In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the Pólya–Szegő inequality (or Szegő inequality) states that the Sobolev energy of a function in a
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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
does not increase under
symmetric decreasing rearrangement In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.
Definition for sets
Given a measurable set, A, in \R ...
.
The inequality is named after the
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
s
George Pólya
George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental ...
and
Gábor Szegő
Gábor Szegő () (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician. He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and T ...
.
Mathematical setting and statement
Given a
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
the symmetric decreasing rearrangement
is the unique function such that for every
the sublevel set
is an
open ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defin ...
centred at the origin
that has the same
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
as
Equivalently,
is the unique
radial
Radial is a geometric term of location which may refer to:
Mathematics and Direction
* Vector (geometric)
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) ...
and radially
nonincreasing function, whose
strict sublevel sets are open and have the same measure as those of the function
.
The Pólya–Szegő inequality states that if moreover
then
and
:
Applications of the inequality
The Pólya–Szegő inequality is used to prove the
Rayleigh–Faber–Krahn inequality In spectral geometry, the Rayleigh–Faber–Krahn inequality, named after its conjecturer, Lord Rayleigh, and two individuals who independently proved the conjecture, G. Faber and Edgar Krahn, is an inequality concerning the lowest Dirichlet eigen ...
, which states that among all the domains of a given fixed volume, the ball has the smallest first
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
for the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
with
Dirichlet boundary condition
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
s. The proof goes by restating the problem as a minimization of the
Rayleigh quotient
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the con ...
.
The
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
can be deduced from the Pólya–Szegő inequality with
.
The optimal constant in the
Sobolev inequality
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Re ...
can be obtained by combining the Pólya–Szegő inequality with some integral inequalities.
Equality cases
Since the Sobolev energy is invariant under translations, any translation of a radial function achieves equality in the Pólya–Szegő inequality. There are however other functions that can achieve equality, obtained for example by taking a radial nonincreasing function that achieves its maximum on a ball of positive radius and adding to this function another function which is radial with respect to a different point and whose support is contained in the maximum set of the first function. In order to avoid this obstruction, an additional condition is thus needed.
It has been proved that if the function
achieves equality in the Pólya–Szegő inequality and if the set
is a
null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...
for Lebesgue's measure, then the function
is radial and radially nonincreasing with respect to some point
.
Generalizations
The Pólya–Szegő inequality is still valid for symmetrizations on the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
or the
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. Th ...
.
The inequality also holds for partial symmetrizations defined by foliating the space into planes (Steiner symmetrization) and into spheres (cap symmetrization).
There are also Pólya−Szegő inequalities for rearrangements with respect to non-Euclidean norms and using the
dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous dual space. The dual n ...
of the gradient.
Proofs of the inequality
Original proof by a cylindrical isoperimetric inequality
The original proof by Pólya and Szegő for
was based on an isoperimetric inequality comparing sets with cylinders and an asymptotics expansion of the area of the area of the graph of a function.
The inequality is proved for a smooth function
that vanishes outside a compact subset of the Euclidean space
For every
, they define the sets
:
These sets are the sets of points who lie between the domain of the functions
and
and their respective graphs. They use then the geometrical fact that since the horizontal slices of both sets have the same measure and those of the second are balls, to deduce that the area of the boundary of the cylindrical set
cannot exceed the one of
. These areas can be computed by the
area formula yielding the inequality
:
Since the sets
and
have the same measure, this is equivalent to
:
The conclusion then follows from the fact that
:
Coarea formula and isoperimetric inequality
The Pólya–Szegő inequality can be proved by combining the
coarea formula,
Hölder’s inequality and the classical
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
.
If the function
is smooth enough, the coarea formula can be used to write
:
where
denotes the
–dimensional
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ass ...
on the Euclidean space
. For almost every each
, we have by Hölder's inequality,
:
Therefore, we have
:
Since the set
is a ball that has the same measure as the set
, by the classical isoperimetric inequality, we have
:
Moreover, recalling that the sublevel sets of the functions
and
have the same measure,
:
and therefore,
:
Since the function
is radial, one has
:
and the conclusion follows by applying the coarea formula again.
Rearrangement inequalities for convolution
When
, the Pólya–Szegő inequality can be proved by representing the Sobolev energy by the
heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectru ...
.
One begins by observing that
:
where for
, the function
is the heat kernel, defined for every
by
:
Since for every
the function
is radial and radially decreasing, we have by the
Riesz rearrangement inequality In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions f : \mathbb^n \to \mathbb^+, g : \mathbb^n \to \mathbb^+ and h : \mathbb^n \to \mathbb^+ satisfy the inequ ...
:
Hence, we deduce that
:
References
{{DEFAULTSORT:Szego inequality
Sobolev spaces
Geometric inequalities
Rearrangement inequalities