The L(log L)e endpoint estimate for maximal singular integral operators

arXiv:1503.04008v1 [math.CA] 13 Mar 2015 The L(log L)ǫendpoint estimate for maximal singular integral operators Tuomas Hyt¨onen and Carlos P´erez Abstract We prove in this paper the following estimate for the maximal operator T∗associated to the singular integral operator T: kT∗fkL1,∞(w).1 ǫZRn |f(x)|ML(log L)ǫ(w)(x)dx,w≥0,0< ǫ ≤1. This follows from the sharp Lpestimate kT∗fkLp(w).p′(1 δ)1/p′kfkLp(ML(log L)p−1+δ(w)),1<p<∞,w≥0,0< δ ≤1. As as a consequence we deduce that kT∗fkL1,∞(w).[w]A1log(e+[w]A∞)ZRn |f|wdx, extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate. 1 Introduction and main results Very recently, the so called Muckenhoupt-Wheeden conjecture has been disproved by Reguera-Thiele in [RT]. This conjecture claimed that there exists a constant csuch that for any function fand any weight w(i.e., a nonnegative locally integrable function), there holds kH fkL1,∞(w)≤cZR |f|Mwdx.(1) where His the Hilbert transform. The failure of the conjecture was previously obtained by M.C. Reguera in [Re] for a special model operator Tinstead of H. This conjecture was motivated by a similar inequality by C. Fefferman and E. Stein [FS] for the Hardy-Littlewood maximal function: kM fkL1,∞(w)≤cZRn|f|Mwdx.(2) 2000 Mathematics Subject Classification: 42B20, 42B25, 46E30. Key words and phrases: maximal operators, Calder´on–Zygmund operators, weighted estimates. The first author is supported by the European Union through the ERC Starting Grant “Analytic–probabilistic methods for borderline singular integrals”. He is part of the Finnish Centre of Excellence in Analysis and Dynamics Research. The second author was supported by the Spanish Ministry of Science and Innovation grant MTM2012-30748. 1 The importance of this result stems from the fact that it was a central piece in the approach by Fefferman-Stein to derive the following vector-valued extension of the classical LpHardy-Littlewood maximal theorem: for every 1 <p,q<∞, there is a finite constant c=cp,qsuch that X j (M fj)q1 qLp(Rn)≤cX j |fj|q1 qLp(Rn).(3) This is a very deep theorem and has been used a lot in modern harmonic analysis explaining the central role of inequality (2). Inequality (1) was conjectured by B. Muckenhoupt and R. Wheeden during the 70’s. That this conjecture was believed to be false was already mentioned in [P2] where the best positive result in this direction so far can be found, and where Mis replaced by ML(log L)ǫ, i.e., a maximal type operator that is “ǫ-logarithmically” bigger than M: kT fkL1,∞(w)≤cεZRn|f|ML(log L)ε(w)dx w ≥0. where Tis the Calder´on-Zygmund operator T. Until very recently the constant of the estimate did not play any essential role except, perhaps, for the fact that it blows up. If we check the computations in [P2] we find that cε≈e1 ε. It turns out that improving this constant would lead to understanding deep questions in the area. One of the main purposes of this paper is to improve this result in several ways. A first main direction is to improve the exponential blow up e1 εby a linear blow up 1 ε. The second improvement consists of replacing Tby the maximal singular integral operator T∗. The method