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2新东方雅思培训
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A SIMPLE PROOF OF THE A 2CONJECTURE ANDREI K.LERNER Abstract.We give a simple proof of the A 2conjecture proved recently by T.Hyt¨o nen.Our proof avoids completely the notion of the Haar shift operator,and it is bad only on the “local mean oscillation decomposition”.Also our proof yields a simple proof of the “two-weight conjecture”as well.1.Introduction Let T be an L 2bounded Calder´o n-Zygmund operator.We say that w ∈A 2if w A 2=sup Q ⊂R n w (Q )w −1(Q )/|Q |2<∞.In this note we give a rather simple proof of the A 2conjecture recently ttled by T.Hyt¨o nen [7].Theorem 1.1.For any w ∈A 2,(1.1) T L 2(w )≤c (n,T ) w A 2.Below is a partial list of important contributions to this result.First,(1.1)was proved for the following operators:•Hardy-Littlewood maxim
al operator (S.Buckley [3],1993);•Beurling transform (S.Petermichl and A.Volberg [24],2002);•Hilbert transform (S.Petermichl [22],2007);•Riesz transform (S.Petermichl [23],2008);
•dyadic paraproduct (O.Beznosova [2],2008);
•Haar shift (M.Lacey,S.Petermichl and M.Reguera [16],2010).After that,the following works appeared with very small intervals:•a simplified proof for Haar shifts (D.Cruz-Uribe,J.Martell and
C.P´e rez [5,6],2010);
•the L 2(w )bound for general T by w A 2log(1+ w A 2)(C.P´e rez,S.Treil and A.Volberg [21],2010);
•(1.1)in full generality (T.Hyt¨o nen [7],2010);
2ANDREI K.LERNER
•a simplification of the proof(T.Hyt¨o nen et al.[12],2010);•(1.1)for the maximal Calder´o n-Zygmund operator T♮(T.Hyt¨o nen et al.[9],2010).
The“Bellman function”proof of the A2conjecture in a geometri-cally doubling metric space was given by F.Nazarov,A.Reznikov and A.Volberg[19](e also[20]).
All currently known proofs of(1.1)were bad on the reprentation
.Such reprentations of T in terms of the Haar shift operators S m,k
D
定义英文also have a long history;for general T it was found in[7].The cond
in place of T key element of all known proofs was showing(1.1)for S m,k
D
with the corresponding constant depending linearly(or polynomially) on the complexity.Obrve that over the past year veral different proofs of this step appeared(,[15,25]).
In a very recent work[18],we have proved that for any Banach function space X(R n),
A D,S|f| X,
(1.2) T♮f X≤c(T,n)sup
D,S
where
A D,S f(x)= j,k f Q k jχQ k j(x)
(this operator is defined by means of a spar family S={Q k j}from a general dyadic grid D;for the notions e Section2below). Obrve that for the operator A D,S f inequality(1.1)follows just in few lines by a very simple argument.This wasfirst obrved in[5,6] (e also[18]).Hence,in the ca when X=L2(w),inequality(1.2) easily implies the A2conjecture.Also,(1.2)yields the“two-weight conjecture”by D.Cruz-Uribe and C.P´e rez;we refer to[18]for the details.
The proof of(1.2)in[18]still depended on the reprentation of T in terms of the Haar shift operators.In this note we will show that this difficult step can be completely avoided.Our new proof of(1.2)is bad only on the“local mean oscillation decomposition”proved by the author in[17].It is interesting that we apply this decomposition twice. First it is applied directly to T♮,and we obtain that T♮is esntially pointwi dominated by the maximal operator M and a ries of dyadic type operators T m.In order to handle T m,we apply the decomposition again to the adjoint operators T⋆m.After this step we obtain a pointwi domination by the simplest dyadic operators A D,S.
Note that all our estimates are actually pointwi,and they do not depend on a particular function spa
ce.This explains why we prefer to write(1.2)with a general Banach function space X.
A 2CONJECTURE 3
2.Preliminaries
2.1.Calder´o n-Zygmund operators.By a Calder´o n-Zygmund op-erator in R n we mean an L 2bounded integral operator reprented as T f (x )= R n
K (x,y )f (y )dy,x ∈supp f,
儿童节快乐的英语with kernel K satisfying the following growth and smoothness condi-tions:
(i)|K (x,y )|≤cxiaopi
|x −y |n +δ,
whenever |x −x ′|<|x −y |/2.
Given a Calder´o n-Zygmund operator T ,define its maximal truncated version by T ♮f (x )=sup 0<ε<ν ε<|y |<ν
K (x,y )f (y )dy .2.2.Dyadic grids.Recall that the standard dyadic grid in R n consists of the cubes
2−k ([0,1)n +j ),k ∈Z ,j ∈Z n .
Denote the standard grid by D .
By a general dyadic grid D we mean a collection of cubes with the following properties:(i)for any Q ∈D its sidelength ℓQ is of the form 2k ,k ∈Z ;(ii)Q ∩R ∈{Q,R,∅}for any Q,R ∈D ;(iii)the cubes of a fixed sidelength 2k form a partition of R n .
