Ap weights for nondoubling measures in Rn and applications

Transactions of the American mathematical society, 354 (2002), 2013-2033. Apweights for nondoubling measures in Rnand applications Joan Orobitg §and Carlos P´ erez ¶ Abstract: We study an analogue of the classical theory of Ap(µ) weights in Rn without assuming that the underlying measure µis doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calder´on-Zygmund operators (in the sense of [NTV1]). We also consider commutators of those Calder´on-Zygmund operators with bounded mean oscillation functions (BMO), extending the main result from [CRW]. Finally, we study self–improving properties of Poincar´e–B.M.O. type inequalities within this context, more precisely we show that if fis a locally integrable function satisfying 1 µ(Q)RQ|f−fQ|dµ ≤a(Q) for all cubes Q, then it is possible to deduce higher Lpintegrability result of fassuming certain simple geometric condition on the functional a. 1 Introduction The classical theory of harmonic analysis for maximal functions and singular integrals on (Rn, µ) has been developed under the assumption that the underlying measure µsatisfies the doubling property, i.e., there exists a constant C > 0 such that µ(B(x, 2r)) ≤C µ(B(x, r)) for every x∈Rnand r > 0. However, some recent results on Calder´on-Zygmund operators ([NTV1], [NTV2], [T1], [T2]) and functions of bounded mean oscillation ([MMNO], [T3]) show that it should be possible to dispense with the doubling condition for most of the classical theory. The purpose of this paper is to present some results which strengthen this point of view. The use of doubling measures in Rnhas two main advantages: (a) one can work with the nested property of dyadic cubes and (b) the faces (or edges) of the cubes have measure zero. The easiness and utility of the dyadic scheme is well known. The profit of (b) is the continuity of the measure µon cubes. That is, given cubes R0⊂R1one can find a monotone family of cubes {Rs},s∈[0,1], such that Rs⊂Rtif s < t and the map L(s) = µ(Rs) is continuos on [0,1]. Following the previous paper [MMNO], we will renounce (a) but we shall maintain property (b). Thus, our point of departure is to consider nonnegative Radon measures µin Rn without mass-points. Then a result of geometrical measure type (see [MMNO, Theorem 2]) assures that we may choose an orthonormal system in Rnso that any cube Qwith sides parallel to the coordinates axes satisfies the above property (b)(µ(∂Q) = 0). Through this work §Research partially supported by CIRIT grant 2000 SGR00059 and by DGICYT grant BFM 2000-0361, Spain. ¶Research partially supported by DGESIC grant PB98-0106, Spain. we shall assume (making a rotation if necessary) that the measure µand the orthonormal system have this property. Moreover, we only shall consider cubes with sides parallel to the coordinate axes. Our measure satisfies a Calder´on-Zygmund decomposition (see [MMNO] or Section 2 later), which is one of the basic and most frequently used tools in the classical theory. This fact and the argument of its proof will allow us to recover many results without assuming that the measure µis doubling. Related to the Calder´on-Zygmund decomposition, the Hardy- Littlewood maximal operator also plays a central role. Given a locally integrable function f one defines the (centered) Hardy-Littlewood maximal function Mas Mf(x) = sup r>0 1 µ(Q(x, r)) ZQ(x,r) |f|dµ, where Q(x, r) denotes the cube centered at xwith sidelength equals to r. The noncentered maximal function Nis defined as Nf(x) = sup Q3x 1 µ(Q)ZQ |f|dµ, where the supremum is taken over all cubes Qcontaining x. Clearly, Mf(x)≤Nf(x) and when the measure µis doubling it also holds Nf(x)≤C