THE MUCKENHOUPT A CLASS AS A METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES. Nikolaos Pattakos and Alexander Volberg

Transcription

1 Math. Res. Lett. 9 (202), no. 02, c International Press 202 THE MUCKENHOUPT A CLASS AS A METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES Nikolaos Pattakos and Alexander Volberg Abstract. We sho ho the A class of eights can be considered as a metric space. As far as e kno this is the first time that a metric d is considered on this set. We use this metric to generalize the results obtained in [8]. Namely, e sho that for any Calderón Zygmund operator T and an A p,<p<, eight 0,thenumbers T L p () L p () converge to T L p ( 0 ) L p ( 0 ) as d (, 0 ) 0. We also find the rate of this convergence and prove that it is sharp.. Introduction and useful results The main purpose of this paper is to define a natural metric structure on the classical Muckenhoupt A p classes, and generalize a continuity result obtained in [8]. As far as e kno, this is the first time that such metric has been studied in the context of continuity of norms of Calderón Zygmund operators. Classically, the A p spaces have only been treated as sets ith no additional structure on them. Weighted inequalities have been studied extensively during the last 5 years and it is interesting that all L p () norms of Calderón Zygmund operators turn out to be continuous ith respect to the eight, as e shall see in this paper. Moreover, e find the rate of this continuity ith respect to the eight and prove that it is sharp (see Theorem.3). In addition, e can realize the completion of the A p metric spaces as subspaces of the BMO space. Many properties of these ne complete metric spaces are going to be considered in Section 2. In [8], the to authors shoed that eighted estimates for classical operators are continuous at the constant eight. Thus, Theorem.3 is a generalization of these results, since e prove the hole continuity of the operator norm ith respect to the eight. At the time that [8] appeared, the authors did not guess that such a metric on the Muckenhoupt classes behaves in a regular manner, i.e., forcing all eighted estimates to be continuous. Such continuity results have been coming up recently in connection ith partial differential equations (PDE) ith random coefficients and continuity of norms of Calderón Zygmund operators. For example, the continuity at =, as used in [2]. The continuity at any eight can also be important in various questions of PDE. This and the fact that the proofs here involve some subtle changes in the approach of [8], made us believe that the main results of the present paper can be of independent interest. The metric A p classes ill be considered in Section 2, here e study many properties of these ne spaces, and the main Theorem.3, in Section 3. Before e state and prove the main Theorems in Sections 2 and 3, e need some definitions and Received by the editors October 9, Mathematics Subject Classification. 30E20, 47B37, 47B40, 30D55. Key ords and phrases. Calderón Zygmund operators, A 2 eights, interpolation. 499

2 500 NIKOLAOS PATTAKOS AND ALEXANDER VOLBERG some already knon results about the eighted theory and its relation ith the BMO space. We are going to ork ith functions L loc (Rn ) that are positive almost everyhere ith respect to Lebesgue measure. Functions like these are knon as eights. The celebrated A p classes of eights are defined in the folloing ay: For <p<, esaythat A p if for all cubes in R n e have that ([] Ap is called the A p characteristic of the eight): [] Ap := sup ( )( p ) p <, here p is the conjugate exponent to p, i.e., p + =. p The class of A eights consists of those such that there is a positive constant c ith the property: M(x) c(x), for almost every x in R n,herem is the Hardy Littleood maximal function. The smallest such constant is denoted by [] A andiscalledthea characteristic. We define the class of A eights as: ( [] A := sup exp( <. log ) It is really easy to see that any A p eight is actually an A eight, and that e have the estimate [] A [] Ap.ItisalsotruethatanyA eight is an A p eight for some <p<. This means that e have the equality: A = A p. <p< Another nice property is that for p q e have A A p A q A, here the inclusions here are strict. All of these sets are different for different values of p and q. The space of BMO functions in R n, consists of locally integrable functions f such that the norm f =sup f f dx is finite. The BMO space and the A space, have many nice properties. First of all, c if f is a BMO function then for any number λ (0, f ], the function e λf is an A p eight, <p<, here the constant c depends on p and the dimension n. Secondly, for small BMO norm, the A p norm of the eight e λf is bounded by the number 2 for example (see e.g., [3]). A subset of BMO that appears in many applications is BLO. It stands for the functions of bounded loer oscillation. A function f L loc (Rn ) is said to belong in BLO if there is a positive constant c such that: f inf f(x) c, x for all cubes, here the infimum is understand as the essential infimum. It can be proved that for any A, the function log is in BLO. Also if a function f BLO )

