Norm inequalities for the minimal and maximal operator, and differentiation of the integral

Publicacions Matem`atiques, Vol 41 (1997), 577–604. NORM INEQUALITIES FOR THE MINIMAL AND MAXIMAL OPERATOR, AND DIFFERENTIATION OF THE INTEGRAL David Cruz-Uribe, SFO, C. J. Neugebauer and V. Olesen Abstract We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse H¨older inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the integral in cases when the associated maximal operator is large, and we give a new condition for this maximal operator to be weak (1,1). 1. Introduction Given a real-valued, measurable function fon Rn, the HardyLittlewood maximal function of fis defined by Mf(x) = sup 1 |I|I |f|dy, where the supremum is taken over all cubes Iwhich contain xand whose sides are parallel to the co-ordinate axes. Intuitively, the maximal operator controls where the function fis large. We define an analogous operator, the minimal operator, which controls where a function is small. Given a function f, define the minimal function of fby mf(x) = inf 1 |I|I |f|dy, Keywords. Minimal operator, maximal operator, weighted norm inequalities, differentiation of the integral. 1991 Mathematics subject classifications: 42B25. 578 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen where again the infimum is taken over all cubes which contain x.By the Lebesgue differentiation theorem, if fis locally integrable then mf(x)≤|f(x)|≤Mf(x) a.e. The first two authors introduced the minimal operator in [2], where we used it to study the fine structure of functions which satisfy the reverse H¨older inequality. We also examined the one-weight, weighted norm inequalities which it satisfies. Further, we have used the minimal operator to study one-sided reverse H¨older inequalities and one-sided (Ap) weights [4], and to study the convergence of convolution operators with “bad” kernels [5]. In a forthcoming paper [3] we apply the weighted norm inequalities for the minimal operator (including the results in this paper) to study the weighted norm inequalities for the geometric maximal operator M0f(x) = sup I exp 1 |I|I log |f(y)|dy, which have been studied previously by Sbordone and Wik [20] and Yin and Muckenhoupt [27]. The purpose of this paper is to study the two-weight, weighted norm inequalities for the minimal operator. As we showed in [2], this problem is complicated by the fact that, unlike the maximal operator, the natural domain of the minimal operator is not Lp:iff∈Lp,p<∞, then mf≡0. Following up on the idea that the minimal operator controls where a function is small, we showed there that the minimal operator has two natural domains: functions fsuch that either log |f|or 1/f is in Lp. The latter case is much more interesting, and the main result in this paper is the following theorem. Theorem 1.1. Given p>0and a pair of non-negative weights (u, v) on R, the following are equivalent: 1.(u, v)∈(Wp): there exists a constant Csuch that given any interval I⊂R, 1 |I|I udx≤C1 |I|I v1/(p+1) dxp+1 ; 2. the weak-type inequality u({x:mf(x)<1/t})≤C tpR v |f|pdx holds for every fsuch that 1/f is in Lp(v); The minimal and maximal operator 579 3. (u, v)∈(Wp)∗: there exists a constant Csuch that given any interval I⊂R, I u m(σ/χI)pdx ≤CI σ dx, where σ=v1/(p+1); 4. the strong-type inequality R u (mf)pdx ≤CR v |f|pdx holds for every fsuch that 1/f is in Lp(v). The equivalence of (1) and (2) and of (3) and (4) in Theorem 1.1 is very similar to results which hold for the maximal operator, with the Muckenhoupt class (Ap) replaced by (Wp) and the Sawyer class (Sp) replaced by (Wp)∗. (Recall that (Sp) is defined by I M(v1−pχI)pudx≤CI v1−pdx < +∞.) However, this theorem has two surprising features. First, the norm inequalities hold for all p>0, and not just for p≥1. For the maximal operator, it is well known that if f∈L1then Mf is never in L1unless f≡0. (See, for example, Stein [23].) Second, the weight classes (Wp) and (Wp)∗are the same. For the maximal