Mf(x). However, if µis nondoubling the maximal functions Mf and Nf may be very different. For instance, it is well known that the operator Macts on Lp(µ), p > 1, and from L1(µ) to L1,∞(µ), whereas this is not the case in general for the operator N. On the other hand, weights for the noncentered case Nhas been studied and characterized by Jawerth [Ja]. The first part of this paper is devoted to develop an analogue of the classical theory of Ap(µ) weights with underlying measure µas above. Then, we will obtain weighted norm inequalities for Mand corresponding weighted estimates for Calder´on-Zygmund operators. This result for singular integrals will follow as a consequence of a version of the classical estimate by Coifman proved in [C]. We will also consider commutators of Calder´on-Zygmund operators with BMO extending the main result from [CRW]. In the second part of the paper we will study BMO–Poincar´e type inequalities. We will obtain similar results to those obtained in [FPW] and [MP] where the underlying measure was assumed to be doubling. The main idea is as follows. Let a:Q → [0,∞) be a functional defined on the family of cubes with sides parallel to the coordinates axes. We want to show that if fis a locally integrable function satisfying 1 µ(Q)ZQ |f−fQ|dµ ≤a(Q) for all cubes Q, then it is possible to deduce higher Lpintegrability result of fassuming certain simple geometric condition on a(see (15) below). Of course, the case a(Q)≡C corresponds to BMO but it is also related to Poincar´e when considering a(Q) = `(Q) µ(Q)ZQ g dµ. 2 The paper is organized as it follows. Section 2 is devoted to define the Ap(µ) class of weights and to study their properties. In Section 3 we shall prove the Lp(w)-boundedness of Mwhen w∈Ap(µ) (Muckenhoupt’s Theorem). Weights and commutators for nonclassical singular integral operators are discussed in Section 4. Self-improving properties are considered in Section 5. Acknowledgment. We are grateful to J. Mateu and J. Verdera for their interest and discussions concerning the results of this paper. We thank E. Saksman for sending us his preprint [Sa]. The second author is very grateful for the hospitality of the Centre de Recerca Matem`atica at the Universitat Aut`onoma de Barcelona where this work was completed. Finally we are very grateful to the referee for the careful reading of the paper and for providing us with some enlightening examples (see Remark 2.4). 2Ap(µ)theory of weights The purpose of this section is to describe a Aptheory of weights adapted to our more general underlying measure µ. For this purpose we state a version of the classical Calder´on-Zygmund type decomposition adapted to our situation. A useful tool to prove the above lemma will be the following auxiliary maximal function. For a given cube Qand for each xin the interior of Qwe define the basis CQ(x) = {Qx(r)} where Qx(r) is the unique cube with sidelength rcontained in Qwhich minimizes the distance from xto the center of Qx(r). The radius of a cube Qis defined to be half of the sidelength. We define the corresponding maximal function MQf(x) = sup R:R∈CQ(x) 1 µ(R)ZR |f(y)|dµ(y), Observe that the properties on µimply that the function hx(r) := 1 µ(Qx(r)RQx(r)|f|dµ is continuous on [0, `(Q)] for all xin the interior of Q. We also denote by Ωt={x∈Q:MQ(g)(x)> t}the level set of MQ(g). Recall that a family of cubes {Qj}is quasidisjoint if there exists a universal constant C such that PjχQj≤C, where χEdenotes the characteristic function of the set E. Lemma 2.1 (The Besicovitch-Calder´on–Zygmund decomposition ). Let Qbe a cube and let g∈L1(µ)(Q)be a non-negative function. Also let tbe a positive number such that t > gQ=1 µ(Q)RQg dµ and such that Ωtis not empty. Then