3 METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES 50 then for sufficiently small λ>0 the function e λf A. The reference for all these resultsis[3]. For the proofs of our theorems interpolation ith change of measure is going to play an important role and for this reason e need some preliminary results on this subject as ell. In the folloing (X, M,μ)and(Y,N,ν) ill denote measure spaces. Suppose T is an operator of a class of functions on X into a class of functions on Y. T is called a sub-linear operator, if it satisfies the folloing properties: (i) If f = f + f 2 and Tf,Tf 2 are defined then Tf is defined, (ii) T (f + f 2 ) Tf + Tf 2, μ almost everyhere, (iii) For any scalar k, ehave T (kf) = k Tf, μ almost everyhere. Let p, q be to real numbers. We say that T is of type (p, q), if T is defined for all functions f in L p (X, M,μ) and there exists a positive number, K, independent of f, such that here and Tf q,ν K f p,μ, ( Tf q,ν = ( f p,μ = Y X ) Tf q q dν ) f p p dμ. Let μ 0,μ be to measures for (X, M). If e define the measure μ = μ 0 + μ,then μ 0,μ are each absolutely continuous ith respect to μ. Thus, by the Radon Nikodym theorem, there exists to functions, α 0,α such that for any E M, μ j (E) = α j dμ, here j =0,. In the folloing e ill assume that α 0,α are never zero. This is equivalent to asserting that the sets of measure zero ith respect to μ j, j =0,, are the same as the sets of measure zero ith respect to μ. Thus, in the various measure spaces that e ill consider, the equivalence classes of functions ill be the same. Let 0 s, and define the measure μ s on X by μ s (E) = α0 s α s dμ, E for each E M. Also assume, that e have to measures ν 0,ν on N, and define the measures ν r, for 0 r, just as e did for μ s above. Given any real numbers p 0,p,q 0,q and any 0 t, e define p t,q t,s(t),r(t) as follos: ( t)p t p 0 + tp t p =, E ( t)q t q 0 s(t) = (tp t), r(t) = (tq t). p q We have the folloing theorem by [9]: + tq t q =,

4 502 NIKOLAOS PATTAKOS AND ALEXANDER VOLBERG Theorem.. Suppose that T is a sub-linear operator satisfying Tf qj,ν j K j f pj,μ j for all f L p j (X, M,μ j ), j =0,. Then, for 0 t, e have Tf qt,ν r(t) for all f L p t (X, M,μ s(t) ). K t 0 K t f pt,μ s(t), In addition to the previous theorem, e need also the folloing proved in [7]: Theorem.2. If the A norm of a eight is small, i.e., [] A +δ<2, then the function f =log, andanycube satisfy f f dx 32 δ. Our purpose is to generalize the main Theorem proved in [8]. This generalization is the folloing: Theorem.3. Let T be a linear operator such that for some <p<, T Lp () L p () F ([] Ap ), for any A p eight in R n,heref is an increasing, real valued function. Fix an A p eight 0.Then: lim T L p () L p () = T Lp ( 0 ) L p ( 0 ), d (, 0 ) 0 and in addition for any sub-linear operator satisfying the hypothesis of the theorem e have the estimate: T L p () L p () T L p ( 0 ) L p ( 0 )( + cd (, 0 )) for all eights A p ith sufficiently small d (, 0 ),herec is a positive constant that depends on p, on the function F, on the dimension n and on [ 0 ] Ap. In order to do that e ill define a metric in the A space and this is going to generalize the convergence [] Ap in the sense that this ill be equivalent to the convergence d (, ) 0 in the metric d. 2. The (A,d )metricspace Let us observe that if e have any eight, any positive constant c>0andany p,then[] Ap =[c] Ap. We define an equivalence relation in A in the folloing ay: for u, v A e ill rite u v if and only if there is a positive constant c such that u = cv almost everyhere in R n. This allos us to define the quotient space: / A = A. Inthesameay,edefinefor p< : A p = A p /.