operator, (Ap) is strictly larger than (Sp). (For an elementary example, see Garc´ıa-Cuerva and Rubio de Francia [6].) We have only been able to prove Theorem 1.1 in full generality on the real line. The central difficulty is that the minimal operator is not equivalent to the centered minimal operator, mc, defined as the minimal operator except the infimum is taken over cubes whose center is x. (An easy example is given by f(x)=ex. By contrast, the maximal operator and the centered maximal operator are equivalent.) Therefore, when proving weighted norm inequalities we are unable to use the Besicovitch covering lemma. On Rthere are special covering lemmas which let us overcome this problem. In higher dimensions we can prove some partial extensions, but only if we assume doubling conditions on the weights. This leads to two open questions. Question 1.2. Does Theorem 1.1 hold in higher dimensions in full generality, or are doubling conditions necessary? For the possible necessity of doubling conditions in higher dimensions, it is worthwhile to consider the behavior of the weighted maximal function on Rand Rn,n>1. (See Sj¨ogren [22] and Vargas [26].) 580 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen Question 1.3. What are the weights which govern the weighted norm inequalities for the centered minimal operator? Following the proof of Theorem 1.1, it is easy to see that for p>0, a necessary condition on (u, v) for the centered minimal operator to satisfy a weak-type inequality is a “weak” (Wp) condition: for every cube I, 1 |I|I udx≤C1 |2I|2I v1/(p+1) dxp+1 , where 2Iis the cube with the same center as Iand twice the side-length. We conjecture that this condition is also sufficient. If true, this result would be especially interesting in the one-weight case, since in that case this condition is equivalent to a weak reverse H¨older inequality which appears frequently in the study of PDE’s and quasi-conformal mappings. (See, for example, Sawyer [19] or Iwaniec and Nolder [10].) The proof of Theorem 1.1 is organized as follows: in Section 2 we show that the (Wp) condition is equivalent to the weak-type inequality; in Section 3 we show that the (Wp)∗condition is equivalent to the strong-type inequality; and in Section 4 we show that the classes (Wp) and (Wp)∗are the same. At the end of each section we describe briefly how these results can be extended to higher dimensions. At the end of Section 3 we also show that, again unlike the maximal operator, mixed norm inequalities hold only trivially for the minimal operator. (See Sawyer [18].) In Section 5 we examine the properties of the (Wp) weights, especially their relation to the (Ap) weights. We have the formal relationship that (Wp)=(A−p), that is, the (Ap) condition with preplaced by −p,p>0. We also have the chain of proper inclusions (A1)⊂(Ar)⊂(As)⊂(W∞)⊂(Wp)⊂(Wq), for all 1 <r<s,0<q<p. (The class (W∞) will be defined below.) These results are mostly elementary —the full structure of (Wp) weights remains to be determined. Sections 6 and 7 contain applications of a generalization of the minimal operator to problems of differentiation of the integral. Let B={Bx:x∈Rn}be a differentiation basis: for each x,Bx={Ejx }, where the sets Ejx →x—that is, for every >0, Ejx ⊂{t:|t−x|≤} for all jsufficiently large. Note that this is more general than the usual definition since we do not assume that x∈Ejx or that Ejx is open. Define the maximal operator relative to Bby MBf(x) = sup Bx 1 |Ejx |Ejx |f|dy. The minimal and maximal operator 581 We will assume that Bis such that MBis measurable; this property is easy to check for the classical differentiation bases. (See de Guzm´an [7].) If MBis weak-type (p, p) for some 1 ≤p<∞, then it is known that B differentiates Lp: given a function fin Lp, the exceptional set EB(f)=x: lim sup j→∞  1 |Ejx |Ejx fdy−f(x)>0 has measure zero. (See de Guzm´an [7], [8].) In Section 6 we consider the case when MBis weak-type (p, p) but fis not in