there is a family of quasidisjoint cubes {Qj}contained in Qsatisfying 1 µ(Qj)ZQj g dµ =t 3 for each jand such that g(x)≤tfor x∈Q\[ j Qj, µ–a.e. (1) In fact we can write [ j Qj= B(n) [ k=1 [ i∈Fk Qi,(2) where each of the family {Qi}i∈Fk,k= 1,· · · , B(n), is formed by pairwise disjoint cubes. B(n)>1is usually called the Besicovitch constant. Proof: Since Ωtis not empty, for any x∈Ωtthere is a cube Px∈CQ(x) such that (µ(Px))−1RPxg dµ > t. Therefore, since hxis continuous, we have a cube Qx∈CQ(x) satisfying 1 µ(Qx)ZQx g dµ =t with Qx(Q. Now, observe that we can not apply directly the Besicovitch Covering Theorem since xmay not be the center of Qx. To overcome this obstacle we proceed as in [MMNO]. For any cube Qxwe define the rectangle Rxin Rnas the unique rectangle in Rncentered at xsuch that Rx∩Q=Qx. Clearly, the ratio of any two sidelengths of Rxis bounded by 2. So, by the Besicovitch Covering Theorem we have a countable collection of rectangles Rjsuch that they cover Ωt, and every point of Rnbelongs to at most B(n) rectangles Ri. Replacing each Rjby its corresponding cube Qjwe get the cubes from the Lemma. Finally, (1) follows by the Lebesgue differentiation theorem since x∈Q\Ωtimplies g(x)≤t,µ-a.e. 2 A nonnegative, locally integrable function is called a weight. We will consider weights which satisfy the following conditions. Definition 2.2 Let 1<p<∞and p0=p/(p−1). We say that a weight wsatisfies the Ap(µ)condition if there exists a constant Ksuch that for all cubes Q 1 µ(Q)ZQ w dµ 1 µ(Q)ZQ w1−p0dµp−1 ≤K. (3) We also say that a weight wsatisfies the As 1(µ) = A1(µ)condition if there exists a constant Ksuch that for all cubes Q, 1 µ(Q)ZQ w dµ ≤Kess inf x∈Q w(x). Finally, we define the A∞(µ)class as A∞(µ) = Sp>1Ap(µ) 4 Observe that trivially, A1(µ)⊂Ap(µ) for all p > 1 and Ap(µ)⊂Aq(µ) if p<q. The reason we use the notation As 1(µ) = A1(µ) (sfrom strong) is to distinguish this class from the class Aw 1(µ) (wfrom weak) of weights wsuch that for some constant C Mw(x)≤C w(x) almost everywhere in x. Observe that As 1(µ)⊂Aw 1(µ) and indeed, in the classical situation, i.e. if the underlying measure is doubling, both conditions are equivalent and hence As 1(µ) = Aw 1(µ). However, and this is a big gap between the two theories, we will show in Section 3 that, in general, we have that As 1(µ)(Aw 1(µ). In fact the class Aw 1(µ) is too large since it is also shown there that it is not a subset of Aw ∞(µ) in general. We start by proving some of the classical results that hold in our more general situation. However, as we shall see in next section not all of them will be true. We will use the standard notation w(E) = REw dµ, for any measurable set E. Lemma 2.3 For a weight wthe following conditions are equivalent: a) w∈A∞(µ). b) For every cube Q 1 µ(Q)ZQ w dµ ≈exp 1 µ(Q)ZQ log w dµ. c) There are constants 0< α, β < 1such that for every cube Q µ({x∈Q:w(x)≤β wQ})≤α µ(Q).(4) d) There are positive constants Cand βsuch that for every cube Qand for every λ > wQ w({x∈Q:w(x)> λ})≤Cλ µ({x∈Q:w(x)> β λ}). e) wsatisfies a reverse H¨older inequality, namely there are positive constants cand δsuch that for every cube Q 1 µ(Q)ZQ w1+δdµ1 1+δ ≤c µ(Q)ZQ w dµ. f) There are positive constants cand ρsuch that for any cube Qand any measurable set E contained in Qthen w(E) w(Q)≤cµ(E) µ(Q)ρ .