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
M(f)(x)=\supr>0
| 1 | |
| rn |
| \int | |
| Br(x) |
|f|,
where is the ball in with radius and center at . Let, we wish to characterise the functions for which we have a bound
\int|M(f)(x)|p\omega(x)dx\leqC\int|f|p\omega(x)dx,
where depends only on and . This was first done by Benjamin Muckenhoupt.[1]
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
| \left( | 1 |
| |B| |
\intB\omega(x)dx\right)\left(
| 1 | |
| |B| |
\intB
| ||||
| \omega(x) |
dx
| ||||
| \right) |
\leqC<infty,
where is the Lebesgue measure of, and is a real number such that: .
We say belongs to if there exists some such that
| 1 | |
| |B| |
\intB\omega(y)dy\leqC\omega(x),
for almost every and all balls .[2]
This following result is a fundamental result in the study of Muckenhoupt weights.
Theorem. Let . A weight is in if and only if any one of the following hold.[2]
(a) The Hardy–Littlewood maximal function is bounded on, that is
\int|M(f)(x)|p\omega(x)dx\leqC\int|f|p\omega(x)dx,
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, and all balls :
| p | |
| (f | |
| B) |
\leq
| c | |
| \omega(B) |
\intBf(x)p\omega(x)dx,
where:
fB=
| 1 | |
| |B| |
\intBf, \omega(B)=\intB\omega(x)dx.
Equivalently:
Theorem. Let, then if and only if both of the following hold:
\supB
| 1 | |
| |B| |
\intB
| \varphi-\varphiB | |
| e |
dx<infty
\supB
| 1 | |
| |B| |
\intB
| ||||
| e |
dx<infty.
This equivalence can be verified by using Jensen's Inequality.
The main tool in the proof of the above equivalence is the following result.[2] The following statements are equivalent
| 1 | |
| |B| |
\intB\omegaq\leq\left(
| c | |
| |B| |
\intB\omega\right)q.
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 .
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 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 smallness assumption on in part (b) is necessary for the result to be true, as, but:
e-log|x|=
| 1 | |
| elog|x| |
=
| 1 | |
| |x| |
is not in any .
Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:
A1\subseteqAp\subseteqAinfty, 1\leqp\leqinfty.
Ainfty=cupp<inftyAp.
If, then defines a doubling measure: for any ball, if is the ball of twice the radius, then where is a constant depending on .
If, then there is such that .
If, then there is and weights
w1,w2\inA1
w=w1
| -\delta | |
| w | |
| 2 |
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.[4] Let us describe a simpler version of this here.[2] Suppose we have an operator which is bounded on, so we have
\forallf\in
| infty | |
| C | |
| c |
:
| \|T(f)\| | |
| L2 |
\leq
| C\|f\| | |
| L2 |
.
Suppose also that we can realise as convolution against a kernel in the following sense: if are smooth with disjoint support, then:
\intg(x)T(f)(x)dx=\iintg(x)K(x-y)f(y)dydx.
Finally we assume a size and smoothness condition on the kernel :
\forallx ≠ 0,\forall|\alpha|\leq1: \left|\partial\alphaK\right|\leqC|x|-n-\alpha.
Then, for each and, is a bounded operator on . That is, we have the estimate
\int|T(f)(x)|p\omega(x)dx\leqC\int|f(x)|p\omega(x)dx,
for all for which the right-hand side is finite.
If, in addition to the three conditions above, we assume the non-degeneracy condition on the kernel : For a fixed unit vector
|K(x)|\geqa|x|-n
whenever
x=t
| u |
0
\int|T(f)(x)|p\omega(x)dx\leqC\int|f(x)|p\omega(x)dx,
for some fixed and some, then .[2]
For, a -quasiconformal mapping is a homeomorphism such that
f\in
| 1,2 | |
| W | |
| loc |
(Rn), and
| \|Df(x)\|n | |
| J(f,x) |
\leqK,
where is the derivative of at and is the Jacobian.
A theorem of Gehring[5] states that for all -quasiconformal functions, we have, where depends on .
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 .[6] (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).