Muckenhoupt weights
Inmathematics,the class ofMuckenhoupt weightsApconsists of those weightsωfor which theHardy–Littlewood maximal operatoris bounded onLp(dω).Specifically, we consider functions f onRnand their associatedmaximal functionsM( f )defined as
whereBr(x)is the ball inRnwith radiusrand center atx.Let1 ≤p< ∞,we wish to characterise the functionsω:Rn→ [0, ∞)for which we have a bound
whereCdepends only onpandω.This was first done byBenjamin Muckenhoupt.[1]
Definition
[edit]For a fixed1 <p< ∞,we say that a weightω:Rn→ [0, ∞)belongs toApifωis locally integrable and there is a constantCsuch that, for all ballsBinRn,we have
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
for almost everyx∈Band all ballsB.[2]
Equivalent characterizations
[edit]This following result is a fundamental result in the study of Muckenhoupt weights.
- Theorem.Let1 <p< ∞.A weightωis inApif and only if any one of the following hold.[2]
- (a) TheHardy–Littlewood maximal functionis bounded onLp(ω(x)dx),that is
- 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:
- where:
Equivalently:
- Theorem.Let1 <p< ∞,thenw=eφ∈Apif and only if both of the following hold:
This equivalence can be verified by usingJensen's Inequality.
Reverse Hölder inequalities andA∞
[edit]The main tool in the proof of the above equivalence is the following result.[2]The following statements are equivalent
- ω∈Apfor some1 ≤p< ∞.
- There exist0 <δ,γ< 1such that for all ballsBand subsetsE⊂B,|E| ≤γ |B|impliesω(E) ≤δ ω(B).
- There exist1 <qandc(both depending onω) such that for all ballsBwe have:
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
[edit]The definition of anApweight 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∈Ap,(p≥ 1),thenlog(w) ∈ BMO(i.e.log(w)hasbounded mean oscillation).
- (b) If f ∈ BMO,then for sufficiently smallδ> 0,we haveeδf∈Apfor 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:
is not in anyAp.
Further properties
[edit]Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:
- Ifw∈Ap,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∈Ap,then there isδ> 1such thatwδ∈Ap.
- Ifw∈A∞,then there isδ> 0and weightssuch that.[3]
Boundedness of singular integrals
[edit]It is not only the Hardy–Littlewood maximal operator that is bounded on these weightedLpspaces. In fact, anyCalderón-Zygmund singular integral operatoris also bounded on these spaces.[4]Let us describe a simpler version of this here.[2]Suppose we have an operatorTwhich is bounded onL2(dx),so we have
Suppose also that we can realiseTas convolution against a kernelKin the following sense: if f ,gare smooth with disjoint support, then:
Finally we assume a size and smoothness condition on the kernelK:
Then, for each1 <p< ∞andω∈Ap,Tis a bounded operator onLp(ω(x)dx).That is, we have the estimate
for all f for which the right-hand side is finite.
A converse result
[edit]If, in addition to the three conditions above, we assume the non-degeneracy condition on the kernelK:For a fixed unit vectoru0
wheneverwith−∞ <t< ∞,then we have a converse. If we know
for some fixed1 <p< ∞and someω,thenω∈Ap.[2]
Weights and quasiconformal mappings
[edit]ForK> 1,aK-quasiconformal mappingis a homeomorphism f :Rn→Rnsuch that
whereDf (x)is thederivativeof f atxandJ( f ,x) = det(Df (x))is theJacobian.
A theorem of Gehring[5]states that for allK-quasiconformal functions f :Rn→Rn,we haveJ( f ,x) ∈Ap,wherepdepends onK.
Harmonic measure
[edit]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∞.[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
[edit]- Garnett, John(2007).Bounded Analytic Functions.Springer.
- ^Muckenhoupt, Benjamin (1972)."Weighted norm inequalities for the Hardy maximal function".Transactions of the American Mathematical Society.165:207–226.doi:10.1090/S0002-9947-1972-0293384-6.
- ^abcdeStein, Elias (1993). "5".Harmonic Analysis.Princeton University Press.
- ^Jones, Peter W. (1980). "Factorization ofApweights ".Ann. of Math.2.111(3): 511–530.doi:10.2307/1971107.JSTOR1971107.
- ^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 Press.