(5) g) wsatisfies the following condition: there are positive constants α, β < 1such that whenever Eis a measurable set of a cube Q µ(E) µ(Q)< α implies w(E) w(Q)< β. (6) 5 Remark 2.4 If one removes our standing assumption on the measure µ(that is, µ(∂Q) = 0 for any cube Q with sides parallel to the coordinates axes) then the Lemma 2.3 may be false, as the following examples show (similar one dimensional examples can be found in the recent paper [GMOPST]). The referee provided us with these examples. We thank his/her anonymous and lucid contribution. First, consider on the line the Radon measure µ=Pk≥12−k2δuk, where ukis a decreasing sequence of positive numbers such that uk↓0 and δxis the point mass at x. Also let wbe the weight which takes on the value 2k2−kat ukfor each k. Then one can easily check that w∈A1(µ) and hence for all Ap(µ), p > 1. Moreover, the maximal operator Macts from L1(w) to L1,∞(w) (because w∈A1(µ)) and so Macts on Lp(w), p > 1. However, it is clear that w /∈L1+ε loc (µ) for any ε > 0 and so wdoesn’t satisfy any reverse H¨older inequality. The second example is a refinement of the previous one. We consider a Radon measure νon the line that is not atom-free but it is fairly nice: it gives positive finite measure to all bounded intervals. Specifically, let νbe the sum of the Lebesgue measure on Rand Pk≥12−k2(δuk+δ−uk) where now uk= 2−k3. Define w(±uk) = 2k2−kfor each kand wequals 1 everywhere else. Then w∈Ap(ν) for all p > 1 (w /∈A1(ν)), as can be checked case-by-case (the point masses make only a bounded difference to the quantities on the left hand side of (3) except for small intervals Qclose to the origin; in the exceptional case, the one or two point masses in the interval that are furthest from the origin dominate). Again, it is clear that w /∈L1+ε loc (µ) for any ε > 0. As a third example, we define µto be the measure on R2which is the product of νand Lebesgue measure on R. Picking cubes oriented in coordinate directions, µhas the same behaviour as ν. The extra subtely is that µis a non-atomic Radon measure to which the results in this paper apply. But it doesn’t contradict our results because now we have chosen the one orientation for cubes that violates condition (b) on the Introduction. Proof of Lemma 2.3: We will write the complete proof of this lemma even though several of the implications are trivial. In the literature (the Lebesgue measure case) there are different ways to prove the main implication a) ⇒e) , but only the one that we present can be adapted to our setting. Here we combine the methods from [CF] and [GCRdF]. a) ⇒b) By Jensen’s inequality it is enough to show that 1 µ(Q)ZQ w dµ ≤Cexp 1 µ(Q)ZQ log w dµ Since the Apclasses are increasing on p, if w∈A∞(µ) there exists some p0>1 such that w∈Apfor p≥p0. Then, there exists a constant Ksuch that for p≥p0 1 µ(Q)ZQ w dµ 1 µ(Q)ZQ w1−p0dµp−1 ≤K. 6 Letting ptend to ∞we obtain b). b) ⇒c) Dividing wby an appropiate constant (to be precise exp 1 µ(Q)RQlog w dµ) we may assume that RQlog w dµ = 0 and, consequently wQ≤C. µ({x∈Q:w(x)≤β wQ})≤µ({x∈Q:w(x)≤β C}) =µ{x∈Q: log(1 + 1 w(x))≥log(1 + 1 β C )} ≤1 log(1 + 1 β C )ZQ log(1 + 1 w)dµ =1 log(1 + 1 β C )ZQ log(1 + w)dµ since RQlog w dµ = 0. Now, since log(1 + t)≤t,t≥0, we get: µ({x∈Q:w(x)≤β wQ})≤1 log(1 + 1 β C )ZQ w dµ ≤C µ(Q) log(1 + 1 β C )≤1 2µ(Q), if we choose βsmall enough. c) ⇒d) Since we assume that λ>wQwe may consider the Besicovitch-Calder´on–Zygmund decomposition {Qj}of wand we find a family of quasidisjoint cubes satisfying λ < 1 µ(Qj)ZQj w dµ ≤2λ for each j. By the properties of the cubes combined with (4) we have w({x∈Q:w(x)> λ})≤ B(n) X k=1 X i∈Fk w(Qi) ≤2λ B(n) X k=1 X i∈Fk µ(Qi)≤2λ 1−α B(n) X k=1 µ({x∈Qi:w(x)> β wQi}) ≤2λ B(n) 1−αµ({x∈Q:w(x)> β wQ}) since wQi> λ > wQ. d) ⇒e) We will be using the formula ZX f(x)pdν =pZ∞ 0 λpν({x∈X:f(x)> λ})dλ λ 7 which holds for every nonnegative measurable function fand in any arbitrary measure space (X, ν) with nonnegative measure ν. Then for arbitrary positive δwe have 1 µ(Q)ZQ w1+δdµ =δ µ(Q)Z∞ 0 λδw({x∈Q:w(x)> λ})dλ λ =δ µ(Q)ZwQ 0 λδw({x∈Q:w(x)> λ})dλ λ+δ µ(Q)Z∞ wQ λδw({x∈Q:w(x)> λ})dλ λ ≤(wQ)1+δ+δ µ(Q)Z∞ wQ λδw({x∈Q:w(x)> λ})dλ λ ≤(wQ)1+δ+C δ µ(Q)Z∞ wQ λδ+1µ({x∈Q:w(x)> β λ})dλ λ ≤(wQ)1+δ+C δ β1+δ 1 µ(Q)Z∞ wQβ λδ+1µ({x∈Q:w(x)> λ})dλ λ ≤(wQ)1+δ+C δ β1+δ 1 µ(Q)ZQ w1+δdµ. If we choose δsmall enough such that C δ β1+δ<1 the last term can be absorbed by the first term of the string of inequalities. e) ⇒f) This is just H¨older’s inequality with r= 1 + δ. Indeed if E⊂Q w(E) µ(Q)=1 µ(Q)ZQ χEw dµ ≤1 µ(Q)ZQ wrdµ1/r µ(E) µ(Q)1/r0 ≤ C µ(Q)ZQ w dµ µ(E) µ(Q)1/r0 and this implies the A∞condition (5) with ρ= 1/r0. f) ⇒g) This is immediate. g) ⇒c) First observe that condition (6) is equivalent to saying that there are positive constants α0, β0<1 such that whenever Eis a measurable set of a cube Q w(E) w(Q)< α0implies µ(E) µ(Q)< β0.(7) Then let E={x∈Q:w(x)> bwQ}where b∈(0,1) is going to be chosen now and let E0=Q\E={x∈Q:w(x)≤bwQ}. Then w(E0)≤b wQµ(E0)≤b w(Q). Then if we take b=β0we have that µ(E0)≤α0µ(Q). This yields (4). Therefore we have shown that c) ⇔d) ⇔e) ⇔f) ⇔g) 8 c) ⇒a) We use again that condition e) is symmetric, namely that condition (7) holds. We also use that the measure w dµ does not see hyperplanes parallel to the axes since the same is true for dµ. Now, if we write dµ =w−1wdµ and since c) ⇔e) we have that there are positive constants cand δsuch that 1 w(Q)ZQ (w−1)1+δwdµ1 1+δ ≤c w(Q)ZQ w−1wdµ. Hence w(Q) µ(Q)1 µ(Q)ZQ w−δdµ1/δ ≤C. Then, if we let δ=1 p−1, that is, p=1 δ+ 1 >1 we have that w∈Ap. The proof of the Lemma is now complete. 2 Now w∈Ap(µ), p > 1, obviously implies that w1−p0∈Ap0(µ). Consequently, it is easy to deduce the following corollary. Corollary 2.5 Let p > 1and let w∈Ap(µ), then: (i) There is ε > 0such that w∈Ap−ε(µ)and therefore Ap(µ) = [ q<p Aq(µ) (ii) There is η > 0such that w1+η∈Ap(µ). It is well known that there is an intimate relationship between the Apweights and the John-Nirenberg space BMO(µ) of locally integrable functions with bounded mean oscillation, namely if sup Q 1 µ(Q)ZQ |f−fQ|dµ < ∞, where the supremum is taken over all cubes Qwith sides parallel to the cordinates axes. As a consequence of the John-Nirenberg property for BMO(µ) ([MMNO]) and the above lemma, we have the following relationship between weights and BMO. Corollary 2.6 (i) If w∈A∞(µ)then log(w)∈BMO(µ). (ii) Fix p > 1and let b∈BMO(µ). Then there exists ε > 0depending upon the BMO(µ) constant of bsuch that exb ∈Ap(µ)for |x|< ε. We will skip the proof because the classical one (e.g. [GCRdF, chapter IV]) also works in our setting. 9 Our proof is very close to the less known proof of the theorem in [CRW], see for instance [GCRdF, p. 473]. Tolsa [T3] has proved that the same result holds when w≡1 if one replaces b∈BMO(µ) by b∈RBMO(µ), another space of functions of bounded mean oscillation more adapted to work with singular integrals and sharp maximal functions. His proof doesn’t use weights, but it is based on the use of a sharp maximal operator. Proof: To simplify notation we will write Tinstead of Tε. By Corollary 2.5 ii) there is η > 0 such that w1+η∈Ap(µ). Then using Corollary 2.6 (ii) we choose δ > 0 such that exp(spb(1+η)/η)∈Ap(µ) if 0 ≤s(1+η)/η < δ with uniform constant. For z∈Cwe define the operator Szf=ezbT(e−zbf). We claim that kSzfkLp(w)≤CkfkLp(w) uniformly on |z| ≤ s < δη/(1 + η). The function z7→ Szfis analytic, and by the Cauchy theorem, if s < δη/(1 + η), d dz Szf|z=0 =1 2πi Z|z|=s Szf z2dz. Observing that d dz Szf|z=0 = [b, T]f and applying the Minkowski inequality to the previous equality we conclude k[b, T]fkLp(w)≤1 2πZ|z|=s kSzfkLp(w) s2|dz| ≤ C skfkLp(w). Thus, we are left with proving the claim, which is equivalent to Z|Tf(x)|pexp(<(z)pb(x)) w(x)dµ(x)≤CZ|f(x)|pexp(<(z)pb(x)) w(x)dµ(x) (14) We write w0:= exp(<(z)p b(1 + η)/η) and w1:= w1+η. Since w0and w1belong to Ap(µ) we have Z|Tf(x)|pw0(x)dµ(x)≤CZ|f(x)|pw0(x)dµ(x) and Z|Tf(x)|pw1(x)dµ(x)≤CZ|f(x)|pw1(x)dµ(x). Now, by the Stein-Weiss interpolation theorem (e.g. [BeL, p.115]) we have that Z|Tf(x)|pw1−θ 0wθ 1dµ(x) ≤CZ|f(x)|pw1−θ 0wθ 1dµ(x) and taking θ= (1 + η)−1we get (14). 