Muckenhoupt weights

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

( 1 | B | ∫ B ω ( x ) d x ) ( 1 | B | ∫ B ω ( x ) − q p d x ) p q ≤ C < ∞ ,

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

1 | B | ∫ B ω ( x ) d x ≤ C ω ( 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.

Equivalently:

Theorem. Let 1 < p < ∞, then w = e^φ ∈ A_p if and only if both of the following hold:

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:

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 | = 1 e log ⁡ | x | = 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 ⊆ A p ⊆ A ∞ , 1 ≤ p ≤ ∞ . A ∞ = ⋃ p < ∞ 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 ∈ A 1 such that w = w 1 w 2 − δ .

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

∀ f ∈ C c ∞ : ∥ T ( f ) ∥ L 2 ≤ 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:

∫ g ( x ) T ( f ) ( x ) d x = ∬ g ( x ) K ( x − y ) f ( y ) d y d x .

Finally we assume a size and smoothness condition on the kernel K:

∀ x ≠ 0 , ∀ | α | ≤ 1 : | ∂ α K | ≤ C | x | − n − α .

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

∫ | T ( f ) ( x ) | p ω ( x ) d x ≤ C ∫ | f ( x ) | p ω ( x ) d x ,

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 ) | ≥ a | x | − n

whenever x = t u ˙ 0 with −∞ < t < ∞, then we have a converse. If we know

∫ | T ( f ) ( x ) | p ω ( x ) d x ≤ C ∫ | f ( x ) | p ω ( x ) d x ,

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 ∈ W l o c 1 , 2 ( R n ) ,  and  ∥ D f ( x ) ∥ n J ( f , x ) ≤ 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).