5 METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES 503 For to elements u, v A, e define the distance function d as: d (u, v) = log u log v. Again it is obvious that all the requirements of a metric are satisfied and the reason for defining the equivalence relation is exactly because e need to have: d (u, v) =0 u v. So e define a metric in A, going through the BMO space. We can check that for an A p eight, [] Ap isequivalenttod (, ) 0 and since A p A, the restriction of the d metric to A p, makes the class a metric space. The draback of these ne metric spaces is that none of them is complete. Hoever, the folloing is an obvious remark that gives more informations about this ne spaces. It states that small balls around the constant eight, are complete in the d metric. Theorem 2.. Consider a closed ball B(,r) of sufficiently small radius r>0 and center at the eight, in the metric space (A,d ), i.e., B(,r) = { A : d (, ) r}. Then B(,r) is a complete metric space ith respect to the metric d. Proof. Consider a Cauchy sequence { n } n N in ( B(,r),d ). This means that the sequence {log n } n N is Cauchy in the BMO space. But BMO is a Banach space and so there is a function f BMO such that log n f in BMO as n. By the John Nirenberg inequality e kno that there is a dimensional constant c>0 such that for c all λ (0, f ] the function e λf A 2.But log n f log n f 0 as n. Here, e use the fact that n B(,r). This means that log n = log n log r and r is sufficiently small. Therefore, the number f is small and so the number is really big. We are no alloed to choose for λ =ande c f get that e f A 2 or equivalently there is a eight A 2 A ith f =log. Itis trivial no to see that d ( n,) 0asn. Of course in the previous Theorem, e can replace the A space by any of the other A p spaces. We already mentioned that none of the A p spaces is complete. The proof of this fact is very simple. Let us prove that A is not complete by finding a Cauchy sequence in the space that has no limit inside A. It ill follo that this example orks for anyone of the A p spaces. Consider a decreasing sequence <r n < 0ith lim n r n =. Define the A eights n = x r n. Then: d ( rn, rm )= r n log x r m log x = r n r m log x and since r n eseethat{ n } n N is Cauchy in A, or equivalently the sequence {log n } n N is Cauchy in BMO. Its limit in the BMO space is obviously the function f(x) = log x. This means that for (x) = x e have d ( n,) 0asn, but since is not in L loc (Rn )itcannotbeana eight. So the space (A,d )isnot complete.

6 504 NIKOLAOS PATTAKOS AND ALEXANDER VOLBERG Let us also mention the folloing result in [4], by Garnett and Jones, that helps to understand better hen a ball in (A p,d ) is complete. It states that for a function f BMO, dist BMO (f,l ):=inf{ f g : g L } sup{λ >0:e λf A 2 }. This means that if e have a Cauchy sequence in A p, the closer the sequence is to the L space, the more chances it has to have a limit in A p. So no e can try and find the completion of these spaces under the metric d.by definition the completion of (A p,d )isthespaceāp that consists of the equivalence classes of all Cauchy sequences of A p. We can identify this space as a subspace of BMO. Indeed: Ā p = {f BMO : { n } n N A p : lim log n f =0}, n and e can think of the A p classasasubsetofā p, by identifying every eight ith its logarithm, log, in BMO. Since the classical A p spaces form an increasing sequence of the variable p (and of course the same is true for the A p spaces), the same is true for this ne subspaces of BMO, Ā Āp Āq A BMO, for p q. They are also convex subsets of BMO. Indeed, consider <p<, andf,g Ā p. This means that there are sequences { n } n N, {v n } n N A p such that: f = lim n n,g = lim n v n,inbmo.let0<t<befixed. We ill sho that tf+( t)g Āp. For this, e only need to see that tf+( t)g = lim n log(nv t n t ), in BMO, and check using Hölder that the eight nv t n t A p, for all n, since: [ t v t ] Ap [] t A p [v] t A p, for all, v A p.thus,tf +( t)g Āp. It is trivial to see no that A is also a convex subset of BMO. For Ā the same holds, since if e have to A eights,, v, it is trivial to see that t v t A and actually that [ t v t ] A [] t A [v] t A. Here, let us observe that for any <p<, ehavethatl Āp. Thereisa nice result of eighted theory (see [3]) that states the folloing (e ill present the statement only for A 2 ): there are dimensional constants c,c 2 > 0, such that for a function φ in R n e have: (a) e φ A 2 provided inf{ φ g : g L } c and (b) inf{ φ g : g L } c 2 provided e φ A 2. This means that all functions f BMO that satisfy the assumption (a), belong to the Ā2 space. Equivalently, there is a small neighborhood of L inside BMO, that lies inside the Ā 2 space. We should also mention that since: BLO = {α log : α 0, A }, e can ask the question if the spaces Ā, BLO are equal. Let us assume that they are. A classical result of eighted theory is that BMO = BLO BLO. By our assumption e have that BMO = Ā Ā. No consider a function f BMO. There are functions φ, ψ Ā such that f = φ ψ. We kno that there are sequences of A eights {φ n } n N, {ψ n } n N such that f = lim n log φ n lim n

