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 Rⁿ 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 Rⁿ with radius r and centre x. Let 1 ≤ p < ∞, we wish to characterise the functions ω : Rⁿ → [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 : Rⁿ → [0, ∞) belongs to A_p if ω is locally integrable and there is a constant C such that, for all balls B in Rⁿ, 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 : Rⁿ → [0, ∞) belongs to A₁ if there exists some C such that

$\frac{1}{|B|} \int_B \omega(x) \, dx \leq C\omega(x),$

for all x ∈ B and all balls B.

Equivalent characterizations

This following result is a fundamental result in the study of Muckenhoupt weights.

Theorem. 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 Rⁿ, 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 |B| implies ω(E) ≤ δ 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  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 L²(dx), 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  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 u₀

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

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  : Rⁿ →Rⁿ 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 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  : Rⁿ →Rⁿ, 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 z₀ in Ω, the harmonic measure w( ⋅ ) = w(z₀, 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).