in [P2] cannot be used directly since the linearity of Tplayed a crucial role. We refer to Section 2.3 for the definition of the maximal function MA=MA(L). We remark that the operator ML(log L)εis pointwise smaller than Mr=MLr,r>1, which is an A1weight and for which the result was known. Theorem 1.1. Let T be a Calder´on-Zygmund operator with maximal singular integral operator T∗. Then for any 0< ǫ ≤1, kT∗fkL1,∞(w).cT ǫZRn|f(x)|ML(log L)ǫ(w)(x)dx w ≥0 (4) If we formally optimize this inequality in ǫwe derive to the following conjecture: kT∗fkL1,∞(w)≤cTZRn|f(x)|MLloglog L(w)(x)dx w ≥0,f∈L∞ c(Rn).(5) To prove Theorem 1.1 we need first an Lpversion of this result, which is fully sharp, at least in the logarithmic case. The result will hold for all p∈(1,∞) but for proving Theorem 1.1 we only need it when pis close to one. There are two relevant properties properties that will be used (see Lemma 4.2). The first one establishes that for appropriate Aand all γ∈(0,1), we have (MAf)γ∈A1with constant [(MAf)γ]A1 independent of Aand f. The second property is that M¯ Ais a bounded operator on Lp′(Rn) where ¯ Ais the complementary Young function of A. The main example is A(t)=tp(1 +log+t)p−1+δ, p∈(1,∞), δ ∈(0,∞) since kM¯ AkB(Lp′(Rn)) .p2(1 δ)1/p′ by (25). 2 Theorem 1.2. Let 1<p<∞and let A be a Young function, then kT∗fkLp(w)≤cTp′kM¯ AkB(Lp′(Rn)) kfkLp(MA(w1/p)p)w≥0.(6) In the particular case A(t)=tp(1 +log+t)p−1+δwe have kT∗fkLp(w)≤cTp′(1 δ)1/p′kfkLpML(logL)p−1+δ(w)w≥0,0< δ ≤1. Another worthwhile example is given by ML(log L)p−1(loglog L)p−1+δinstead of ML(logL)p−1+δfor which: kT∗fkLp(w)≤cTp′(1 δ)1/p′kfkLpML(logL)p−1(loglog L)p−1+δ(w)w≥0,0< δ ≤1. There are some interesting consequences from Theorem 1.1, the first one is related to the one weight theory. We first recall that the definition of the A∞constant considered in [HP] and where is shown it is the most suitable one. This definition was originally introduced by Fujii in [F1] and rediscovered later by Wilson in [W1]. Definition 1.3. [w]A∞:=sup Q 1 w(Q)ZQM(wχQ)dx. Observe that [w]A∞≥1 by the Lebesgue differentiation theorem. When specialized to weights w∈A∞or w∈A1, Theorem 1.1 yields the following corollary. It was formerly known for the linear singular integral T[HP], and this was used in the proof, which proceeded via the adjoint of T; the novelty in the corollary below consists of dealing with the maximal singular integral T∗. Corollary 1.4. kT∗fkL1,∞(w).log(e+[w]A∞)ZRn|f|Mwdx,(7) and hence kT∗fkL1,∞(w).[w]A1log(e+[w]A∞)ZRn|f|wdx,(8) The key result that we need is the following optimal reverse H¨older’s inequality obtained in [HP] (see also [HPR] for a better proof and [DMRO] for new characterizations of the A∞class of weights). Theorem 1.5. Let w ∈A∞, then there exists a dimensional constant τnsuch that − ZQwrw1/rw≤2− ZQw where rw=1+1 τn[w]A∞ Proof of Corollary 1.4.To apply (4), we use logt≤tα αfor t>1 and α > 0 to deduce that ML(logL)ǫ(w).1 αǫML1+ǫα (w) Hence, if w∈A∞we can choose αsuch that αǫ =1 τn[w]A∞. Then, applying Theorem 1.5 1 ǫML(logL)ǫ(w).1 ǫ(ǫτ[w]A∞)ǫMLrw(w).1 ǫ[w]ǫ A∞M(w) and optimizing with ǫ≈1/log(e+[w]A∞) we obtain (7).  