Muckenhoupt weights

In mathematics, the class of Muckenhoupt weights A_p consists of those weights Template:Mvar for which the Hardy–Littlewood maximal operator is bounded on L^p(dω). Specifically, we consider functions  f  on Rⁿ and their associated maximal functions M( f ) defined as

$M(f)(x)=\underset{r>0}{\sup }\frac{1}{r^n}\underset{B_r(x)}{\int }|f|,$

where B_r(x) is the ball in Rⁿ with radius Template:Mvar and center at Template:Mvar. Let 1 ≤ p < ∞, we wish to characterise the functions ω : Rⁿ → [0, ∞) for which we have a bound

$\int |M(f)(x){|}^p\omega (x)dx\le C\int |f{|}^p\omega (x)dx,$

where Template:Mvar depends only on Template:Mvar and Template:Mvar. This was first done by Benjamin Muckenhoupt.

Definition

For a fixed 1 < p < ∞, we say that a weight ω : Rⁿ → [0, ∞) belongs to A_p if Template:Mvar is locally integrable and there is a constant Template:Mvar such that, for all balls Template:Mvar in Rⁿ, we have

$(\frac{1}{|B|}\underset{B}{\int }\omega (x)dx){(\frac{1}{|B|}\underset{B}{\int }\omega (x{)}^{-\frac{q}{p}}dx)}^{\frac{p}{q}}\le C<\mathrm{\infty },$

where |B| is the Lebesgue measure of Template:Mvar, and Template:Mvar is a real number such that: Template:Sfrac + Template:Sfrac = 1.

We say ω : Rⁿ → [0, ∞) belongs to A₁ if there exists some Template:Mvar such that

$\frac{1}{|B|}\underset{B}{\int }\omega (y)dy\le C\omega (x),$

for almost every x ∈ B and all balls Template:Mvar.

Equivalent characterizations

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

Theorem. Let 1 < p < ∞. A weight Template:Mvar is in A_p if and only if any one of the following hold.

(a) The Hardy–Littlewood maximal function is bounded on L^p(ω(x)dx), that is

$\int |M(f)(x){|}^p\omega (x)dx\le C\int |f{|}^p\omega (x)dx,$

for some Template:Mvar which only depends on Template:Mvar and the constant Template:Mvar in the above definition.

(b) There is a constant Template:Mvar such that for any locally integrable function  f  on Rⁿ, and all balls Template:Mvar:

$(f_B{)}^p\le \frac{c}{\omega (B)}\underset{B}{\int }f(x{)}^p\omega (x)dx,$

where:

$f_B=\frac{1}{|B|}\underset{B}{\int }f,\omega (B)=\underset{B}{\int }\omega (x)dx.$

Equivalently:

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

$\underset{B}{\sup }\frac{1}{|B|}\underset{B}{\int }{e}^{\phi -{\phi }_B}dx<\mathrm{\infty }$

$\underset{B}{\sup }\frac{1}{|B|}\underset{B}{\int }{e}^{-\frac{\phi -{\phi }_B}{p-1}}dx<\mathrm{\infty }.$

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 Template:Mvar and subsets E ⊂ B, |E| ≤ γ |B| implies ω(E) ≤ δ ω(B).
3. There exist 1 < q and Template:Mvar (both depending on Template:Mvar) such that for all balls Template:Mvar we have:

$\frac{1}{|B|}\underset{B}{\int }{\omega }^q\le {\left(\frac{c}{|B|}\underset{B}{\int }\omega \right)}^q.$

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 Template:Mvar 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:

$e^{-\log |x|}=\frac{1}{e^{\log |x|}}=\frac{1}{|x|}$

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:

$A_1\subseteq A_p\subseteq A_{\mathrm{\infty }},1\le p\le \mathrm{\infty }.$

$A_{\mathrm{\infty }}=\bigcup _{p<\mathrm{\infty }}{A}_p.$

If w ∈ A_p, then w dx defines a doubling measure: for any ball Template:Mvar, if 2B is the ball of twice the radius, then w(2B) ≤ Cw(B) where C > 1 is a constant depending on Template:Mvar.

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 $w_1,w_2\in A_1$ such that $w=w_1{w}_2^{-\delta }$.

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 Template:Mvar which is bounded on L²(dx), so we have

$\mathrm{\forall }f\in C_c^{\mathrm{\infty }}:\Vert T(f){\Vert }_{L^2}\le C\Vert f{\Vert }_{L^2}.$

Suppose also that we can realise Template:Mvar as convolution against a kernel Template:Mvar in the following sense: if  f , g are smooth with disjoint support, then:

$\int g(x)T(f)(x)dx=\iint g(x)K(x-y)f(y)dydx.$

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

$\mathrm{\forall }x\ne 0,\mathrm{\forall }|\alpha |\le 1:\left|{\partial }^{\alpha }K\right|\le C|x{|}^{-n-\alpha }.$

Then, for each 1 < p < ∞ and ω ∈ A_p, Template:Mvar is a bounded operator on L^p(ω(x)dx). That is, we have the estimate

$\int |T(f)(x){|}^p\omega (x)dx\le C\int |f(x){|}^p\omega (x)dx,$

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 Template:Mvar: For a fixed unit vector u₀

$|K(x)|\ge a|x{|}^{-n}$

whenever $x=t{\dot{u}}_0$ with −∞ < t < ∞, then we have a converse. If we know

$\int |T(f)(x){|}^p\omega (x)dx\le C\int |f(x){|}^p\omega (x)dx,$

for some fixed 1 < p < ∞ and some Template:Mvar, then ω ∈ A_p.

Weights and quasiconformal mappings

For K > 1, a Template:Mvar-quasiconformal mapping is a homeomorphism  f  : Rⁿ →Rⁿ such that

$f\in W_{loc}^{1,2}({\mathbf{𝐑}}^n),\text{ and }\frac{\Vert Df(x){\Vert }^n}{J(f,x)}\le K,$

where Df (x) is the derivative of  f  at Template:Mvar and J( f , x) = det(Df (x)) is the Jacobian.

A theorem of Gehring states that for all Template:Mvar-quasiconformal functions  f  : Rⁿ →Rⁿ, we have J( f , x) ∈ A_p, where Template:Mvar depends on Template:Mvar.

Harmonic measure

If you have a simply connected domain Ω ⊆ C, we say its boundary curve Γ = ∂Ω is Template:Mvar-chord-arc if for any two points z, w in Γ there is a curve γ ⊆ Γ connecting Template:Mvar and Template:Mvar 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).