Given a cube Q 0,denote by D (Q 0)the t of all dyadic cubes with respect to Q 0,that is,the cubes from D (Q 0)are formed by repeated subdivision of Q 0and each of its descendants into 2n congruent sub-cubes.Obrve that if Q 0∈D ,then each cube from D (Q 0)will also belong to D .
A well known principle says that there are ξn general dyadic grids D αsuch that every cube Q ⊂R n is contained in some cube Q ′∈D αsuch that |Q ′|≤c n |Q |.For ξn =3n this is attributed in the literature to M.Christ and,independently,to J.Garnett and P.Jones.For ξn =2n it can be found in a recent work by T.Hyt¨o nen and C.P´e rez [11].Very recently it was shown by J.Conde et al.[4]that one can take ξn =n +1,and this number is optimal.For our purpos any of such variants is suitable.We will u the one from [11].
4ANDREI K.LERNER
Proposition 2.1.There are 2n dyadic grids D αsuch that for any cube Q ⊂R n there exists a cube Q α∈D αsuch that Q ⊂Q αand ℓQ α≤6ℓQ .The grids D αhere are the following:
D α={2−k ([0,1)n +j +α)},α∈{0,1/3}n .
We outline briefly the proof.First,it is easy to e that it suffices to consider the one-dimensional ca.Take an arbitrary interval I ⊂R .Fix k 0∈Z such that 2−k 0−1≤3ℓI <2−k 0.If I does not contain any point 2−k 0j,j ∈Z ,then I is contained in some I ′=[2−k 0j,2−k 0(j +1))(since such intervals form a partition of R ),and ℓI ′≤6ℓI .On the other hand,if I contains some point j 02−k 0,then I does not contain any point 2−k 0(j +1/3),j ∈Z (since ℓI <2−k 0/3),and therefore I is contained in some I ′′=[2−k 0(j +1/3),2−k 0(j +4/3)),and ℓI ′′≤6ℓI .
2.3.Local mean oscillations.Given a measurable function f on R n and a cube Q ,the local mean oscillation of f on Q is defined by
ωλ(f ;Q )=inf c ∈R
(f −c )χQ ∗ λ|Q | (0<λ<1),where f ∗denotes the non-increasing rearrangement of f .
By a median value of f over Q we mean a possibly nonunique,real number m f (Q )such that
中餐英语菜单max |{x ∈Q :f (x )>m f (Q )}|,|{x ∈Q :f (x )<m f (Q )}| ≤|Q |/2.It is easy to e that the t of all median values of f is either one point or the clod interval.In the latter ca we will assume for the definiteness that m f (Q )is the maximal median value.Obrve that it follows from the definitions that
(2.1)|m f (Q )|≤(fχQ )∗(|Q |/2).
Given a cube Q 0,the dyadic local sharp maximal function M #,d五十步笑百步译文
λ;Q 0f is defined by
M #,d
λ;Q 0f (x )=sup x ∈Q ′∈D (Q 0)
ωλ(f ;Q ′).
We say that {Q k j }is a spar family of cubes if:(i)the cubes Q k j are disjoint in j ,with k fixed;(ii)if Ωk =∪j Q k j ,then Ωk +1⊂Ωk ;(iii)|Ωk +1∩Q k j |≤1
A2CONJECTURE5 Theorem2.2.Let f be a measurable function on R n and let Q0be a fixed cube.Then there exists a(possibly empty)spar family of cubes Q k j∈D(Q0)such that ∈Q0,
|f(x)−m f(Q0)|≤4M#,d1
2n+2(f;Q k j)χQ k
j
(x).
The following proposition is well known,and it can be found in a slightly different form in[13].We give its proof here for the sake of the completeness.The proof is a classical argument ud,for example,to show that T is bounded from L∞to BMO.Also the same argument is ud to prove a good-λinequality relating T and M.
Proposition2.3.For any cube Q⊂R n,
(2.2)ωλ(T f;Q)≤c(T,λ,n)
∞ m=01|2m Q| 2m Q|f(y)|dy and
(2.3)ωλ(T♮f;Q)≤c(T,λ,n)
四级辅导∞ m=01|2m Q| 2m Q|f(y)|dy . Proof.Let f1=fχ2√
nQ
英文翻译网|f(y)||K(x,y)−K(x0,y)|dy ≤cℓδQ R n\2Q|f(y)|(2mℓQ)n+δ 2m+1Q\2m Q|f(y)|dy ≤c
∞ m=01|2m Q| 2m Q|f(y)|dy .
From this and from the weak type(1,1)of T,
T f−T(f2)(x0) χQ ∗ λ|Q|
≤(T(f1))∗(λ|Q|)+ T(f2)−T(f2)(x0) L∞(Q)
≤c
1
n|Q||f(y)|dy+c
∞ m=01|2m Q| 2m Q|f(y)|dy
≤c′fluctuation
∞ m=01|2m Q| 2m Q|f(y)|dy , which proves(2.2).dmcl