3 As a consequence of Theorem 1.1 we have, by using some variations of the ideas from [CP1], the following: Corollary 1.6. Let u, σ be a pair of weights and let p ∈(1,∞). We also let δ, δ1, δ2∈(0,1]. Then (a) If K=sup Qku1/pkLp(log L)p−1+δ,Q 1 |Q|ZQσdx!1/p′ <∞,(9) then kT∗(fσ)kLp,∞(u).1 δK(1 δ)1/p′kfkLp(σ)(10) (The boundedness in the case δ=0is false as shown in [CP1].) (b) As consequence, if K=sup Qku1/pkLp(log L)p−1+δ1,Q 1 |Q|ZQσdx!1/p′ +sup Q 1 |Q|ZQudx!1/p kσ1/p′kLp′(log L)p′−1+δ2,Q<∞, (11) then kT∗(fσ)kLp(u).K 1 δ1 1 δ1!1 p′ +1 δ2 1 δ2!1 pkfkLp(σ).(12) The first qualitative result as in (10) was obtained in [CP1], Theorem 1.2 and its extension Theorem 4.1. We remark that this result holds for any operator Twhich satisfies estimate (4). We also remark that this corollary improves the main results from [CRV] (see also [ACM]) by providing very precise quantitative estimates. We refer to these papers for historical information about this problem. We don’t know whether the factors 1 δi,i=1,2 can be removed or improved from the estimate (12). Perhaps our method is not so precise to prove the conjecture formulated in Section 7. However, it is clear from our arguments that these factors are due to the appearance of the factor 1 ǫin (4). Acknowledgments We would like to thank the anonymous referee for detailed comments that improved the presentation. 2 Basic definitions and notation 2.1 Singular integrals In this section we collect some notation and recall some classical results. By a Calder´on-Zygmund operator we mean a continuous linear operator T:C∞ 0(Rn)→ D′(Rn) that extends to a bounded operator on L2(Rn), and whose distributional kernel Kcoincides away from the diagonal x=yin Rn×Rnwith a function Ksatisfying the size estimate |K(x,y)| ≤ c |x−y|n 4 and the regularity condition: for some ε > 0, |K(x,y)−K(z,y)|+|K(y,x)−K(y,z)| ≤ c|x−z|ε |x−y|n+ε, whenever 2|x−z|<|x−y|, and so that T f(x)=ZRnK(x,y)f(y)dy, whenever f∈C∞ 0(Rn) and x<supp(f). Also we will denote by T∗the associated maximal singular integral: T∗f(x)=sup ε>0Z|y−x|>ε K(x,y)f(y)dyf∈C∞ 0(Rn) More information can be found in many places as for instance in [G] or [Duo]. 2.2 Orlicz spaces and normalized measures We will also need some basic facts from the theory of Orlicz spaces that we state without proof. We refer to the book of Rao and Ren [RR] for the proofs and more information on Orlicz spaces. Another interesting recent book is [W2]. A Young function is a convex, increasing function A: [0,∞)→[0,∞) with A(0) =0, such that A(t)→ ∞ as t→ ∞. Such a function is automatically continuous. From these properties it follows that A: [t0,∞)→[0,∞) is a strictly increasing bijection, where t0=sup{t∈[0,∞) : A(t)=0}. Thus A−1(t) is well-defined (single-valued) for t>0, but in general it may happen that A−1(0) =[0,t0] is an interval. The properties of Aeasily imply that for 0 < ε < 1 and t≥0 A(εt)≤εA(t).(13) The A-norm of a function fover a set Ewith finite measure is defined by kfkA,E=kfkA(L),E=inf{λ > 0 : − ZEA |f(x)| λ!dx ≤1} where as usual we define the average of fover a cube E,− REf=1 |E|REf dx. In many situations the convexity does not play any role and basically the monotonicity is the fundamental property. The convexity is used for proving that k kA,Eis a norm which is often not required. We will use the fact that kfkA,E≤1 if and only if − ZEA(|f(x)|)dx ≤1.(14) Associated with each Young function A, one can define a complementary function ¯ A(s)=sup t>0{st −A(t)}s≥0.(15) Then ¯ Ais finite-valued if and only if limt→∞ A(t)/t=supt>0A(t)/t=∞, which we henceforth assume; otherwise, ¯ A(s)=∞for all s>supt>0A(t)/t. Also, ¯ Ais strictly increasing on [0,∞) if and only if limt→0A(t)/t=inft>0A(t)/t=0; otherwise ¯ A(s)=0 for all s≤inft>0A(t)/t. 5 Such ¯ Ais also a Young function and has the property that st ≤A(t)+¯ A(s),t,s≥0.(16) and also t≤A−1(t)¯ A−1(t)≤2t,t>0.(17) The main property is the following generalized H¨older’s inequality 1 |E|ZE|fg|dx ≤2kfkA,Ekgk¯ A,E.(18) As we already mentioned, the following Young functions play a main role in the theory: A(t)=tp(1 +log+t)p−1+δt, δ > 0,p>1. 2.3 General maximal functions and Lpboundedness: precise versions of old results Given a Young function Aor more generally any positive function A(t) we define the following maximal operator ([P1],[P2]) MA(L)f(x)=MAf(x)=sup Q∋xkfkA,Q. This operator satisfies the following distributional type estimate: there are finite dimensional constants cn,dnsuch that |{x∈Rn:MAf(x)>t}| ≤ cnZRnA dnf t!dx f ≥0,t>0 (19) This follows from standard methods and we refer to [CMP, Remark A.3] for details. A first consequence of this estimate is the following Lpestimate of the operator, which is nothing more than a more precise version of one the main results from [P1]. A second application will be used in the proof of Lemma 4.2. Lemma 2.1. Let A be a Young function, then kMAkB(Lp(Rn)) ≤cnαp(A) (20) where αp(A)is the following tail condition that plays a central role in the sequel αp(A)= Z∞ 1 A(t) tp dt t!1/p <∞.(21) Examples of functions satisfying the Bpcondition are A(t)=tq, 1 ≤q<p. More interesting examples are given by A(t)=tp (1 +log+t)1+δA(t)≈tplog(t)−1loglog(t)−(1+δ),p>1, δ > 0. Often we need to consider instead of the function Ain (21) the complementary ¯ A. We also record a basic estimate between a Young function and its derivative: A(t)≤tA′(t) (22) which holds for any t∈(0,∞) such that A′(t) does exist. 6 There is the following useful alternative estimate of (20) that will be used in the sequel. Although variants of this lemma are well known in the literature (cf. [CMP], Proposition 5.10), we would like to stress the fact that we avoid the doubling condition on the Young functions Band ¯ B, which is important in view of the quantitative applications to follow: even if our typical Young functions are actually doubling, we want to avoid the appearance of their (large) doubling constants in our estimates. Lemma 2.2. Let B a Young function. Then kMBkB(Lp(Rn)) ≤cnβp(B) (23) where βp(B)= Z∞ B(1) t ¯ B(t)pd¯ B(t)!1/p Proof. We first prove that for a>0 Z∞ B−1(a) dB(t) tp≤Z∞ ¯ B−1(a)t ¯ B(t)pd¯ B(t).