Lp. Using the minimal operator, we develop a type of convergence such that if a sequence {gk}converges to fthen the measure of the exceptional set EB(f)is controlled by the measures of the EB(gk)’s. This can be thought of as a generalization of a result by Hayes and Pauc [9] on the “transmission” of differentiability. (Also see de Guzm´an [7], [8].) Our approach appears to be different from earlier work in this area and needs to be explored further. Because of the importance of the assumption that MBbe weak-type (p, p) for some p, in Section 7 we give a new result on the weak-type behavior of MBin the case where each Bxis the translate by xof a fixed collection {Ej}. As a corollary we generalize results of Nagel and Stein [16], [24]: if the sets Ej={x:|x−xj|≤rj}tend to zero (whether or not |xj|/rj→∞) then there is a subsequence B={Ejk}such that the maximal operator MBis weak (1,1). Throughout this paper all notation is standard or will be defined as needed. All cubes are assumed to have their sides parallel to the coordinate axes. Given a Borel set Eand a function f, let |E|denote the Lebesgue measure of E,f(E)=Efdxand E(f)=f(E)/|E|.By f/χEwe denote the function equal to fon Eand infinity elsewhere. The letter Cwill denote a positive constant whose value may change at each appearance. Finally, we want to thank the referee for the many insightful corrections and comments contained in the report. 2. Weak-Type Norm Inequalities In this section we prove that statements (1) and (2) in Theorem 1.1 are equivalent. The proof requires two classical covering lemmas which are special to R. We sketch their proofs as a convenience to the reader. Lemma 2.1. Let Fbe a collection of intervals in R. Then there exists a countable subcollection F0such that {I:I∈F}={I:I∈F 0}. 582 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen Proof: If I∈F, let Iodenote the interior of I. Then we can find a countable subcollection F1such that {Io:I∈F}={Io:I∈F 1}. To finish the proof, note that the set {I:I∈F}\{Io:I∈F}must be countable. Lemma 2.2. Let Fbe a finite collection of intervals in R. Then there exist two subcollections F1and F2, each consisting of pairwise disjoint intervals, such that {I:I∈F}={I:I∈F 1∪F 2}. Proof: Let F={Ij,1≤j≤N}. Without loss of generality, we may assume that each Ijcontains a point xjnot contained in any other element of F. By re-ordering the elements of Fwe may also assume that x1<x 2<···<x N. Now let F1={I1,I 3,...},F2={I2,I 4,...}. To prove that (1) implies (2), fix non-negative fsuch that 1/f ∈Lp(v). We first assume that f(x) is everywhere positive. For each t>0 let Et={x:mf(x)<1/t}. Then for each x∈Etthere exists an interval Ixcontaining xsuch that (1) 1 |Ix|Ix fdy<1/t. Clearly Ix⊂Et,soEtis equal to the union of the Ix’s. By Lemma 2.1 {Ix}has a countable subcollection {Ij}whose union is Et. Let EN={Ij:1≤j≤N}and apply Lemma 2.2 to this finite collection. This gives us a disjoint subcollection FNof {Ij:1≤j≤N}such that u(EN)≤2 I∈FN u(I). If we combine this with inequality (1) and the (Wp) condition we see that, since f(I)>0, u(EN)≤2 tp I∈FN u(I)1 |I|I fdx −p ≤C tp I∈FNI v1/(p+1) dxp+1 I fdx −p . By H¨older’s inequality, I v1/(p+1) dx =I v1/(p+1) fp/(p+1) ·fp/(p+1) dx ≤I v fpdx1/(p+1) I fdx p/(p+1) . The minimal and maximal operator 583 If we combine this with the fact that the intervals in FNare disjoint, we see that u(EN)≤C tpR v fpdx. The constant Cis independent of N, so the weak-type inequality follows if we let Ntend to infinity. Now for arbitrary fdefine the sequence fn=f+1/n. Then each fn is strictly positive and 1/fn∈Lp(v)if1/f is. Further, a straightforward argument shows that the mfn(x)’s decrease to mf(x). By the above argument the weak-type inequality holds for each fn, so by the monotone convergence theorem it holds for arbitrary f. To show that (2) implies (1) is straightforward. Fix an interval Iand let f=v1/(p+1)/χI.Ifv1/(p+1)(I)=+∞then (1) is immediate. If it is finite, then for x∈I, mf(x)≤1 |I|I v1/(p+1) dx ≡1/t. If we substitute this into the weak-type inequality we get u(I)≤C1 |I|I v1/(p+1) dxp ·I v1−p/(p+1) dx, and this is the (Wp) condition. Remark. In higher dimensions, the (Wp) condition is still necessary for the weights (u, v) to satisfy the weak-type inequality; the above proof goes through without change. We can prove that it is sufficient if we assume that uis a doubling weight: there exists a constant Csuch that u(2I)≤Cu(I) for every cube I. To adapt the proof, note that Et⊂{2Ix:x∈Et}, and hence by the Besicovitch covering theorem Et⊂∪2Ij, where the Ij’s have bounded overlap and tIjfdx≤|Ij|. The rest of the proof goes through without change. 3. Strong-Type Norm Inequalities In this section we prove that conditions (3) and (4) in Theorem 1.1 are equivalent. To do so, we need to introduce two auxiliary operators. Let σbe a Borel measure on R. Define the weighted maximal and minimal operators Mσf(x) = sup 1 σ(I)I |f|dσ, mσf(x) = inf 1 σ(I)I |f|dσ, 584 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen where the supremum and infimum are taken over all Icontaining xsuch that σ(I)>0. For 1 <p≤∞,||Mσf||p,σ ≤Cp||f||p,σ (See Sj¨ogren [22]. Also, this can be proved by using Lemma 2.2 to obtain a weak (1,1) inequality and then using interpolation to get the strong-type inequality.) We will use this fact to prove a norm inequality for the weighted minimal operator. Lemma 3.1. There exists a constant Csuch that for every 0<p<∞ and fsuch that 1/f ∈Lp(σ), R dσ (mσf)p≤CR dσ |f|p. Proof: Fix 0 <p<∞and r>1 such that (r−1)p=2. ByH¨older’s inequality, (mσf)−1≤Mσ(f1−r)r−1. Then by the above remark, dσ (mσf)p≤Mσ(f1−r)(r−1)pdσ ≤C(|f|1−r)(r−1)pdσ =Cdσ |f|p. We also need to show that the (Wp)∗condition implies an apparently stronger condition. Lemma 3.2. If (u, v)∈(Wp)∗then there exists a constant Csuch that if Fis a finite union of intervals, then F u m(σ/χF)pdx ≤Cσ(F). Proof: If Iand Jare disjoint, closed intervals, then for each x∈I, m(σ/χI∪J)(x)=m(σ/χI)(x). For if Kis an interval, x∈Kand |K\I|>0, then 1 |K|K σ χI∪J =∞. Therefore, when calculating m(σ/χI∪J)(x) for x∈I, we can restrict ourselves to intervals Ksuch that K⊂I. Hence I∪J u m(σ/χI∪J)pdx =I u m(σ/χI)pdx +J u m(σ/χJ)pdx ≤C(σ(I)+σ(J)) =Cσ(I∪J). The minimal and maximal operator 585 In general, if Fis a finite union of intervals then Fis (up to a set of measure zero) the finite union of disjoint, closed intervals, and the proof follows by induction. We can now prove the desired result. For any p>0, the proof that (4) implies (3) is essentially the same as the proof that (2) implies (1): let f=v1/(p+1)/χIand substitute this into the strong-type inequality. The (Wp)∗condition follows at once. The proof that (3) implies (4) is similar to the proof of the strongtype inequality for the maximal operator given by Jawerth [11]. We first assume that vis everywhere positive; we treat the general case at the end. Fix p>0 and fnon-negative such that 1/f ∈Lp(v). For each integer k, let Ak={x:2 −(k+1) ≤mf(x)<2−k}and let Kkbe a compact subset of Ak. We can cover Kkby a finite collection of intervals Ik j,1≤j≤jk, such that 1 2k+1 ≤1 |Ik j|Ik j fdx< 1 2k. By induction, define the disjoint sets Ek 1=Ik 1∩Kk,Ek 2=(Ik 2\Ik 1)∩ Kk,... . Then Kk=∪Ek j. Given an integer N>0, we have ∪N −NKk u (mf)pdx = k,j Ek j u (mf)pdx ≤2p k,j 2kpu(Ek j) ≤2p k,j u(Ek j)|Ik j|pIk j fdx −p . Since vis positive, σ(Ik j)>0. Further, by H¨older’s inequality, σ(Ik j)≤R v fpdx1/(p+1) Ik j fdx p/(p+1) ≤R v fpdx1/(p+1) (2−k|Ik j|)p/(p+1) <+∞. 592 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen averages are taken over sets which are the finite union of cubes. Minor modifications to the proof in Section 2 shows that this condition is also necessary, but it is not clear if it is equivalent to the (Wp) condition. These modifications give a version of the equivalence of the (Wp) and (Wp)∗conditions in higher dimensions with the assumption that σis a doubling weight. As we noted earlier, it is not clear how this relates to the assumption (used to extend the weak-type inequality to higher dimensions) that uis a doubling weight. 