The Stein Strömberg Covering Theorem in metric spaces
aa r X i v : . [ m a t h . C A ] J a n THE STEIN STR ¨OMBERG COVERING THEOREM IN METRIC SPACES
J. M. ALDAZ
Abstract.
In [NaTa] Naor and Tao extended to the metric setting the O ( d log d ) boundsgiven by Stein and Str¨omberg for Lebesgue measure in R d , deriving these bounds first froma localization result, and second, from a random Vitali lemma. Here we show that the Stein-Str¨omberg original argument can also be adapted to the metric setting, giving a third proof.We also weaken the hypotheses, and additionally, we sharpen the estimates for Lebesguemeasure. Introduction
In [StSt], Stein and Str¨omberg proved that for Lebesgue measure in R d , and with ballsdefined by an arbitrary norm, the centered maximal function has weak type (1,1) bounds oforder O ( d log d ), which is much better than the exponential bounds obtained via the Vitalicovering lemma. Naor and Tao extended the Stein-Str¨omberg result to the metric settingin [NaTa]. There, a localization result is proven (using the notion of microdoubling, whichbasically entails a very regular growth of balls) from which the Stein-Str¨omberg bounds areobtained (using the notion of strong microdoubling, which combines microdoubling with localcomparability). Also, a second argument is given, via a random Vitali Theorem that has itsorigin in [Li].Here we note that the Stein-Str¨omberg original proof, which is shorter and conceptuallysimpler, can also be used in the metric setting, yielding a slightly more general result. We willdivide the Stein-Str¨omberg argument into two parts, one with radii separated by large gaps,and the second, with radii inside an interval, bounded away from 0 and ∞ . This will allow usto obtain more precise information about which hypotheses are needed in each case. We shallsee that under the same hypotheses used by Naor and Tao, the Stein-Str¨omberg coveringtheorem for sparse radii (cf. Theorem 4.1 below) suffices to obtain the d log d bounds in themetric setting. But Theorem 4.1 itself is presented in a more general version. In particular,one does not need to assume that the metric space is geometrically doubling.We also show that the Stein-Str¨omberg method, applied to balls with no restriction in theradii, yields the O ( d log d ) bounds in the metric context, for doubling measures where thegrowth of balls can be more irregular than is allowed by the microdoubling condition. Finally,we lower the known weak type (1,1) bounds in the case of Lebesgue measure: For d lacunarysets of radii, from ( e + 1)( e + 1) to ( e /d + 1)(1 + 2 e /d ) (to 6 in the specific case of ℓ ∞ balls), Mathematical Subject Classification. and for unrestricted radii, from e ( e + 1)(1 + o (1)) d log d to (2 + 3 ε ) d log d , where ε > d = d ( ε ) is sufficiently large.2. Notation and background material
Some of the definitions here come from [A2]; we refer the interested reader to that paperfor motivation and additional explanations.We will use B ( x, r ) := { y ∈ X : d ( x, y ) < r } to denote open balls, B ( x, r ) to denote theirtopological closure, and B cl ( x, r ) := { y ∈ X : d ( x, y ) ≤ r } to refer to closed balls. Recall thatin a general metric space, a ball B , considered as a set, can have many centers and manyradii. When we write B ( x, r ) we mean to single out x and r , speaking respectively of thecenter and the radius of B ( x, r ). Definition 2.1.
A Borel measure is τ -smooth if for every collection { U α : α ∈ Λ } of opensets, µ ( ∪ α U α ) = sup µ ( ∪ ni =1 U α i ), where the supremum is taken over all finite subcollectionsof { U α : α ∈ Λ } . We say that ( X, d, µ ) is a metric measure space if µ is a Borel measure onthe metric space ( X, d ), such that for all balls B ( x, r ), µ ( B ( x, r )) < ∞ , and furthermore, µ is τ -smooth.The assumption of τ -smoothness does not represent any real restriction, since it is consistentwith standard set theory (Zermelo-Fraenkel with Choice) that in every metric space, everyBorel measure which assigns finite measure to balls is τ -smooth (cf. [Fre, Theorem (a), pg.59]). Definition 2.2.
