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(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 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 x ∈ B 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 = eφ ∈ Ap 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
- ω ∈ Ap for some 1 ≤ p < ∞.
- 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).
- 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 w ∈ Ap, (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 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|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 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.
- If w ∈ Ap, then there is δ > 1 such that wδ ∈ Ap.
- If w ∈ A∞, 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 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.[2]
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[5] 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 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∞.[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.
- ↑ Muckenhoupt, Benjamin (1972). "Weighted norm inequalities for the Hardy maximal function". Transactions of the American Mathematical Society, vol. 165: 207–226.
- 1 2 3 4 5 Stein, Elias (1993). "5". Harmonic Analysis. Princeton University Press.
- ↑ Jones, Peter W. (1980). "Factorization of Ap weights". Ann. Of Math. (2) 111 (3): 511–530. doi:10.2307/1971107.
- ↑ Grafakos, Loukas (2004). "9". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc.
- ↑ Gehring, F. W. (1973). "The Lp-integrability of the partial derivatives of a quasiconformal mapping". Acta Math. 130: 265–277. doi:10.1007/BF02392268.
- ↑ Garnett, John; Marshall, Donald (2008). Harmonic Measure. Cambridge University Measure.