In mathematics, the class of Muckenhoupt weights Ap consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on Lp(dω) . Specifically, we consider functions  f  on Rn and their associated maximal functions M( f ) defined as where Br(x) is the ball in Rn with radius r and center at x. Let 1 ≤ p < ∞ , we wish to characterise the functions ω : Rn → [0, ∞) for which we have a bound 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 Ap if ω is locally integrable and there is a constant C such that, for all balls B in Rn , we have where 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 for almost every x ∈ B and all balls B.

Equivalent characterizations

This following result is a fundamental result in the study of Muckenhoupt weights. 1 < p < ∞ . A weight ω is in Ap if and only if any one of the following hold. Lp(ω(x)dx) , that is  f  on Rn , and all balls B: Equivalently: 1 < p < ∞ , then if and only if both of the following hold: This equivalence can be verified by using Jensen's Inequality.

Reverse Hölder inequalities and

A{{sub|∞}} The main tool in the proof of the above equivalence is the following result. The following statements are equivalent ω ∈ Ap for some 1 ≤ p < ∞ . 0 < δ, γ < 1 such that for all balls B and subsets E ⊂ B , implies ω(E) ≤ δ ω(B) . 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 Ap 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: w ∈ Ap, (p ≥ 1), then log(w) ∈ BMO (i.e. log(w) has bounded mean oscillation).  f  ∈ BMO , then for sufficiently small δ > 0 , we have eδf ∈ Ap 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 , but: is not in any Ap .

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: w ∈ Ap , 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. w ∈ Ap , then there is δ > 1 such that wδ ∈ Ap . w ∈ A∞ , then there is δ > 0 and weights such that.

Boundedness of singular integrals

It is not only the Hardy–Littlewood maximal operator that is bounded on these weighted Lp 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 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: Finally we assume a size and smoothness condition on the kernel K: Then, for each 1 < p < ∞ and ω ∈ Ap , T is a bounded operator on Lp(ω(x)dx) . That is, we have the estimate 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 whenever with −∞ < t < ∞ , then we have a converse. If we know for some fixed 1 < p < ∞ and some ω, then ω ∈ Ap .

Weights and quasiconformal mappings

For K > 1 , a K-quasiconformal mapping is a homeomorphism  f  : Rn →Rn such that 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) ∈ Ap , where p depends on K.

Harmonic measure

If you have a simply connected domain Ω ⊆ C , we say its boundary curve 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 . 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).