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 $&thinsp;f&thinsp;$ on $R^{n}$ and their associated maximal functions $M(&thinsp;f&thinsp;)$ 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 $&thinsp;f&thinsp;$ 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


 * 1) $ω ∈ A_{p}$ for some $1 ≤ p < ∞$.
 * 2) There exist $0 < δ, γ < 1$ such that for all balls $B$ and subsets $E ⊂ B$, $|E| ≤ γ&thinsp;|B|$ implies $ω(E) ≤ δ&thinsp;ω(B)$.
 * 3) 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 $&thinsp;f&thinsp; ∈ BMO$, 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 $w&thinsp;dx$ 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 $A_{p}$ 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 $L^{p}$, so we have


 * $$\forall f \in C^{\infty}_c : \qquad \|T(f)\|_{L^2} \leq C\|f\|_{L^2}.$$

Suppose also that we can realise $T$ as convolution against a kernel $K$ in the following sense: if $L^{2}(dx)$ 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 $&thinsp;f&thinsp;, g$ and $1 < p < ∞$, $T$ is a bounded operator on $ω ∈ A_{p}$. 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 $L^{p}(ω(x)dx)$ 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 $&thinsp;f&thinsp;$


 * $$|K(x)| \geq a |x|^{-n}$$

whenever $$x = t \dot u_0$$ with $u_{0}$, 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 $−∞ < t < ∞$ and some $ω$, then $1 < p < ∞$.

Weights and quasiconformal mappings
For $ω ∈ A_{p}$, a $K$-quasiconformal mapping is a homeomorphism $K > 1$ such that


 * $$f\in W^{1,2}_{loc}(\mathbf{R}^n), \quad \text{ and } \quad \frac{\|Df(x)\|^n}{J(f,x)}\leq K, $$

where $&thinsp;f&thinsp; : R^{n} →R^{n}$ is the derivative of $Df&thinsp;(x)$ at $x$ and $&thinsp;f&thinsp;$ is the Jacobian.

A theorem of Gehring states that for all $K$-quasiconformal functions $J(&thinsp;f&thinsp;, x) = det(Df&thinsp;(x))$, we have $&thinsp;f&thinsp; : R^{n} →R^{n}$, where $p$ depends on $K$.

Harmonic measure
If you have a simply connected domain $J(&thinsp;f&thinsp;, x) ∈ A_{p}$, we say its boundary curve $Ω ⊆ C$ is $K$-chord-arc if for any two points $Γ = ∂Ω$ in $z, w$ there is a curve $Γ$ connecting $z$ and $w$ whose length is no more than $γ ⊆ Γ$. For a domain with such a boundary and for any $K|z − w|$ in $z_{0}$, the harmonic measure $Ω$ is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym derivative is in $w( ⋅ ) = w(z_{0}, Ω, ⋅)$. (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).