5. The Structure of (Wp)Weights In this section we gather together a number of observations about the structure of (Wp) weights and their relationship to (Ap) weights. First, we have the following inclusion. Theorem 5.1. If (u, v)∈(Ap),1<p<∞, then (u, v)∈(Wq)for all q>0, with constant independent of q. Conversely, if u=vthen for every q>0,(Wq)=(A∞). Proof: By H¨older’s inequality, for all r>1,( mf)−1≤M(f1−r)r−1. Fix rsuch that q(r−1)=p0>p. Since (u, v)∈(Ap) implies that (u, v)∈(Sp0), u (mf)qdx ≤M(f1−r)(r−1)qudx ≤C(|f|1−r)(r−1)qvdx≤Cv |f|qdx. The constant Cdepends on the (Ap) constant of (u, v) and is independent of q. The proof of the converse is found in Theorem 3.1 of [2]. For completeness we include the proof here. If (v,v)∈(Wp) then v1/(p+1) satisfies a reverse H¨older inequality with exponent p+ 1. But then by a result of Str¨omberg and Wheeden [25], v=(v1/(p+1))p+1 ∈(A∞). In the two-weight case the converse of Theorem 5.1 is false. A simple counter-example on the real line is given by the pair (e|x|,e 2|x|). In fact, between the (Ap) and (Wq) classes we can interpolate another class of weights. We say that the pair of weights (u, v)isin(W∞) if it satisfies the two-weight, reverse Jensen inequality: there exists a constant Csuch that for every interval I, 1 |I|I udx≤Cexp 1 |I|I log vdx . The minimal and maximal operator 593 (If u=vthis inequality characterizes (A∞). See Garc´ıa-Cuerva and Rubio de Francia [6].) This is the formal limit of the (Ap) condition as p→∞and the formal limit of the (Wq) condition as q→∞. Further, it follows at once from Jensen’s inequality that (Ap)⊂(W∞)⊂(Wq) for all p>1 and all q>0. Both of these inclusions are proper: the pair (e|x|,e 2|x|)isin(W∞) but not in any (Ap) class; the pair (e|x|,e 3|x|/2)is in (Wq) for all q>0 but is not in (W∞). By H¨older’s inequality the (Wp) classes are themselves nested: (Wp)⊂ (Wq) for all p>q>0. This inclusion is proper. To show this we will actually show more: we will construct an example to show that for any p>0 there exists (u, v)∈(Wp) such that (u, v)/∈(Wp+) for any >0. It will suffice to construct an example on [0,∞) since we can extend it to Ras an even function. Let vbe an increasing function on [0,∞) such that for any r>0, vr is not a doubling weight. (For example, let v(x)=e−1/x.) Define uby the integral equation 1 tt 0 udx=1 tt 0 v1/(p+1) dxp+1 . We claim that (u, v)∈(Wp). (Intuitively, uis the largest function such that (u, v)isin(Wp).) To see this, fix I=[a, b]⊂[0,∞). Then for all t>0, m(v1/(p+1)/χI)(t)≥m(v1/(p+1))(t) =1 tt 0 v1/(p+1) dx =(U(t)/t)1/(p+1), where U(t)=t 0udx. Hence, integrating by parts we see that I u m(v1/(p+1)/χI)pdx ≤I xp/(p+1)u Up/(p+1) dx =(p+1)xp/(p+1)U(x)1/(p+1) b a−pI U1/(p+1) x1/(p+1) dx ≤(p+1)x 0 v1/(p+1) dx b a =(p+1)I v1/(p+1) dx. Therefore (u, v)isin(Wp)∗and so in (Wp). 594 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen However, (u, v) cannot be in (Wp+) for any >0. For if it were, then for all t>0 we would have the inequality 1 tt 0 v1/(p+1) dxp+1 ≤C1 tt 0 v1/(p++1) dxp++1 . Since vis increasing, by Lemma 7.1 of Cruz-Uribe [1], this reverse H¨older type inequality implies that v1/(p++1) is in (A∞) and so a doubling weight. However this contradicts our choice of v,so(u, v) cannot be in (Wp+). The property that (Wp) implies (Wp+) is the analogue of the central property of (Ap) weights when u=v; initially we conjectured that it held for all (Wp) weights as well. This led to the following question. Question 5.2. For which pairs (u, v)∈(Wp)does there exist an >0 such that (u, v)∈(Wp+)? The analogous result is known for the maximal operator: see Leckband and Neugebauer [13], [14]. Based on their work we conjecture that this problem is related to weighted norm inequalities for the iterated minimal operator. We can summarize the above results in the chain of proper