Muckenhoupt weights

In mathematics, the class of Muckenhoupt weights Ap consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on Lp(). 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 centre 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.[1]

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 |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

for all xB and all balls B.[2]

Equivalent characterizations

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

Theorem. A weight ω is in Ap if and only if any one of the following hold.[2]
(a) The Hardy–Littlewood maximal function is bounded on Lp(ω(x)dx), that is
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:
where:

Equivalently:

Theorem. Let 1 < p < ∞, then w = eφAp 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.[2] The following statements are equivalent

  1. ωAp for some 1 ≤ p < ∞.
  2. There exist 0 < δ, γ < 1 such that for all balls B and subsets EB, |E| ≤ γ|B| implies ω(E) ≤ δω(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 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:

(a) If wAp, (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 eδfAp 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:

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:

If wAp, then wdx 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 wAp, then there is δ > 1 such that wδAp.
If wA, then there is δ > 0 and weights such that .[3]

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.[4] Let us describe a simpler version of this here.[2] 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 ωAp, 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 ωAp.[2]

Weights and quasiconformal mappings

For K > 1, a K-quasiconformal mapping is a homeomorphism f : RnRn 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[5] states that for all K-quasiconformal functions f : RnRn, 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 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|zw|. 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.[6] (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).

References

  • Garnett, John (2007). Bounded Analytic Functions. Springer.
  1. Muckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function". Transactions of the American Mathematical Society, vol. 165: 207–226.
  2. Stein, Elias (1993). "5". Harmonic Analysis. Princeton University Press.
  3. Jones, Peter W. (1980). "Factorization of Ap weights". Ann. of Math. 2. 111 (3): 511–530. doi:10.2307/1971107.
  4. Grafakos, Loukas (2004). "9". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc.
  5. Gehring, F. W. (1973). "The Lp-integrability of the partial derivatives of a quasiconformal mapping". Acta Math. 130: 265–277. doi:10.1007/BF02392268.
  6. Garnett, John; Marshall, Donald (2008). Harmonic Measure. Cambridge University Press.
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