(24) We discretize the integrals with a sequence ak:=ηka, where η > 1 and eventually we pass to the limit η→1. Then Z∞ B−1(a) dB(t) tp= ∞ X k=1ZB−1(ak+1) B−1(ak) dB(t) tp≤ ∞ X k=1 1 B−1(ak)pZB−1(ak+1) B−1(ak)dB(t)= ∞ X k=1 1 B−1(ak)p(ak+1−ak). Similarly, Z∞ ¯ B−1(a)t ¯ B(t)pd¯ B(t)= ∞ X k=0Z¯ B−1(ak+1) ¯ B−1(ak)t ¯ B(t)pd¯ B(t) ≥ ∞ X k=0¯ B−1(ak+1) ¯ B(¯ B−1(ak+1))pZ¯ B−1(ak+1) ¯ B−1(ak)d¯ B(t)= ∞ X k=0¯ B−1(ak+1) ak+1p(ak+1−ak), where we used the fact that t7→ ¯ B(t)/tis increasing, so its reciprocal is decreasing. Moreover, B−1(ak+1) ak+1≥¯ B−1(ak) ak+1 B−1(ak) B−1(ak) (17) ≥ak ak+1 1 B−1(ak)=1 ηB−1(ak) and hence Z∞ B−1(a) dB(t) tp≤ηpZ∞ ¯ B−1(a)t ¯ B(t)pd¯ B(t). Since this is valid for any η > 1, we obtain (24). Now, let t1=max(1,t0), where t0=max{t:B(t)=0}. Using B(t)dt/t≤dB(t) and applying (24) with a=B(t1+ǫ)>0 αp(B)=lim ǫ→0Z∞ t1+ǫ B(t) tp dt t1/p≤lim ǫ→0Z∞ B−1(B(t1+ǫ)) dB(t) tp1/p (24) ≤lim ǫ→0Z∞ ¯ B−1(B(t1+ǫ)) t ¯ B(t)pd¯ B(t)1/p≤Z∞ B(1) t ¯ B(t)pd¯ B(t)1/p, where in the last step we used (17) with t=B(t1+ǫ) to conclude that ¯ B−1(B(t1+ǫ)) ≥B(t1+ǫ) t1+ǫ≥B(t1) t1≥B(1), since B(t)/tis increasing and t1≥1.  7 In this paper we will consider Bso that ¯ B(t)=A(t)=tp(1 +log+t)p−1+δ, δ > 0. Then, for 0< δ ≤1 A′(t)≤2pA(t) tt>1 and ¯ A(1) =sup t∈(0,1)(t−tp)=(t−tp)t=p−1/(p−1) =(p−1)p−p′. Thus, by the lemma kM¯ AkB(Lp′(Rn)) ≤cnZ∞ (p−1)p−p′ t A(t)!p′ A′(t)dt 1/p′ ≤cnp2 1 δ!1/p′ (25) Similarly for the smaller functional: ¯ B(t)=A(t)=tp(1 +log+t)p−1(1 +log+(1 +log+t))p−1+δδ > 0. Then, using that A′(t)≤3pA(t) tt>1, when 0 < δ ≤1 and hence by the lemma kM¯ AkB(Lp′(Rn)) ≤cnp2 1 δ!1/p′ 2.4 The iteration lemma We will need the following variation of the Rubio de Francia algorithm. Lemma 2.3. Let 1<s<∞and let v be a weight. Then there exists a nonnegative sublinear operator R satisfying the following properties: (a) h ≤R(h) (b) kR(h)kLs(w)≤2khkLs(v) (c) R(h)v1/s∈A1with [R(h)v1/s]A1≤cs′ Proof. We consider the operator S(f)=M(f v1/s) v1/s Since kMkLs∼s′, we have kS(f)kLs(v)≤cs′kfkLs(v). Now, define the Rubio de Francia operator Rby R(h)= ∞ X k=0 1 2k Sk(h) (kSkLs(v))k. It is very simple to check that Rsatisfies the required properties.  8 2.5 Two weight maximal function Our main new result is intimately related to a sharp two weight estimate for M. Theorem 2.4. Given a pair of weights u, σ and p, 1<p<∞, suppose that K=sup Q 1 |Q|ZQu(y)dy!1/pσ1/p′X,Q<∞.(26) where X is a Banach function space such that its corresponding associate space X′satisfies MX′: Lp(Rn)→Lp(Rn). Then kM(fσ)kLp(u).KkMX′kB(Lp(Rn)) kfkLp(σ)(27) In particular if X =LBwith B(t)=tp′(1 +log+t)p′−1+δ,δ > 0, then by (25) kMX′kB(Lp(Rn)) =kM¯ BkB(Lp(Rn)) ≈(p′)2(1 δ)1/p. where the last ≈is valid for δ≤1. This result together with some improvements can be found in [PR]. 3 Dyadic theory In this section we define an important class of dyadic model operators and recall a general result by which norm inequalities for maximal singular integral operators can be reduced to these dyadic operators. The result is due to Lerner [Le2], and comes from his approach to prove the A2theorem proved by the first author [H]. We say that a dyadic grid, denoted D, is a collection of cubes in Rnwith the following properties: 1) each Q∈ D satisfies |Q|=2nk for some k∈Z; 2) if Q,P∈ D then Q∩P=∅,P,or Q; 3) for each k∈Z, the family Dk={Q∈ D :|Q|=2nk}forms a partition of Rn. We say that a family of dyadic cubes S ⊂ D is sparse if for each Q∈ S, [ Q′∈S Q′(Q Q′≤1 2|Q|. Given a sparse family, S, if we define E(Q) :=Q\[ Q′∈S Q′(Q Q′, then 1) the family {E(Q)}Q∈S is pairwise disjoint 2) E(Q)⊂Q, and 3) |Q| ≤ 2|E(Q)|. If S ⊂ D is a sparse family we define the sparse Calder´on-Zygmund operator associated to Sas TSf:=X Q∈S − ZQf dx ·χQ. As already mentioned the key idea is to “transplant” the continuous case to the discrete version by means of the following theorem. 