inclusions (A1)⊂(Ar)⊂(As)⊂(W∞)⊂(Wp)⊂(Wq), 1<r<s,0<q<p. As we noted in the Introduction, we have the formal relationship that (Wp)=(A−p), (i.e. the (Ap) condition with p replaced by −p) for all p>0. This lets us extend the chain of (Ap) inclusions to negative indices. (There is no similar relationship between the (Sp) and (Wp)∗conditions. A straightforward calculation shows that the conditions (S−p) and (W−p)∗,p>0, are both equivalent to the class {(u, v):u≤Cv}.) Finally, we note that while it is easy to show that (u, Mu)∈(A1) for any u, and so (u, Mu)∈(Sp) for any p>1, the analogous result is not true for the (Wp) classes. Let v(x)=e|x|. Then an easy calculation shows that (mv,v) is not in (Wp) for any p>0. While there ought to be some way of constructing (Wp) weights using the minimal operator, it is unclear how to go about it. 6. The Minimal Operator and Differentiability of the Integral In this section we apply the minimal operator to problems of differentiation of the integral in Rn. For completeness we repeat the definitions The minimal and maximal operator 595 given in the Introduction. Let B={Bx:x∈Rn}be a differentiation basis: for each x,Bx={Ejx }where the sets Ejx →x—that is, for every >0, Ejx ⊂{t:|t−x|≤}for all jsufficiently large. Note that this is more general than the usual definition since we do not assume that x∈Ejx or that Ejx is open. Define the maximal operator relative to Bby MBf(x) = sup Bx 1 |Ejx |Ejx |f|dy. We will assume that Bis such that MBis measurable; this property is easy to check for the classical differentiation bases. (See de Guzm´an [7].) If MBis weak-type (p, p) for some 1 ≤p<∞, then it is known that B differentiates Lp: given a function fin Lp, the exceptional set EB(f)=x: lim sup j→∞  1 |Ejx |Ejx fdy−f(x)>0 has measure zero. (See de Guzm´an [7], [8].) Throughout this section we assume that MBis weak-type (p0,p 0), for some p0≥1. We are interested in the differentiability of fdxwhen f is not in Lp0(that is, when MBfis too large to give any information). In particular, we want to characterize the class of functions which B differentiates. To state our main result we need to define an (A2)-type condition relative to a differentiation basis. We say that a function wis in (A2)B if A2B(w)≡sup j,x 1 |Ejx |Ejx wdy·1 |Ejx |Ejx w−1dy < ∞. (By convention we assume that 0 ·∞= 0.) Theorem 6.1. Suppose that MBis weak-type (p0,p 0)for some p0≥1.Letfbe a non-negative function on Rnsuch that 1/f ∈Lp(Rn) for some 0<p<∞. Suppose that {gk}is a sequence of non-negative, measurable functions such that: 1/gkconverges to 1/f in the metric of Lp; and A2B(|f−gk|)≤C<∞for all k.If|EB(f)|>λthen for all >0there exists k=k(λ, )such that |EB(gk)|>λ−. This theorem has the following immediate corollary. Corollary 6.2. Under the hypotheses of Theorem 6.1, if |EB(gk)|=0 for all kthen |EB(f)|=0. Before proving Theorem 6.1 we make the following observations. 596 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen First, If |EB(f)|= 0 then the existence of a sequence of gk’s which converge in the specified manner is immediate: let gk=f+1/k. More generally, if φis in (A2)Bthen the sequence gk=f+φ/k converges in this manner to f. Second, Theorem 6.1 is an example of the “transmission” of differentiability. A classical result of this kind is due to Hayes and Pauc [9]: if B is a density basis, Bdifferentiates fand |g|≤fthen Bdifferentiates g. To see the relation between their result and Theorem 6.1, let gkbe a sequence which decreases monotonically to fand such that A2B(|f−gk|)is uniformly bounded. By the monotone convergence theorem the sequence {1/gk}converges to 1/f in Lp, so by Theorem 6.1 Bdifferentiates fif it differentiates the gk’s. On the other hand, if Bis a density basis, then by the theorem of Hayes and Pauc, we have the same conclusion without having to assume the (A2)Bcondition. Thus Theorem 6.1 can be thought of as a generalization of their result, but the exact role of the (A2)Bcondition is not well