Let (
X, d, µ ) be a metric measure space and let g be a locally integrablefunction on X . For each x ∈ X , the centered Hardy-Littlewood maximal operator M µ isgiven by(1) M µ g ( x ) := sup { r :0 <µ ( B ( x,r )) } µ ( B ( x, r )) Z B ( x,r ) | g | dµ. Maximal operators can be defined using closed balls instead of open balls, and this doesnot change their values, as can be seen by an approximation argument. When the measureis understood, we will omit the subscript µ from M µ .A sublinear operator T satisfies a weak type (1 ,
1) inequality if there exists a constant c > µ ( { T g > s } ) ≤ c k g k L ( µ ) s , where c = c ( T, µ ) depends neither on g ∈ L ( µ ) nor on s >
0. The lowest constant c thatsatisfies the preceding inequality is denoted by k T k L → L , ∞ . Definition 2.3.
A Borel measure µ on ( X, d ) is doubling if there exists a
C > r > x ∈ X , µ ( B ( x, r )) ≤ Cµ ( B ( x, r )) < ∞ . Definition 2.4.
A metric space is D -geometrically doubling if there exists a positive integer D such that every ball of radius r can be covered with no more than D balls of radius r/ tein Str¨omberg covering theorem 3 If a metric space supports a doubling measure, then it is geometrically doubling. Regardingweak type inequalities for the maximal operator, in order to estimate µ { M f > s } , oneconsiders balls B ( x, r ) over which | f | has average larger than s . Now, while in the uncenteredcase any such ball is contained in the corresponding level set, this is not so for the centeredmaximal function. Thus, using the balls B ( x, r ) to cover { M f > s } can be very inefficient.A key ingredient in the Stein-Str¨omberg proof is to cover { M f > s } by the much smallerballs B ( x, tr ), 0 < t <<
1, something that leads to the “microdoubling” notion of Naorand Tao. We slightly modify their notation, using 1 /n -microdoubling to denote what theseauthors call n -microdoubling. Definition 2.5. ([NaTa, p. 735] Let 0 < t < K ≥
1. Then µ is said to be t -microdoubling with constant K if for all x ∈ X and all r >
0, we have µB ( x, (1 + t ) r ) ≤ KµB ( x, r ) . The next property is mentioned in [NaTa], and more extensively studied in [A2].
Definition 2.6.
A measure µ satisfies a local comparability condition if there exists a constant C ∈ [1 , ∞ ) such that for all pairs of points x, y ∈ X , and all r >
0, whenever d ( x, y ) < r , wehave µ ( B ( x, r )) ≤ Cµ ( B ( y, r )) . We denote the smallest such C by C ( µ ) or C µ . Remark 2.7. If µ is doubling with constant K then it is microdoubling and satisfies alocal comparability condition with the same constant K , while if it is t -microdoubling withconstant K and 2 ≤ (1 + t ) M , then µ is doubling and satisfies a local comparability conditionwith constant K M . Thus, the difference between doubling and microdoubling lies in the size ofthe constants, it is quantitative, not qualitative: The microdoubling condition adds somethingnew only when K < K , in which case it entails a greater regularity in the growth of themeasure of balls, as the radii increase. Likewise, bounds of the form µB ( x, T r ) ≤ KµB ( x, r )for T >
2, allow a greater irregularity in the growth of balls than standard doubling ( T = 2)or than microdoubling.We mention that while local comparability is implied by doubling, it is a uniformity condi-tion, not a growth condition. Thus, it is compatible with the failure of doubling, and even fordoubling measures, it is compatible with any rate of growth for the volume of balls. Consider,for instance, the case of d -dimensional Lebesgue measure λ d : A doubling constant is 2 d , a1 /d -microdoubling constant is e , and the smallest local comparability constant is C ( λ d ) = 1.The next definition combines the requirement that the microdoubling and the local com-parability constants be “small” simultaneously. Definition 2.8. ([NaTa, p. 737] Let 0 < t < K ≥
1. Then µ is said to be strong t -microdoubling with constant K if for all x ∈ X , all r >
0, and all y ∈ B ( x, r ), µB ( y, (1 + t ) r ) ≤ KµB ( x, r ) . J. M. Aldaz
Thus, if µ is strong t -microdoubling with constant K , then C ( µ ) ≤ K . Also, local compa-rability is the same as strong 0-microdoubling. To get a better understanding of how boundsdepend on the different constants, it is useful to keep separate C ( µ ) and K . Definition 2.9.
