Muckenhoupt weights
Encyclopedia
In mathematics
, the class of Muckenhoupt weights
consists of those weights 
for which the Hardy–Littlewood maximal operator is bounded on 
. Specifically, we consider functions 
on 
and their associated maximal function
s
defined as
where
is a ball in 
with radius 
and centre 
. We wish to characterise the functions 
for which we have a bound
where
depends only on 
and 
. This was first done by Benjamin Muckenhoupt.
, we say that a weight 
belongs to 
if 
is locally integrable and there is a constant 
such that, for all balls 
in 
, we have
where
and 
is the Lebesgue measure
of
. We say 
belongs to 
if there exists some 
such that
for all
and all balls 
.
is in 
if and only if any one of the following hold.
(a) The Hardy–Littlewood maximal function is bounded on
, that is
for some
which only depends on 
and the constant 
in the above definition.
(b) There is a constant
such that for any locally integrable function 
on 
for all balls
. Here
is the average of
over 
and
Equivalently,
, where 
, if and only if
and
This equivalence can be verified by using Jensen's Inequality
.
Reverse Hölder inequalities and
The main tool in the proof of the above equivalence is the following result. The following statements are equivalent
(a)
belongs to 
for some 
(b) There exists an
and a 
(both depending on 
such that
for all balls
(c) There exists
so that for all balls 
and subsets 
We call the inequality in (b) a reverse Hölder inequality as the reverse inequality follows for any non-negative function directly from Hölder's inequality
. If any of the three equivalent conditions above hold we say
belongs to 
.
weight and the reverse Hölder inequality indicate that such a weight cannot degenerate or grow too quickly. This property can be phrased equivalently in terms of how much the logarithm of the weight oscillates:
(a) If
, then 
(i.e. 
has bounded mean oscillation
).
(b) If
, then for sufficiently small 
, we have 
for some 
.
This equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the John–Nirenberg inequality. Note that the
condition in part (b) is necessary for the result to be true, as 
is a BMO function, but 
is not in any 
.
(i)
(ii)
(iii) If
, then 
defines a doubling measure: for any ball B, if 2B is the ball of twice the radius, then 
where C > 1 is a constant depending on 
.
(iv) If
, then there is 
such that 
.
(v) If
then there is 
and weights 
such that 
.
spaces. In fact, any Calderón-Zygmund singular integral operator is also bounded on these spaces. Let us describe a simpler version of this here. Suppose we have an operator 
which is bounded on 
, so we have
for all smooth and compactly supported
. Suppose also that we can realise 
as convolution against a kernel 
in the sense that, whenever 
and 
are smooth and have disjoint support
Finally we assume a size and smoothness condition on the kernel
:
for all
and multi-indices 
. Then, for each 
and 
, we have that 
is a bounded operator on 
. That is, we have the estimate
for all
for which the right-hand side is finite.
: For a fixed unit vector 
whenever
with 
, then we have a converse. If we know
for some fixed
and some 
, then 
.
, a K-quasiconformal mapping
is a homeomorphism
with 
and
where
is the derivative
of
at 
and 
is the Jacobian
.
A theorem of Gehring states that for all K-quasiconformal functions
, we have 
where 
depends on 
.
, we say its boundary curve 
is K-chord-arc if for any two points 
there is a curve 
connecting 
and 
whose length is no more than 
. For a domain with such a boundary and for any 
, the harmonic measure
is absolutely continuous with respect to one-dimensional Hausdorff measure
and its Radon–Nikodym derivative is in
. (Note that in this case, one needs to adapt the definition of weights to the case where the underlying measure is one-dimensional Hausdorff measure).