2 16 5BMO–Poincar´e inequalities In this section we shall apply the results from Section 2 to extend the main theorem from [FPW] and [MP] to our setting. It has been shown in these papers that there is a unified theory of some well known classical results concerning Lpproperties of functions with some kind of smoothness, more precisely with control on the oscillation. In particular it is shown for instance both the classical Sobolev theorem and the Lpproperty of BMO functions are part of a more general phenomenon of self–improving properties. As in [FPW] and [MP] we impose the following discrete condition on the functional a relative to a locally integrable weight function w. Recall that a functional ais a function a:Q → [0,∞) where Qdenotes the family of cubes with sides parallel to the coordinates axes. Recall that we use the notation w(E) = REw dµ. Definition 5.1 Let 0< r < ∞and wbe a weight function. We say that the functional a satisfies the weighted Drcondition if there exists a finite constant Csuch that for each cube Qand any family ∆of pairwise disjoint subcubes of Q, X P∈∆ a(P)rw(P)≤Cra(Q)rw(Q).(15) We denote by kakthe best constant C. We introduce the notation kgkLr,∞(Q,w)= sup t>0 tw({x∈Q:|g(x)|> t}) w(Q)1/r for the normalized weak or Marcinkiewicz Lrnorm. Before we present our result we need to make some observations in order to adapt to our setting some well known properties of the so–called optimal polynomials defined as follows. Fix a cube Qand a nonnegative integer m. The space Pmof real-valued polynomials of degree at most mis a Hilbert space with the inner product 1 µ(Q)ZQ fg dµ. Consider the orthonormal basis {ϕν},|ν| ≤ m, obtained by applying the Gram–Schmidt orthonormalization process to the power functions {xν},|ν| ≤ m. Observe that kϕνkL∞(Q)≤C1 µ(Q)ZQ |ϕν|2dµ1/2 =C(16) since the space Pmis finite dimensional, and so all norms on it are equivalent. The constant Cdepends only on m. 17 We let PQbe the operator defined by PQf(x) = X |ν|≤m 1 µ(Q)ZQ fϕνdµ ϕν(x), which is a projection from L1(Q) onto Pm. By (16) we have the following key property: kPQfkL∞(Q)≤γ µ(Q)ZQ |f(y)|dµ(y),(17) where γ=C2. Observe that when m= 0, PQf=fQ=1 µ(Q)RQf dµ. These polynomials PQfare optimal in the sense that inf π∈Pm 1 µ(Q)ZQ |f−π|dµ ≈1 µ(Q)ZQ |f−PQf|dµ. In fact we may replace the L1norm by any Lpnorm, 1 <p<∞. Indeed, the inequality in the direction “≤” is trivial. To prove the opposite inequality, observe that since PQis a projection we have PQπ=πfor any polynomial of degree at most m, and therefore 1 µ(Q)ZQ |f−PQf|dµ ≤1 µ(Q)ZQ (|f−π|+|PQ(f−π)|)dµ ≤1 µ(Q)ZQ |f−π|dµ +kPQ(f−π)kL∞(Q)≤1 + γ µ(Q)ZQ |f−π|dµ by (17). THEOREM 5.2 Let µbe a measure as above and let wbe an A∞(µ)weight. Let abe a functional satisfying the Drcondition (15). Suppose that fis a locally integrable function such that for all cubes Qin Rn 1 µ(Q)ZQ |f−PQf|dµ ≤a(Q) (18) Then there exists a constant Csuch that for all the cubes Qin Rn kf−PQfkLr,∞(Q,w)≤Ckaka(Q). Corollary 5.3 Under the same hypothesis of the Theorem, if 0< p < r, then there exists a constant C=C(p)independent of fand Qsuch that 1 w(Q)ZQ |f−PQf|pwdµ1/p ≤Ckaka(Q). 