Given a set S we define its s - blossom as the enlarged set(3) Bl ( S, s ) := ∪ x ∈ S B ( x, s ) , and its uncentered s -blossom as the set(4) Blu ( S, s ) := ∪ x ∈ S ∪ { B ( y, s ) : x ∈ B ( y, s ) } . When S = B ( x, r ), we simplify the notation and write Bl ( x, r, s ), instead of Bl ( B ( x, r ) , s ),and likewise for uncentered blossoms. We say that µ blossoms boundedly if there exists a K ≥ r > x ∈ X , µ ( Blu ( x, r, r )) ≤ Kµ ( B ( x, r )) < ∞ .Blossoms can be defined using closed instead of open balls, in an entirely analogous way.To help understand the relationship between blossoms and balls, we include the followingdefinitions and results. Definition 2.10.
A metric space has the approximate midpoint property if for every ε > x, y , there exists a point z such that d ( x, z ) , d ( z, y ) < ε + d ( x, y ) / Definition 2.11.
A metric space X is quasiconvex if there exists a constant C ≥ x, y , there exists a curve with x and y as endpoints, such that itslength is bounded above by Cd ( x, y ). If for every ε > C = 1 + ε , then we saythat X is a length space .It is well known that for a complete metric space, having the approximate midpoint prop-erty is equivalent to being a length space. Example 2.12.
The s -blossom of an r -ball may fail to contain a strictly larger ball, even inquasiconvex spaces.For instance, let X ⊂ R be the set { } × [0 , ∪ [0 , × { } with metric defined byrestriction of the ℓ ∞ norm; then we can take C = 2. Now B ((1 , ,
1) = (0 , × { } , whilefor every r > B ((1 , , r ) = X , which is not contained in Blu ((1 , , , / Blu ((1 , , , /
6) nor Bl ((1 , , , /
6) are balls, i.e., given any x ∈ X and any r > B ( x, r ) = Blu ((1 , , , /
6) and B ( x, r ) = Bl ((1 , , , / X has the approximate midpoint property, thenblossoms and balls coincide (as we show next) so in this case considering blossoms givesnothing new. Theorem 2.13.
Let ( X, d ) be a metric space. The following are equivalent:a) X has the approximate midpoint property.b) For all x ∈ X , and all r, s > , Bl ( x, r, s ) = B ( x, r + s ) . c) For all x ∈ X , and all r > , Bl ( x, r, r ) = B ( x, r ) . tein Str¨omberg covering theorem 5 Proof.
Suppose first that X has the approximate midpoint property. Since Bl ( x, r, s ) ⊂ B ( x, r + s ) , to prove b) it is enough to show that if y ∈ B ( x, r + s ) , then y ∈ Bl ( x, r, s ), orequivalently, that there is a z ∈ X such that d ( x, z ) < r and d ( z, y ) < s . If either d ( x, y ) < s or d ( x, y ) < r we can take z = x and there is nothing to prove, so assume otherwise. Let ( ˆ X, ˆ d )be the completion of ( X, d ); then ˆ X is a length space, since it has the approximate midpointproperty. Let Γ : [0 , → ˆ X be a curve with Γ(0) = x , Γ(1) = y , and length ℓ (Γ) < r + s .Then Γ([0 , ⊂ B ( x, r ) ∪ B ( y, s ), for if there is a w ∈ [0 ,
1] with Γ( w ) / ∈ B ( x, r ) ∪ B ( y, s ),then ℓ (Γ) ≥ r + s . Now let c ∈ [0 ,
1] be the time of first exit of Γ( t ) from B ( x, r ), that is,for all t < c , Γ( t ) ∈ B ( x, r ) and Γ( c ) / ∈ B ( x, r ) . Then Γ( c ) ∈ B ( y, s ), so by continuity of Γ,there is a δ ∈ [0 , c ) such that Γ( δ ) ∈ B ( y, s ). Thus, the open set B ( x, r ) ∩ B ( y, s ) = ∅ in ˆ X .But X is dense in ˆ X , so there exists a z ∈ X such that d ( x, z ) < r and d ( z, y ) < s , as wewanted to show.Part c) is a special case of part b). From part c) we obtain a) as follows. Let x, y ∈ X ,and let r > d ( x, y ) < r . By hypothesis, y ∈ Bl ( x, r, r ) = B ( x, r ), so thereis a z ∈ X such that d ( x, z ) < r and d ( z, y ) < r . Thus, X has the approximate midpointproperty. (cid:3) Example 2.14.
