# Muckenhoupt weights

In mathematics, the class of Muckenhoupt weights A$_p$ consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on L$^p$(dω). Specifically, we consider functions  f  on R$^n$ and their associated maximal functions M( f ) defined as
    $M(f)(x)=\sup _{r>0}{\frac {1}{r^{n}}}\int _{B_{r}(x)}|f|,$
where B$_r$(x) is the ball in R$^n$ with radius r and center at x. Let 1 ≤ p < ∞, we wish to characterise the functions ω : R$^n$ → [0, ∞) for which we have a bound
    $\int |M(f)(x)|^{p}\,\omega (x)dx\leq C\int |f|^{p}\,\omega (x)\,dx,$
where C depends only on p and ω. This was first done by Benjamin Muckenhoupt.

## Definition

For a fixed 1 < p < ∞, we say that a weight ω : R$^n$ → [0, ∞) belongs to A$_p$ if ω is locally integrable and there is a constant C such that, for all balls B in R$^n$, we have
    $\left({\frac {1}{|B|}}\int _{B}\omega (x)\,dx\right)\left({\frac {1}{|B|}}\int _{B}\omega (x)^{-{\frac {q}{p}}}\,dx\right)^{\frac {p}{q}}\leq C<\infty ,$
where |B| is the Lebesgue measure of B, and q is a real number such that: ⁠1/p⁠ + ⁠1/q⁠ = 1.
We say ω : R$^n$ → [0, ∞) belongs to A$_1$ if there exists some C such that
    ${\frac {1}{|B|}}\int _{B}\omega (y)\,dy\leq C\omega (x),$
for almost every x ∈ B and all balls B.

## Equivalent characterizations

This following result is a fundamental result in the study of Muckenhoupt weights.
    Theorem. Let 1 < p < ∞. A weight ω is in A$_p$ if and only if any one of the following hold.
       (a) The Hardy–Littlewood maximal function is bounded on L$^p$(ω(x)dx), that is
          $\int |M(f)(x)|^{p}\,\omega (x)\,dx\leq C\int |f|^{p}\,\omega (x)\,dx,$
       for some C which only depends on p and the constant A in the above definition.
       (b) There is a constant c such that for any locally integrable function  f  on R$^n$, and all balls B:
          $(f_{B})^{p}\leq {\frac {c}{\omega (B)}}\int _{B}f(x)^{p}\,\omega (x)\,dx,$
       where:
          $f_{B}={\frac {1}{|B|}}\int _{B}f,\qquad \omega (B)=\int _{B}\omega (x)\,dx.$
Equivalently:
    Theorem. Let 1 < p < ∞, then w = e$^φ$ ∈ A$_p$ if and only if both of the following hold:
       $\sup _{B}{\frac {1}{|B|}}\int _{B}e^{\varphi -\varphi _{B}}dx<\infty$
       $\sup _{B}{\frac {1}{|B|}}\int _{B}e^{-{\frac {\varphi -\varphi _{B}}{p-1}}}dx<\infty .$
This equivalence can be verified by using Jensen's Inequality.

## Reverse Hölder inequalities and A$_∞$

The main tool in the proof of the above equivalence is the following result. The following statements are equivalent
* ω ∈ A$_p$ for some 1 ≤ p < ∞.
* There exist 0 < δ, γ < 1 such that for all balls B and subsets E ⊂ B, |E| ≤ γ |B| implies ω(E) ≤ δ ω(B).
* There exist 1 < q and c (both depending on ω) such that for all balls B we have:
       ${\frac {1}{|B|}}\int _{B}\omega ^{q}\leq \left({\frac {c}{|B|}}\int _{B}\omega \right)^{q}.$
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 ω belongs to A$_∞$.

## Weights and BMO

The definition of an A$_p$ 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 w ∈ A$_p$, (p ≥ 1), then log(w) ∈ BMO (i.e. log(w) has bounded mean oscillation).
    (b) If  f  ∈ BMO, then for sufficiently small δ > 0, we have e$^{δf}$ ∈ A$_p$ for some p ≥ 1.
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 δ > 0 in part (b) is necessary for the result to be true, as −log|x| ∈ BMO, but:
    $e^{-\log |x|}={\frac {1}{e^{\log |x|}}}={\frac {1}{|x|}}$
is not in any A$_p$.

## 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:
    $A_{1}\subseteq A_{p}\subseteq A_{\infty },\qquad 1\leq p\leq \infty .$
    $A_{\infty }=\bigcup _{p<\infty }A_{p}.$
    If w ∈ A$_p$, then w dx defines a doubling measure: for any ball B, if 2B is the ball of twice the radius, then w(2B) ≤ Cw(B) where C > 1 is a constant depending on w.
    If w ∈ A$_p$, then there is δ > 1 such that w$^δ$ ∈ A$_p$.
    If w ∈ A$_∞$, then there is δ > 0 and weights $w_{1},w_{2}\in A_{1}$ such that $w=w_{1}w_{2}^{-\delta }$.

## Boundedness of singular integrals

It is not only the Hardy–Littlewood maximal operator that is bounded on these weighted L$^p$ 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 T which is bounded on L$^2$(dx), so we have
    $\forall f\in C_{c}^{\infty }:\qquad \|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}}.$
Suppose also that we can realise T as convolution against a kernel K in the following sense: if  f , g are smooth with disjoint support, then:
    $\int g(x)T(f)(x)\,dx=\iint g(x)K(x-y)f(y)\,dy\,dx.$
Finally we assume a size and smoothness condition on the kernel K:
    $\forall x\neq 0,\forall |\alpha |\leq 1:\qquad \left|\partial ^{\alpha }K\right|\leq C|x|^{-n-\alpha }.$
Then, for each 1 < p < ∞ and ω ∈ A$_p$, T is a bounded operator on L$^p$(ω(x)dx). That is, we have the estimate
    $\int |T(f)(x)|^{p}\,\omega (x)\,dx\leq C\int |f(x)|^{p}\,\omega (x)\,dx,$
for all  f  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 K: For a fixed unit vector u$_0$
    $|K(x)|\geq a|x|^{-n}$
whenever $x=t{\dot {u}}_{0}$ with −∞ < t < ∞, then we have a converse. If we know
    $\int |T(f)(x)|^{p}\,\omega (x)\,dx\leq C\int |f(x)|^{p}\,\omega (x)\,dx,$
for some fixed 1 < p < ∞ and some ω, then ω ∈ A$_p$.

## Weights and quasiconformal mappings

For K > 1, a K-quasiconformal mapping is a homeomorphism  f  : R$^n$ →R$^n$ such that
    $f\in W_{loc}^{1,2}(\mathbf {R} ^{n}),\quad {\text{ and }}\quad {\frac {\|Df(x)\|^{n}}{J(f,x)}}\leq K,$
where Df (x) is the derivative of  f  at x and J( f , x) = det(Df (x)) is the Jacobian.
A theorem of Gehring states that for all K-quasiconformal functions  f  : R$^n$ →R$^n$, we have J( f , x) ∈ A$_p$, where p depends on K.

## Harmonic measure

If you have a simply connected domain Ω ⊆ C, we say its boundary curve Γ = ∂Ω is K-chord-arc if for any two points z, w in Γ there is a curve γ ⊆ Γ connecting z and w whose length is no more than K|z − w|. For a domain with such a boundary and for any z$_0$ in Ω, the harmonic measure w( ⋅ ) = w(z$_0$, Ω, ⋅) is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in A$_∞$. (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).