18 This is a consequence of the well-known inequality 1 ν(E)ZE |h|pwdµ1/p ≤Cp,r khkLr,∞(E), which holds in any measure space of finite measure, and whenever plies between zero and r. Proof of Theorem 5.2. To prove the theorem we adapt the method considered in [FPW, Appendix] which again is based on the good–λmethod. The first step. For a fixed cube Qwe let E(Q, t) = {x∈Q:|f(x)−PQf(x)|> t} and Ω(Q, t) = {x∈Q:MQ(f−PQf)(x)> t}. Observe that by the Lebesgue differentation theorem E(Q, t)⊂Ω(Q, t), µalmost everywhere. We want to prove that sup t>0 trw(Ω(Q, t)) w(Q)≤C a(Q)r(19) with a constant Cindependent of t > 0. First observe that we may assume that t > a(Q) since otherwise (19) is easy. With this assumption the Besicovitch-Calder´on–Zygmund decomposition of |f−PQf|gives us a family {Qt i}of cubes strictly contained in Q, such that 1 µ(Qt i)ZQt i |f−PQf|dµ =t(20) and such that Ω(Q, t)⊂ ∪iQt iµ–almost everywhere. Since wis absolutely continuous with respect to µwe have w(E(Q, t)) ≤X i w(Qt i). To any of these cubes Qt iwe perform again the corresponding Besicovitch-Calder´on–Zygmund decomposition of |f−PQf|at level qt with q > 1 and we obtain another family of subcubes {Qqt j}, strictly contained in Qt i, such that for each j 1 µ(Qqt j)ZQqt j |f−PQf|dµ =q t (21) and hence for each i,{x∈Qt i:|f(x)−PQf(x)|> qt} ⊂ SjQq t j(µ–almost everywhere). On the other hand, if x∈Q\ ∪iQt ithen x /∈ ∪jQq t jand hence |f(x)−PQf(x)| ≤ qt µ–almost everywhere. Therefore E(Q, qt) = [ i {x∈Qt i:|f(x)−PQf(x)|> qt} ⊂ [ j Qqt j(22) µ–almost everywhere and consequently w(E(Q, qt)) ≤X j w(Qq t j). The second step (good-lambda inequality). 19 Recall that kak, γ, B(n)≥1 denote the constants in (15), (17) and the Besicovitch constant respectively. Also, ρdenotes the constant in the A∞condition (5). We claim the following: There exists a constant csuch that for each q > γ and 0 <  ≤ kakand t > 0 the following estimate holds X j w(Qq t j)≤[B(n)]2 ρc (q−γ)ρX i w(Qt i) + kakr rtra(Q)rw(Q)!.(23) Let us observe first that for t≤a(Q) the inequality (23) trivially holds. Indeed, since the family {Qqt j}has bounded overlap with constant B(n)2we have X j w(Qq t j)≤B(n)2w(Q)≤[B(n)]2kakr rtra(Q)rw(Q) which is smaller than the right side of the inequality. ¿From now on we fix  > 0 and t > a(Q). Using (2) it is easy to see that we may assume that the cubes {Qt i}are pairwise disjoint as well as the cubes {Qtq j}inside each cube Qt i. Therefore we only have to prove (23) without the constant (B(n))2on its right hand side. We split the family {Qt i}in two: (i) i∈Iif 1 |Qt i|ZQt i |f−PQt if|<  t or (ii) i∈II if 1 |Qt i|ZQt i |f−PQt if| ≥  t. Then X j w(Qq t j) = X i∈IX Qqt j⊂Qt i w(Qqt j) + X i∈II X Qqt j⊂Qt i w(Qqt j) = I+II To estimate the second sum II we will use (18) and (15): II ≤X i∈II w(Qt i)≤X i∈II 1 t|Qt i|ZQt i |f−PQt if|!r w(Qt i) ≤1 rtrX i a(Qt i)rw(Qt i)≤kakr rtra(Q)rw(Q). To estimate Iwe use that w∈A∞(µ) and therefore it satisfies (5). Thus 20 I=X i∈I w([ Qqt j⊂Qt i Qqt j)≤cX i∈I µ(SQqt j⊂Qt iQqt j) µ(Qt i)!ρ w(Qt i) Now to estimate the inner unweighted part we first observe that by (17) and (20) |PQt if−PQf|=|PQt i(f−PQf)| ≤ γ µ(Qt i)ZQt i |f−PQf|dµ =γt and hence by (21) qt =1 µ(Qqt j)ZQqt j |f−PQf|dµ ≤1 µ(Qqt j)ZQqt j |f−PQt if|dµ +γt and then µ(Qqt j)≤1 (q−γ)tZQqt j |f−PQt if|dµ. Therefore µ([ Qqt j⊂Qt i Qqt j) = X Qqt j⊂Qt i µ(Qqt j)≤X Qqt j⊂Qt i 1 (q−γ)tZQqt j |f−PQt if|dµ ≤1 (q−γ)tZQt i |f−PQt if|dµ ≤ q−γµ(Qt i) since i∈I. Combining this estimate we get I≤ρc (q−γ)ρX i∈Fk w(Qt i) and finally X j w(Qq t j)≤ ρc (q−γ)ρX i w(Qt i) + kak rtra(Q)rw(Q)!, as desired. The third step. We now are ready to prove (19). Let t0>0 be a constant to be chosen in a moment and let qsuch that q−γ > 1. Also, we denote E(Q, t0) = E0and E(Q, qmt0) = Emfor m= 1,2,· · · . We have the inclusions E0⊃E1⊃ · · · Em⊃ · · · . To each set Emwe assign and fix a family of cubes {Qm i}following the method described above in the first step of the proof of the theorem. We start by taking the family {Q0 i}= 21 {Qt0 i}, corresponding to the first level set E0, and from this we obtain {Q1 i}and we keep repeating the procedure. Then we have [ i Q0 i⊃[ i Q1 i⊃[ i Q2 i⊃ · · · ⊃ [ i Qm i⊃ · · · and by (22) we have, µ–almost everywhere, Em⊂[ i Qm i. Therefore w(Em)≤X i w(Qm i) = I(m). Now, since q > γ we have by the good-λinequality (23) that I(m)≤C ρI(m−1) + Ckakr r(qm−1t0)ra(Q)rw(Q), and hence for each m= 1,2,· · · (qmt0)rI(m) w(Q)≤C qrρ(qm−1t0)rI(m−1) w(Q)+C qrkakr ra(Q)r.