understood. Third, if fis a continuous function then EB(f) is empty. Thus, via Moore-Smith convergence techniques (see Kelley [12]) we can use the type of convergence defined in Theorem 6.1 to construct a topological space in which all functions in the closure of the continuous functions are differentiated by B. It is reasonable to conjecture that this set contains all functions which Bdifferentiates. However, we have not been able to prove this. Moreover, this topological space is not a topological vector space and it does not seem to correspond to any well-known space. Again the role of the (A2)Bcondition is not well understood. To prove Theorem 6.1 we need a definition and two preliminary lemmas. Given a function fdefine the minimal function of fwith respect to Bby mBf(x) = inf Bx 1 |Ejx |Ejx |f|dy. Lemma 6.3. There exists a constant Csuch that for all 0<p<∞, Rn dx mB(f)p≤CRn dx |f|p. Proof: Since we are assuming that MBis weak-type (p0,p 0), we know that it is strong-type (q,q) for q>p 0. Using this the proof is identical to that of Theorem 5.1. Lemma 6.4. Let f,g be functions such that f≥1,g≥1and The minimal and maximal operator 597 A2B(|f−g|)≤C0<∞. Then (3) 1 |Ejx |Ejx fdy −1 −1 |Ejx |Ejx g dy−1 ≤C01 |Ejx |Ejx fg |f−g|1/3 dy−3 , where if g=+∞we define fg/|f−g|=f. Proof: We first consider the case when 0<Ejx fdx<+∞and 0 <Ejx g dx < +∞. In this case it will suffice to show that (4) 1 |Ejx |Ejx fg |f−g|1/3 dy3 1 |Ejx |Ejx fdy−1 |Ejx |Ejx g dy is less than or equal to C0 1 |Ejx |Ejx fdy·1 |Ejx |Ejx g dy. If we apply H¨older’s inequality twice to (4) we see that it is dominated by 1 |Ejx |Ejx fdy  1 |Ejx |Ejx g dy 1 |Ejx |Ejx dy |f−g| 1 |Ejx |Ejx |f−g|dy, and by our hypothesis the product of the last two factors is at most C0. Now suppose that neither fnor gis integrable on Ejx . Then the left-hand side of (3) is 0 so there is nothing to prove. Finally, suppose that Ejx fdx<∞but Ejx g dx =+∞. 598 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen Then Ejx |f−g|dx =+∞, so by the (A2)Bcondition Ejx dx |g−f|=0. Hence |g−f|=+∞almost everywhere on Ejx and, since we assume that fis finite almost everywhere on Ejx , this implies that g=+∞ almost everywhere on Ejx . But then inequality (3) reduces to H¨older’s inequality. Remark. The proof of Lemma 6.4 makes strong use of the (A2)B condition: we cannot replace it with the weaker condition (Ap)B,p>2. We can now prove Theorem 6.1. Since EB(f)=EB(f+ 1) and 1/(f+1) ∈Lpif 1/f is, we may assume without loss of generality that f≥1. For i>0 define Ei=   x: lim sup j1 |Ejx |Ejx fdy −1 −1 f(x) >1/i   . Then the Ei’s are nested and their union is all of EB(f). Therefore we may choose isufficiently large that |Ei|>λ. Now for any k,by Lemma 6.4 lim sup j1 |Ejx |Ejx fdy −1 −1 f(x) ≤lim sup j1 |Ejx |Ejx fdy −1 −1 |Ejx |Ejx gkdy−1 + lim sup j1 |Ejx |Ejx gkdy−1 −1 gk(x) + 1 gk(x)−1 f(x) ≤CmBfgk |f−gk|1/3 (x)−3 + lim sup j1 |Ejx |Ejx gkdy−1 −1 gk(x) + 1 gk(x)−1 f(x) =Ak(x)+Bk(x)+Ck(x). The minimal and maximal operator 599 Therefore Ei⊂{x:Ak(x)>1/3i}∪{x:Bk(x)>1/3i}∪{x:Ck(x)>1/3i}, so by Lemma 6.3 (with exponent 3p), |Ei|≤(3iC)pRn dy mB|fgk| |f−gk|1/33p +|E(gk)|+(3i)pRn 1 gk −1 f p dy ≤(3iC)pRn dy |fgk| |f−gk|p+|E(gk)|+(3i)pRn 1 gk −1 f p dy =(3i)p(Cp+1)Rn 1 gk −1 f p dy +|E(gk)|. Now choose kso large that the first term is less than , and the proof is complete. 7. Weak-Type Inequalities for the Maximal Operator In this section we give a condition on a collection of sets B={Ej} such that Ej→0 so that Bhas a subsequence {Ejk}such that the basis B∗={Bx={Ejk+x}} differentiates Lp,p≥1. (For brevity we will write B∗={Ejk}.) In order to state our main result, we need a definition. If E⊂Rn, define the set E∗=E−E={x−y:x, y ∈E}. The measure of E∗could be much larger than that of E, though in some cases it is not. For example, if Eis convex, then as a consequence of the Brunn-Minkowski theorem, |E∗|≤2n n|E|. (For details, see Schneider [21, p. 409].) We show that if the Ej’s are such that |E∗ j|≤C|Ej| then a “good” subsequence can always be extracted from it. More precisely, we prove the following. Theorem 7.1. Let B={Ej}be a collection of sets such that Ej→0. Then there exists a subsequence B∗={Ejk},jk<j k+1 for all k, such that the maximal function M∗f(x) = sup k 1 |E∗ jk|Ejk+x |f|dy 600 D. Cruz-Uribe, C. J. Neugebauer, V. Olesen is weak (1,1). In particular, if there exists a constant Csuch that |E∗ j|≤C|Ej|for all j, then the maximal operator MB∗is weak (1,1). Thus it is also weak (p, p)for p≥1, and so B∗differentiates Lp. Before proving Theorem 7.1, we first make some observations. Consider the special case where Ej=B(xj,r j), that is, each Ejis a ball of radius rjand center xj. If the ratio |xj|/rjis uniformly bounded, then the balls approach the origin non-tangentially. This is the classical case, and the maximal operator MBis dominated by the Hardy-Littlewood maximal operator and so is always weak (1,1). (See Stein [24].) If the ratio |xj|/rj→∞then the balls approach the origin tangentially. In this case there is always a subsequence of the balls which forms a differentiation basis. This extends results by Nagel and Stein [16], [24], who showed that there exists a differentiation basis of balls approaching the origin non-tangentially. The existence of such a subsequence is the best possible result, as the following example shows. We will construct a sequence of balls {Bj}, Bj→0, and a function f∈L∞such that 1 |Bj|Bj f(x+y)dy → f(x), for all xin a set of positive measure. Let Cbe a compact, nowhere dense subset of R2of positive measure. We construct the balls {Bj}as follows. Fix n>0 and let σ1,... ,σ kbe a finite set of balls of radius 1/n such that C⊂σ1∪···∪σk. For 1 ≤i≤k, let σ ibe the ball concentric with σiof radius 1/(n+ 1). Now choose a closed ball Aiin (σi\σ i)osuch that Ai∩C=∅. For each x∈σi∩Cthere exists ηx>0 such that, if |z−x|≤ηxthen Ai+(z−x)⊂(σi\σ i)oand (Ai+(z−x)) ∩C=∅. For each i, the collection of balls {σ(x, ηx):x∈C∩σi}covers C∩σi, so there exists a finite subcover {σ(xj,η j):1≤j≤ki}. We form the collection of balls {Ai−xj,1≤j≤ki,1≤i≤k}. The union of all such sets of balls over all ngives a sequence of balls {Bj}. Now for each x∈Cthere is a subsequence Bji→0 such that (x+Bji)∩C=∅. For at the n-th stage of our construction x∈σi for some 1 ≤i≤k,sox∈σ(xj,η xj). Hence Ai−xj+xis a ball in σi\σ iwhich is disjoint from C. Further, Ai−xj⊂{|x|≤2/n}.Now let f=χC; then 1 |Bji|Bji f(x+y)dy =0. This shows that {Bj}is the desired sequence. The minimal and maximal operator 601 Finally, we note that we have relied on the fact that balls are convex. This raises the following question, for which we have no good intuition as to the correct answer. Question 7.2. Given a collection of sets B={Ej},Ej→0, does a “good” subsequence always exist without the hypothesis that |E∗ j|≤C|Ej|? We now prove Theorem 7.1. To do so we need two lemmas. Lemma 7.3. If E⊂Rnis compact and |E∗|>0, then there exists δ>0such that |(Bδ∪E)∗|≤3|E∗|, where Bδ={x:|x|≤δ}. Proof: Let δj=1/j. Then |(Bδj∪E)∗|→|∩ j(Bδj∪E)∗|. Furthermore,  j (Bδj∪E)∗=({0}∪E)∗. One inclusion is obvious. To see the reverse inclusion, let z∈∩ j(Bδj∪ E)∗. Then for each j,z=xj−yjfor some xj,y j∈Bδj∪E. There are three cases. If xj,y j∈Efor some jthen z∈E∗⊂({0}∪E)∗. If there exist an infinite number of j’s such that xj∈Bδj, then (by passing to a subsequence) xj→0. Since Eis compact (again passing to a subsequence) yj→y, where y∈{0}∪E. Hence z=0−y∈({0}∪E)∗. By symmetry the same argument holds if an infinite number of yj’s are in Bδj. To complete the proof, note that |({0}∪E)∗|≤|E∗|+2|E|<3|E∗|, and so the desired inequality holds if we choose δsufficiently small. Lemma 7.4. Let {Ej}be a sequence of sets such that 0<|Ej|<∞, and let j=|Ej|/2j.IfCj⊂Ejwith |Cj|≤j, then M∗f(x) = sup j 1 |Ej|Cj+x |f|dy satisfies ||M∗f||p≤||f||p,p≥1. Proof: Clearly M∗f(x)≤ j≥1 1 |Ej|Cj |f(x+y)|dy, so by Minkowski’s integral inequality we see that ||M∗f||p≤||f||p|Cj| |Ej|≤||f||p.