Muckenhoupt weights

Inmathematics,the class ofMuckenhoupt weightsA_pconsists of those weightsωfor which theHardy–Littlewood maximal operatoris bounded onL^p(dω).Specifically, we consider functions f onRⁿand their associatedmaximal functionsM( f )defined as

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

whereB_r(x)is the ball inRⁿwith radiusrand center atx.Let1 ≤p< ∞,we wish to characterise the functionsω:Rⁿ→ [0, ∞)for which we have a bound

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

whereCdepends only onpandω.This was first done byBenjamin Muckenhoupt.

Definition

For a fixed1 <p< ∞,we say that a weightω:Rⁿ→ [0, ∞)belongs toA_pifωis locally integrable and there is a constantCsuch that, for all ballsBinRⁿ,we have

$\left({\frac {1}{|B|}}\int _{B}\omega (x)\,dx\right)\left({\frac {1}{|B|}}\int _{B}\omega (x)^{-{\frac {q}{p}}}\,dx\right)^{\frac {p}{q}}\leq C<\infty,$

where|B|is theLebesgue measureofB,andqis a real number such that:⁠1/p⁠+⁠1/q⁠= 1.

We sayω:Rⁿ→ [0, ∞)belongs toA₁if there exists someCsuch that

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

for almost everyx∈Band all ballsB.

Equivalent characterizations

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

Theorem.Let1 <p< ∞.A weightωis inA_pif and only if any one of the following hold.

(a) TheHardy–Littlewood maximal functionis bounded onL^p(ω(x)dx),that is

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

for someCwhich only depends onpand the constantAin the above definition.

(b) There is a constantcsuch that for any locally integrable function f onRⁿ,and all ballsB:

$(f_{B})^{p}\leq {\frac {c}{\omega (B)}}\int _{B}f(x)^{p}\,\omega (x)\,dx,$

where:

$f_{B}={\frac {1}{|B|}}\int _{B}f,\qquad \omega (B)=\int _{B}\omega (x)\,dx.$

Equivalently:

Theorem.Let1 <p< ∞,thenw=e^φ∈A_pif and only if both of the following hold:

$\sup _{B}{\frac {1}{|B|}}\int _{B}e^{\varphi -\varphi _{B}}dx<\infty$

$\sup _{B}{\frac {1}{|B|}}\int _{B}e^{-{\frac {\varphi -\varphi _{B}}{p-1}}}dx<\infty.$

This equivalence can be verified by usingJensen's Inequality.

Reverse Hölder inequalities andA_∞

The main tool in the proof of the above equivalence is the following result.The following statements are equivalent

1. ω∈A_pfor some1 ≤p< ∞.
2. There exist0 <δ,γ< 1such that for all ballsBand subsetsE⊂B,|E| ≤γ |B|impliesω(E) ≤δ ω(B).
3. There exist1 <qandc(both depending onω) such that for all ballsBwe have:

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

We call the inequality in the third formulation a reverse Hölder inequality as the reverse inequality follows for any non-negative function directly fromHölder's inequality.If any of the three equivalent conditions above hold we sayωbelongs toA_∞.

Weights and BMO

The definition of anA_pweight 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) Ifw∈A_p,(p≥ 1),thenlog(w) ∈ BMO(i.e.log(w)hasbounded mean oscillation).

(b) If f  ∈ BMO,then for sufficiently smallδ> 0,we havee^δf∈A_pfor somep≥ 1.

This equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and theJohn–Nirenberg inequality.

Note that the smallness assumption onδ> 0in 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 anyA_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:

$A_{1}\subseteq A_{p}\subseteq A_{\infty },\qquad 1\leq p\leq \infty.$

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

Ifw∈A_p,thenw dxdefines adoubling measure:for any ballB,if2Bis the ball of twice the radius, thenw(2B) ≤Cw(B)whereC> 1is a constant depending onw.

Ifw∈A_p,then there isδ> 1such thatw^δ∈A_p.

Ifw∈A_∞,then there isδ> 0and 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 weightedL^pspaces. In fact, anyCalderón-Zygmund singular integral operatoris also bounded on these spaces.Let us describe a simpler version of this here.Suppose we have an operatorTwhich is bounded onL²(dx),so we have

$\forall f\in C_{c}^{\infty }:\qquad \|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}}.$

Suppose also that we can realiseTas convolution against a kernelKin the following sense: if f ,gare smooth with disjoint support, then:

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

Finally we assume a size and smoothness condition on the kernelK:

$\forall x\neq 0,\forall |\alpha |\leq 1:\qquad \left|\partial ^{\alpha }K\right|\leq C|x|^{-n-\alpha }.$

Then, for each1 <p< ∞andω∈A_p,Tis a bounded operator onL^p(ω(x)dx).That is, we have the estimate

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

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 kernelK:For a fixed unit vectoru₀

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

whenever$x=t{\dot {u}}_{0}$with−∞ <t< ∞,then we have a converse. If we know

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

for some fixed1 <p< ∞and someω,thenω∈A_p.

Weights and quasiconformal mappings

ForK> 1,aK-quasiconformal mappingis a homeomorphism f :Rⁿ→Rⁿsuch that

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

whereDf (x)is thederivativeof f atxandJ( f ,x) = det(Df (x))is theJacobian.

A theorem of Gehringstates that for allK-quasiconformal functions f :Rⁿ→Rⁿ,we haveJ( f ,x) ∈A_p,wherepdepends onK.

Harmonic measure

If you have a simply connected domainΩ ⊆C,we say its boundary curveΓ = ∂ΩisK-chord-arc if for any two pointsz,winΓthere is a curveγ⊆ Γconnectingzandwwhose length is no more thanK|z−w|.For a domain with such a boundary and for anyz₀inΩ,theharmonic measurew( ⋅ ) =w(z₀,Ω, ⋅)is absolutely continuous with respect to one-dimensionalHausdorff measureand itsRadon–Nikodym derivativeis inA_∞.(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.