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. [1]
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 almost every x ∈ B and all balls B. [2]
This following result is a fundamental result in the study of Muckenhoupt weights.
Equivalently:
This equivalence can be verified by using Jensen's Inequality.
The main tool in the proof of the above equivalence is the following result. [2] The following statements are equivalent
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∞.
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:
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.
Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:
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.
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]
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.
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).
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. [1]
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 almost every x ∈ B and all balls B. [2]
This following result is a fundamental result in the study of Muckenhoupt weights.
Equivalently:
This equivalence can be verified by using Jensen's Inequality.
The main tool in the proof of the above equivalence is the following result. [2] The following statements are equivalent
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∞.
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:
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.
Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:
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.
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]
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.
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).