7 METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES 505 log ψ n = lim n log φ n ψn, here the limit is in BMO. But φ n ψn is an A 2 eight for all n. SoegetthatĀ2 = BMO. But this is obviously false. Note that from the argument follos the inclusion, Ā Ā Ā2. Trivially, e have the more general fact, that for any <p<, Ā +( p)ā Āp. Also,sincee have that A p p A p, e get the equivalence f Āp ( p )f A p. For p = 2 e have f Ā2 f Ā2, hich means that the Ā2 class is symmetric ith respect to the origin in the BMO space. No other Āp class has this property. Here e should remember the folloing about poer eights. A function of the form x α is an A p eight in R n, if and only if n <α<n(p ). The interval for α is symmetric ith respect to the origin, if and only if p = 2. No e can see that there is a correspondence beteen the Ā2 spaceandtheinterval( n, n). 3. The continuity in the eight Our goal in this section is to prove the main Theorem: Theorem 3.. Let T be a linear operator such that for some <p<, T L p () L p () F ([] Ap ), for any A p eight in R n,heref is an increasing, real valued function. Fix an A p eight 0.Then: lim T L p () L p () = T Lp ( 0 ) L p ( 0 ), d (, 0 ) 0 and in addition for any sub-linear operator satisfying the hypothesis of the theorem e have the estimate: T L p () L p () T L p ( 0 ) L p ( 0 )( + cd (, 0 )) for all eights A p ith sufficiently small d (, 0 ),herec is a positive constant that depends on p, on the function F, on the dimension n and on [ 0 ] Ap. Here let us mention something that is important. Say that our A eight is of the order [] A < +δ. Then by Theorem.2 e get that d (, ) c δ. Since Theorem.3 is going to be a generalization of the main Theorem in [8], the rate of convergence that e have in both theorems should agree. The above observation explains exactly this. Remark 3.. Notice that the second half of the previous theorem is true for the Maximal function (since it is true for all sub-linear operators that are bounded in the ay described in the theorem), i.e., for all eights A p that are sufficiently close to 0 A p,ithd (, 0 ) δ: M Lp () L p () M Lp ( 0 ) L p ( 0 )( + cδ). It is ell knon (see []) that M L p () L p () c[] p A p, hich can be used here. The argument is similar to the one given in [8], but it uses some small ne ideas in order to overcome the problem that e have to ork ith to eights and 0, instead of one eight, as as done in [8]. We are going to present it because e have to point out these nice differences.

8 506 NIKOLAOS PATTAKOS AND ALEXANDER VOLBERG Proof. First e ill sho that for any sub-linear operator that satisfies the assumptions of our theorem e have: T L p () L p () T L p ( 0 ) L p ( 0 )( + cδ) for all eights A p ith d (, 0 ) δ. Let0<δbeasmall number that e consider to be fixed. Fix also an A p eight, ithd (, 0 ) <δ. This means that log 0 δ. We ould like to rite our eight as = 0 t W t, for some small and positive number t (hich is going to be about δ), and some eight W A p.at exactly this point, the justification of this fact is diffirent than the one given in [8]. The argument used there can not be used here. Fortunately, e are able to continue as follos. From the expression e can see that W = t 0. For this, let us consider only the case p = 2, but the general case is identical to this one. Since 0 A 2 e kno that there is a small ɛ>0such that := 0 +ɛ A 2. Then obviously ( ) 0 = s s for small s>0. To continue, consider the function f =log 0.The BMO norm of f is really small since: f = s d (, 0 ) s δ, c and so by the John Nirenberg inequality e have that for all λ (0, f ] the function ( ) λ e λf s = 0 A 2,herecis a positive constant that depends only on the dimension. If e choose λ = c 0 δ, c 0 > 0 is any constant less than or equal to sc, eseethat ( δs 2 := A 2, hich implies that the function s 2 s A 2. Then: ) c0 0 W := t 0 = s 2 s A 2, t 0 here e put t = δ c 0. Here e should mention that the A 2 norm of W can be chosen to be bounded above by a constant that depends only on the A 2 norm of.onthe other hand, [ ] A2 depends only on the A 2 norm of 0, and this is fixed. With this in mind, let us assume that the A 2 characteristic of W is bounded above by c. The important thing here is that it does not depend on δ. No the proof continues as in [8], namely: Write γ = T Lp ( 0 ) L p ( 0 ).Bythe interpolation result of Stein and Weiss, Theorem., for X = Y = R n, M = N = L and μ 0 = ν 0 = 0 dx, μ = ν = Wdx,herebyL e denote the σ-algebra of Lebesgue measurable sets in R n,eget T Lp () L p () γ t T t L p (W ) L p (W ) ) t γ t c t F ([W ] Ap γ t c t F (c) t, and the right-hand side goes to γ as t 0 + or equivalently as δ 0 +.Inother ords, lim sup T Lp () L p () T Lp ( 0 ) L p ( 0 ), d (, 0 ) 0 t0