9 6 Proof of Corollary 1.6 We follow very closely the argument given in [CP1], the essential difference is that we compute in a more precise way the constants involved. We consider the set Ω = {x∈Rn:T∗(fσ)(x)>1} Then by homogeneity it is enough to prove u(Ω)1/p.1 δK(1 δ)1/p′kfkLp(σ)(37) where we recall that K=sup Qku1/pkLp(log L)p−1+δ,Q 1 |Q|ZQσdx!1/p′ <∞(38) Now, by duality, there exists a non-negative function h∈Lp′(Rn), khkLp′(Rn)=1, such that u(Ω)1/p=ku1/pχΩkLp(Rn)=ZΩ u1/phdx =u1/ph(Ω).1 εZRn|f|ML(log L)ε(u1/ph)σdx ≤1 ε ZRn|f|pσdx!1/p ZRnML(log L)ε(u1/ph)p′σdx!1/p′ , where we have used inequality (4) from Theorem 1.1 and then H¨older’s inequality. Therefore everything is reduced to understanding a two weight estimate for ML(logL)ε. We need the following Lemma that can be found in [P1] or in [CMP] Appendix A, Proposition A.1 Lemma 6.1. Given a Young function A, suppose f is a non-negative function such that kfkA,Qtends to zero as l(Q)tends to infinity. Given a >2n+1, for each k ∈Zthere exists a disjoint collection of maximal dyadic cubes {Qk j}such that for each j, ak<kfkA,Qk j≤2nak,(39) and {x∈Rn:MAf(x)>4nak} ⊂ [ j 3Qk j. Further, let Dk=SjQk jand Ek j=Qk j\(Qk j∩Dk+1). Then the Ek j’s are pairwise disjoint for all j and k and there exists a constant α > 1, depending only on a, such that |Qk j| ≤ α|Ek j|. Fix a function hbounded with compact support. Fix a>2n+1; for k∈Zlet Ωk={x∈Rn: 4nak<MAf(x)≤4nak+1}. Then by Lemma 6.1, Ωk⊂[ j 3Qk j,where kfkA,Qk j>ak. We will use a generalization of H¨older’s inequality due to O’Neil [O1]. (Also see Rao and Ren [RR, p. 64].) We include a proof for the reader’s convenience. 16 Lemma 6.2. Let A,B and C be Young functions such that B−1(t)C−1(t)≤κA−1(t),t>0.(40) Then for all functions f and g and all cubes Q, kfgkA,Q≤2κkfkB,QkgkC,Q.(41) Proof. The assumption (40) says that if A(x)=B(y)=C(z), then yz ≤κx. Let us derive a more applicable consequence: Let y,z∈[0,∞), and assume without loss of generality (by symmetry) that B(y)≤C(z). Since Young functions are onto, we can find a y′≥yand x∈[0,∞) such that B(y′)=C(z)=A(x). Then (40) tells us that yz ≤y′z≤κx. Since Ais increasing, it follows that Ayz κ≤A(x)=C(z)=max(B(y),C(z)) ≤B(y)+C(z).(42) Let then s>kfkBand t>kgkC. Then, using (42), − ZQA|fg| κst ≤ − ZQB|f| s+− ZQC|g| t≤1+1, and hence − ZQA|fg| 2κst≤1 2− ZQA|fg| κst ≤1. This proves that kfgkA≤2κst, and taking the infimum over admissible sand tproves the claim.  If A(t)=t(1 +log+t)ε, the goal is to “break” MAin an optimal way, with functions Band Cso that one of them, for instance B, has to be B(t)=tp(1 +log+t)p−1+δcoming from (38). We can therefore estimate MAusing Lemma 6.1 as follows: ZRn(MA(u1/ph))p′σdx =X kZΩk (MA(u1/ph))p′σdx ≤cX k akp′σ(Ωk) ≤cX j,k akp′σ(3Qk j) ≤cX j,k σ(3Qk j)ku1/phkp′ A,Qk j . ≤cX j,k σ(3Qk j)ku1/pkp′ B,Qk j khkp′ C,Qk j , by (41). Now since ku1/pkB,Qk j≤3nku1/pkB,3Qk j, we can apply condition (38), and since the Ek j’s are disjoint, ≤cX j,k 1 |3Qk j|Z3Qk j σdxku1/pkp′ B,3Qk j