Riesz potential
Encyclopedia
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 Iαƒ of a locally integrable function
ƒ on Rn is the function defined by
where the constant is given by
This singular integral
is well-defined provided ƒ decays sufficiently rapidly at infinity, specifically if ƒ ∈ Lp(Rn)
with 1 ≤ p < n/α. If p > 1, then the rate of decay of ƒ and that of Iαƒ are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
More generally, the operators Iα 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 Kα 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 support
is chiefly of interest in potential theory
because Iαμ 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,
Mathematics
Mathematics 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 potential
Potential theory
In 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 Hungarian
Hungary
Hungary , 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...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Marcel Riesz
Marcel Riesz
Marcel 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 operator
Laplace operator
In 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 Iαƒ of a locally integrable function
Locally integrable function
In 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 Rn is the function defined by
where the constant is given by
This singular integral
Singular integral
In 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 ∈ Lp(Rn)
Lp space
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 Iαƒ are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
More generally, the operators Iα are well-defined for complex
Complex number
A 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 sense
Distribution (mathematics)
In 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 convolution
Convolution
In 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 Kα 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 support
Support (measure theory)
In 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 theory
Potential theory
In 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 Iαμ is then a (continuous) subharmonic function
Subharmonic function
In 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 Rn.
Consideration of the Fourier transform
Fourier transform
In 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 theorem
Convolution theorem
In 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 semigroup
Semigroup
In 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 potentialBessel potentialIn mathematics, the Bessel potential is a potential similar to the Riesz potential but with better decay properties at infinity....
- Fractional integration
- Sobolev spaceSobolev spaceIn 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 equationFractional Schrödinger equationThe 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...