9 METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES 507 and in addition e have the desired estimate: T L p () L p () T L p ( 0 ) L p ( 0 )( + cδ), here c is a constant depending on n, p and [ 0 ] Ap, for all eights in A p that are δ close to 0 in the d metric. We can also conclude the folloing: Proposition 3.. The set {log : A p } is open in BMO for all <p<+. Proof. To see this fix 0 A p and choose sufficiently small δ>0. For f BMO ith f log 0 δ, ritef =logu, hereu is a positive function. Then follo the previous reasoning in the beginning of the proof, ith = u and rite u = t 0 W t, for 0 <t<. It follos that W A p,ifδ>0 is small depending only on the A p norm of 0,andsou = t 0 W t is an A p eight, by Hölder s inequality. As e can see, this is exactly the same argument as before. This result is ne and could not be obtained from the techniques used in [8]. No e sho that for a linear operator e have the estimate: T Lp ( 0 ) L p ( 0 ) lim inf T L p () L p (). d (, 0 ) 0 Here again the reasoning requires subtle, but not difficult modification from the one given in [8]. Let us assume for simplicity that p =2andthat T L2 ( 0 ) L 2 ( 0 ) =. Note that other p s can be treated similarly. So far, e have proved that: lim inf T L 2 () L 2 () = d (, 0 ) 0 lim sup T L 2 () L 2 () d (, 0 ) 0 and: d (, 0 ) δ< T L 2 () L 2 () +cδ. Let M φ denote the operation of multiplication by φ. To finish the proof of the continuity at = 0 e are going to assume that: lim inf d (, 0 ) 0 M TM 20 2 L2 ( 0 ) L 2 ( 0 ) < and get a contradiction. This means that there is τ>0 small, and a sequence of A 2 eights n such that d ( n, 0 ) 0asn andinaddition: (3.) n T n g L 2 ( 0 ) ( τ) g L 2 ( 0 ) for all functions g L 2 ( 0 ). Fix no any cube in R n. Here e can make the normalization assumption n 0 dx = for all n N. We claim to things: ( ) 2 n 2 0 L2 ( 0,) 0asn here by L 2 ( 0,) e mean the L 2 ( 0 ) norm over, and (2 ) there exists a subsequence k n such that kn 0 almost everyhere in the cube.

10 508 NIKOLAOS PATTAKOS AND ALEXANDER VOLBERG Obviously (2 ) follos from ( ). For a proof of, see lemma after the end of this proof. No ithout loss of generality e can assume that the subsequence is the original sequence n.notethat( ) implies 2 n f 2 0 f L 2 ( 0,) 0asn for all bounded f, and so for g = f 2 0,eget T( n g) Tg L2 ( 0,) 0as n and this implies that for a subsequence of n (hich again e assume that is the hole sequence), n T n g Tg almost everyhere in the cube. Itis time to apply Fatou s lemma in inequality (3.) and get: lim inf n n T n g lim inf 2 n 0 2 n T n g L2 ( 0,) ( τ) g L2 ( 0,). L2 ( 0,) Here g = f 2 0 ith bounded f form a dense family in L 2 ( 0,). For g from this dense family it follos: Tg L 2 ( 0 ) ( τ) g L 2 ( 0 ) by letting the cube expand to infinity, for g in some dense subclass of L 2 ( 0 ).By assumption T L2 ( 0 ) L 2 ( 0 ) = and this is ho e have our contradiction. All that remains is the folloing lemma: Lemma 3.. Let 0, A 2 such that d (, 0 ) ɛ, hereɛ is sufficiently small. Let us have a normalization assumption 0 dx =.Then 2 n 2 0 L 2 ( 0,) 2 c(ɛ) 2,herec(ɛ) goes to 0 as ɛ goes to 0. Proof. We ant to estimate the expression: = 0 L 2 ( 0,) + 2 ( 0 ) 2. The last integral can be taken care of really easy, since by our normalization assumption and Cauchy Schartz e get the folloing: ( 0 ) 2 = ( ) ( ( ) ) ( ) = Therefore, the quantity that e need to estimate is bounded above by: L 2 ( 0,). It is time to use the fact that d (, 0 ) ɛ. We get that the eight 0 is in the ( ) A 2 class and actually because the BMO norm of log 0 is really small, the A 2 characteristic is bounded by + c(ɛ), here c(ɛ) is a constant that goes to 0 as ɛ goes to 0. So: [ c(ɛ). L 2 ( 0,) 0 ]A 2

