Muckenhoupt weights


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 functions defined as
where is the ball in with radius and centre. Let, we wish to characterise the functions for which we have a bound
where 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 is the Lebesgue measure of, and is a real number such that:.
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.
Equivalently:
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
  1. for some.
  2. There exist such that for all balls and subsets, implies.
  3. There exist and such that for all balls we have:
We call the inequality in the third formulation 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.

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:
This equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the John–Nirenberg inequality.
Note that the smallness assumption on in part is necessary for the result to be true, as, 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:

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
Suppose also that we can realise as convolution against a kernel in the following sense: if are smooth with disjoint support, then:
Finally we assume a size and smoothness condition on the kernel :
Then, for each and, 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 -quasiconformal mapping is a homeomorphism such that
where is the derivative of at and is the Jacobian.
A theorem of Gehring states that for all -quasiconformal functions, we have, where depends on.

Harmonic measure

If you have a simply connected domain, we say its boundary curve is -chord-arc if for any two points in there is a curve connecting and whose length is no more than. For a domain with such a boundary and for any in, the harmonic measure is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in..