

singular integralIn mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator...

is well-defined provided

*ƒ* decays sufficiently rapidly at infinity, specifically if

*ƒ* ∈

L^{p}(**R**^{n})In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

with 1 ≤

*p* <

*n*/α. If

*p* > 1, then the rate of decay of

*ƒ* and that of

*I*_{α}*ƒ* are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)

More generally, the operators

*I*_{α} are well-defined for

complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

α such that 0 < Re α <

*n*.

The Riesz potential can be defined more generally in a

weak senseIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

as the

convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

where

*K*_{α} is the locally integrable function:

The Riesz potential can therefore be defined whenever ƒ is a compactly supported distribution. In this connection, the Riesz potential of a positive

Borel measure μ with

compact supportIn mathematics, the support of a measure μ on a measurable topological space is a precise notion of where in the space X the measure "lives"...

is chiefly of interest in

potential theoryIn mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

because

*I*_{α}μ is then a (continuous)

subharmonic functionIn mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory....

off the support of μ, and is lower semicontinuous on all of

**R**^{n}.

Consideration of the

Fourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

reveals that the Riesz potential is a Fourier multiplier. In fact, one has

and so, by the

convolution theoremIn mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...

,

The Riesz potentials satisfy the following

semigroupIn mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

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

- Bessel potential
In mathematics, the Bessel potential is a potential similar to the Riesz potential but with better decay properties at infinity....

- Fractional integration
- Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

- Fractional Schrödinger equation
The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term fractional Schrödinger equation was coined by...