khkp′ C,Qk j |Ek j| ≤Kp′X j,kZEk j MC(h)p′dx ≤Kp′ZRnMC(h)p′dx. ≤Kp′kMCkp′ B(Lp′(Rn)) ZRnhp′dx. 17 If we choose Csuch that MCis bounded on Lp′(Rn), namely it must satisfy the tail condition (21). We are left with choosing the appropriate C. Now, 1 <p<∞and δ > 0 are fixed from condition (38) but ε > 0 is free and will be chosen appropriately close to 0. To be more precise we need to choose 0< ε < δ/pand let η=δ−pε. Then A−1(t)≈t (1 +log+t)ε =t1/p (1 +log+t)ε+(p−1+η)/p×t1/p′(1 +log+t)(p−1+η)/p =B−1(t)C−1(t), where B(t)≈tp(1 +log+t)(1+ε)p−1+η=tp(1 +log+t)p−1+δ and C(t)≈tp′(1 +log+t)−1−(p′−1)η. These manipulations follow essentially O’Neil [O2] but we need to be careful with the constants. It follows at once from Lemma 2.1 that kMCkB(Lp′(Rn)) .1 η1/p′ =(1 δ−pε)1/p′, where we suppress the multiplicative dependence on p. Finally if we choose ε=δ 2pwe get the desired result: u(Ω)1/p.1 δK(1 δ)1/p′kfkLp(σ)(43) This completes the proof of part (a) of Corollary 6. To prove part (b) we combine Lerner’s theorem 3.1, kT∗fkLp(u)≤cTsup S⊂D kTSfkLp(u), with the characterization of the two-weight inequalities for TSfrom [LSU] by testing conditions: a combination of their characterizations for weak and strong norm inequalities shows in particular that kTS(.σ)kLp(σ)→Lp(u)hkTS(.σ)kLp(σ)→Lp,∞(u)+kTS(.u)kLp′(u)→Lp′,∞(σ) Now, as it is mentioned after the statement of Corollary 1.6, since TSsatisfies estimate (4) (see (35)) we can apply the same argument as the just given to both summands and since that estimate has to be independent of the grid and we must take the two weight constant Kover all cubes, not just for those from the specific grid. This concludes the proof of the corollary. 7 Conjectures A conjecture related to Corollary 1.6 is as follows: Conjecture 7.1. Let T∗, p,u, σ as above. Let X is a Banach function space so that its corresponding associate space X′satisfies MX′:Lp′(Rn)→Lp′(Rn). If K=sup Qku1/pkX,Q 1 |Q|ZQσdx!1/p′ <∞,(44) 18 then kT∗(fσ)kLp,∞(u).KkMX′kB(Lp′(Rn))kfkLp(σ).(45) As a consequence, if Y is another Banach function space with MY′:Lp(Rn)→Lp(Rn)and if K=sup Qku1/pkX,Q 1 |Q|ZQσdx!1/p′ + 1 |Q|ZQudx!1/p kσ1/p′kY,Q<∞,(46) then kT∗(fσ)kLp(u).KkMX′kB(Lp′(Rn)) +kMY′kB(Lp(Rn))kfkLp(σ)(47) This is a generalization of the conjecture stated in [CRV] which arises from the work [CP1,CP2]. We also refer to the recent papers [La,TV] for further results in this direction. If we could prove this, we would get as corollary: Corollary 7.2. kT∗kB(Lp(w)) ≤c[w]1/p Ap[w]1/p′ A∞+[σ]1/p A∞(48) This last result itself is known [HL] (see also [HLP] for a more general case), but not as a corollary of a general two-weight norm inequality. References [ACM] T. C. Anderson, D. Cruz-Uribe and K. Moen, Logarithmic bump conditions for Calder´on– Zygmund operators on spaces of homogeous type, to appear Publicaciones Mathematique. [CAR] J. M. Conde-Alonso and G. 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Box 68 (Gustaf H¨allstr¨omin katu 2b) FI-00014 University of Helsinki, Finland E-mail address:[email protected] Department of Mathematics University of the Basque Country UPV/EHU, Leioa, Spain, and IKERBASQUE, Basque Foundation for Science, Bilbao, Spain E-mail address:[email protected] 21