(24) Also for m= 0 there is a corresponding inequality: tr 0I(0) w(Q)≤C qrkakr ra(Q)r, using that I(0) ≤C w(Q), ≤ kak, and by choosing t0=q a(Q). Now for each N= 1,2,· · · ,we define ϕ(N) = sup m=0,1,··· ,N (qmt0)rI(m) w(Q)<∞. Then, combining the last inequality with (24) we get ϕ(N)≤C qrρϕ(N) + C qrkakr ra(Q)r and we can choose  > 0 sufficiently small such that ϕ(N)≤Ckakra(Q)r. This means that trw(Et) w(Q)≤Ckakra(Q)r, for tof the form t=qmt0,m= 0,1,· · · . This yields immediately the result for arbitrary t concluding the proof of the theorem. 2 22 References [BeL] J. Bergh and J. L¨ofstr¨om, “Interpolation Spaces,” Springer-Verlag, 1976. [ChF] M. Christ and R. Fefferman, A note on weighted norm inequalities for the Hardy- Littlewood maximal operator, Proc. Amer. Math. Soc. 12 (1983), 447–448. [C] R. Coifman, Distribution function inequalities for singular integrals, Proc. Acad. Sci. U.S.A. 69 (1972), 2838–2839. [CF] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals , Studia Math. 51 (1974), 241–250. [CRW] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611–635. [CR] R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), 249–254. [FPW] B. Franchi, C. P´erez and R. L. Wheeden, Self– Improving Properties of John– Nirenberg and Poincar´e Inequalities on Spaces of Homogeneous Type, J. Funct. Anal. 153 (1998), 108–146. [GCRdF] J. Garc´ıa-Cuerva and J.L. Rubio de Francia, “Weighted Norm Inequalities and Related Topics,” North Holland Math. Studies 116, North Holland, Amsterdam, 1985. [GMOPST] P. Gurka, P., F. J. Mart´ın-Reyes, P. Ortega, L. Pick, M. D. Sarri´on and A. de la Torre, Good and bad measures, J. London Math. Soc. (2) 61 (2000), no. 1, 123–138. [Ja] B. Jawerth, Weighted norm inequalities: linearization, localization, and factorization, Amer. J. Math. 108 (1986), 361-414. [MP] P. MacManus and C. P´erez, Generalized Poincar´e inequalities: Sharp self–improving properties, Internat. Math. Res. Notices, 1998 (2), 101– 116. [MMNO] J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for nondoubling measures, Duke Math. J. 102 (2000), 533–565. [NTV1] F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calder´on-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1997 (15), 703–726. [NTV2] F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calder´on-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998 (9), 463–487. [Sa] E. Saksman, The local mapping properties of the Hilbert transform, preprint 1999. 23 [St] E. M. Stein, “Harmonic Analysis,” Princeton Univ. Press, Princeton, NJ, 1995. [T1] X. Tolsa, L2-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), 269–304. [T2] X. Tolsa, Cotlar’s inequality and existence of principal values for the Cauchy integral wihout the doubling condition, J. Reine Angew. Math. 502 (1998), 199–235. [T3] X. Tolsa, BMO,H1and Calder´on-Zygmund operators for non doubling measures, to appear in Math. Ann. [V] J. Verdera, On the T(1)-Theorem for the Cauchy integral, Ark. Mat. 38 (2000), 183–199. Joan Orobitg Carlos Prez Departament de Matem`atiques Departamento de Matem´aticas Universitat Aut`onoma de Barcelona Universidad Aut´onoma de Madird 08193 Bellaterra, Spain 28049 Madrid, Spain [email protected] [email protected] 24