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 class of Muckenhoupt weights
consists of those weights for which the Hardy–Littlewood maximal operator is bounded on . Specifically, we consider functions on and their associated maximal functionMaximal function
Maximal functions appear in many forms in harmonic analysis . One of the most important of these is the Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals and partial differential equations...
s
defined aswhere
is a ball in with radius and centre . We wish to characterise the functions for which we have a boundwhere
depends only on and . This was first done by Benjamin Muckenhoupt.Definition
For a fixed, we say that a weight belongs to if is locally integrable and there is a constant such that, for all balls in , we have
where
and is the Lebesgue measureLebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...
of
. We say belongs to if there exists some such that
for all
and all balls .Equivalent characterizations
This following result is a fundamental result in the study of Muckenhoupt weights. A weight is in if and only if any one of the following hold.(a) The Hardy–Littlewood maximal function is bounded on
, that isfor some
which only depends on and the constant in the above definition.(b) There is a constant
such that for any locally integrable function on 
for all balls
. Here
is the average of
over and
Equivalently,
, where , if and only if
and
This equivalence can be verified by using Jensen's Inequality
Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context,...
.
Reverse Hölder inequalities and 
The main tool in the proof of the above equivalence is the following result. The following statements are equivalent(a)
belongs to for some (b) There exists an
and a (both depending on such that
for all balls
(c) There exists
so that for all balls and subsets 
We call the inequality in (b) a reverse Hölder inequality as the reverse inequality follows for any non-negative function directly from Hölder's inequality
Hölder's inequality
In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....
. If any of the three equivalent conditions above hold we say
belongs to .Weights and BMO
The definition of an weight and the reverse Hölder inequality indicate that such a weight cannot degenerate or grow too quickly. This property can be phrased equivalently in terms of how much the logarithm of the weight oscillates:(a) If
, then (i.e. has bounded mean oscillationBounded mean oscillation
In harmonic analysis, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded...
).
(b) If
, then for sufficiently small , we have for some .This equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the John–Nirenberg inequality. Note that the
condition in part (b) is necessary for the result to be true, as is a BMO function, but is not in any .Further properties
Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:(i)
(ii)
(iii) If
, then defines a doubling measure: for any ball B, if 2B is the ball of twice the radius, then where C > 1 is a constant depending on .(iv) If
, then there is such that .(v) If
then there is and weights such that .Boundedness of singular integrals
It is not only the Hardy–Littlewood maximal operator that is bounded on these weighted spaces. In fact, any Calderón-Zygmund singular integral operator is also bounded on these spaces. Let us describe a simpler version of this here. Suppose we have an operator which is bounded on , so we have
for all smooth and compactly supported
. Suppose also that we can realise as convolution against a kernel in the sense that, whenever and are smooth and have disjoint support
Finally we assume a size and smoothness condition on the kernel
:
for all
and multi-indices . Then, for each and , we have that is a bounded operator on . That is, we have the estimate
for all
for which the right-hand side is finite.A converse result
If, in addition to the three conditions above, we assume the non-degeneracy condition on the kernel: For a fixed unit vector 
whenever
with , then we have a converse. If we know
for some fixed
and some , then .Weights and quasiconformal mappings
For, a K-quasiconformal mappingQuasiconformal mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity....
is a homeomorphism
with and
where
is the derivativeDerivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of
at and is the JacobianJacobian
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space...
.
A theorem of Gehring states that for all K-quasiconformal functions
, we have where depends on .Harmonic measure
If you have a simply connected domain, we say its boundary curve is K-chord-arc if for any two points there is a curve connecting and whose length is no more than . For a domain with such a boundary and for any , the harmonic measureHarmonic measure
In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, harmonic measure of a bounded domain in Euclidean space R^n, n\geq 2 is the probability...
is absolutely continuous with respect to one-dimensional Hausdorff measureHausdorff measure
In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rn or, more generally, in any metric space. The zero dimensional Hausdorff measure is the number of points in the set or ∞ if the set is infinite...
and its Radon–Nikodym derivative is in
. (Note that in this case, one needs to adapt the definition of weights to the case where the underlying measure is one-dimensional Hausdorff measure).