11 METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES Comments and observations Let us have a closer look to hat the previous Theorem tells us. Consider any linear operator T that satisfies the assumptions of Theorem.3. This means that for any A p e have a number T Lp () L p (). SoehaveamapF T : A p R defined by the formula: F T () = T Lp () L p (). First, e should observe that for A p this is ell defined since T L p () L p () = T Lp (c) L p (c) for all A p and all positive constants c>0. By Theorem.3 e have that this map is continuous, since: lim F T () F T ( 0 ) =0 d (, 0 ) 0 for all eights, 0 A p. In [8] the authors shoed that for the Hilbert transform, H, ins e have that for all sufficiently small δ s, there is a eight A 2, ith the properties that [] A2 +δ<2 and: c δ H L2 () L 2 (). This means that Theorem.3 is sharp for p = 2 at the constant eight 0 =.This is true also for the Hilbert transform in the line and for the Martingale transform. It is a good point to mention that there are singular operators like the Riesz projection P + in S, that converge faster to their L 2 (dx) norm than the previous mentioned operators (see [8]). Namely, there is universal constant c>0 such that for all eights [] A2 +δ<2, e have: P + L2 () L 2 () cδ. In addition, there is a universal constant c > 0, such that for all sufficiently small δ s, there is eight A 2 ith the properties [] A2 +δ and: c δ P + L2 () L 2 (). We should also mention that in the proof of Theorem.3, there is only one time (namely in the second step) that e really need to use the fact that our operator is linear in order to get that: T L p ( 0 ) L p ( 0 ) lim inf d (, 0 ) 0 T L p () L p (). It is used hen e claim that the convergence 2 n f 2 0 f L2 ( 0,) 0asn, implies the convergence T ( n f) Tf L2 ( 0,) 0asn. By [5, 6], e kno that for any Calderón Zygmund operator T,any<p<, and any A p eight, e have the estimate: T L p () L p () c[] max{, p } A p, here c is a universal constant that does not depend on the eight, and so e see that Theorem.3 can be applied for this class of operators, since the function F that appears in the statement of this Theorem can be chosen to be equal to F (x) = cx max{, p }.

12 50 NIKOLAOS PATTAKOS AND ALEXANDER VOLBERG References [] Stephen M. Buckley, Estimates for operator norms on eighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340(), November 993. [2] J. G. Conlon and T. Spencer, A strong central limit theorem for a class of random surfaces, 20, arxiv: [3] J. Garcia-Cuerva and J. Rubio De Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud. 6, North-Holland, Amsterdam, 985. [4] J. Garnett and P. W. Jones, The distance in BMO to L. Ann. Math. 08 (978), [5] T. Hytönen, The sharp eighted bound for general Calderón Zygmund operators, 200, arxiv: [6] T. Hytönen, C. Pérez, S. Treil and A. Volberg, Sharp eighted estimates of the dyadic shifts and A 2 conjecture, 200, arxiv: , 200. [7] Michael Brian Korey, Ideal eights: asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation, J. Fourier Anal. Appl. 4(4,5) (998). [8] N. Pattakos and A. Volberg, Continuity of eighted estimates in A p norm, Proc. Amer. Math. Soc. S (20)65-. [9] E. Stein and G. Weiss, Interpolation of operators ith change of measures, Trans. Amer. Math. Soc. 87() (958), Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA address: pattakos@msu.edu address: volberg@math.msu.edu