Let X be the unit sphere (unit circumference) in the plane, with the chordalmetric, that is, with the restriction to X of the euclidean metric in the plane. While this spacedoes not have the approximate midpoint property, blossoms are nevertheless geodesic balls.However, the equality Bl ( x, r, s ) = B ( x, r + s ) no loger holds. For instance, Bl ((1 , , , = Bl ((1 , , √ , √
2) = B ((1 , ,
2) = X \ { ( − , } .3. Microblossoming and related conditions
Definition 3.1.
Let 0 < t < K ≥
1. Then µ is said to t -microblossom boundedlywith constant K , if for all x ∈ X and all r >
0, we have(5) µ ( Blu ( x, r, tr )) ≤ KµB ( x, r ) . We shall say µ is a measure that ( t, K )-microblossoms, instead of using the longer expres-sion. Example 3.2.
Microblossoming (even together with doubling) is more general than mi-crodoubling, in a quantitative sense. Consider ( Z d , ℓ ∞ , µ ), where µ is the counting measure.Then µ is doubling, and “microdoubling in the large”, since for large radii ( r > d ), µ canbe regarded as a discrete approximation to Lebesgue measure. However, µB (0 ,
1) = 1, andfor every t > µB (0 , t ) ≥ d , no matter how small t is. Thus, the measure µ is not( t, K )-microdoubling, for any K < d , 0 < t <<
1. However, µ is 1 /d -microblossoming, sincefor r > d , µ behaves as a microdoubling measure, and for r ≤ d , Blu ( x, r, r/d ) = B ( x, r ).A less natural but stronger example is furnished by the measure µ given by [A2, Theorem5.9]. Since µ satisfies a local comparability condition, and is defined in a geometricallydoubling space, it blossoms boundedly, so it microblossoms boundedly (at least with theblossoming constant). But µ is not doubling, and hence it is not microdoubling. J. M. Aldaz
Example 3.3.
While ( t, K )-microdoubling entails (2 , K )-doubling for some K ≥ K , theanalogous statement is not true for microblossoming. The following example shows that(1 / , X = { , , } with theinherited metric from R , and let µ = δ . Then B (0 , ∩ B (3 ,
3) = { } , but µB (0 ,
3) = 0while µB (3 ,
3) = 1, so local comparability fails. Since bounded blossoming implies localcomparability, all we have to do is to check that µ is (1 / , t ≤ B (0 , t ) ⊂ Blu (0 , t, t/ ⊂ { , } , so µB (0 , t ) = µBlu (0 , t, t/
2) = 0, and for t > B (0 , t ) = Blu (0 , t, t/
2) = X . Likewise, for t ≤ B (1 , t ) = Blu (1 , t, t/ ⊂ { , } , so µB (1 , t ) = µBlu (1 , t, t/
2) = 0, and for t > B (1 , t ) = Blu (1 , t, t/
2) = X . Definition 3.4.
Given a metric measure space (
X, d, µ ), and denoting the support of µ by supp ( µ ), the relative increment function of µ , ri µ : supp ( µ ) × (0 , ∞ ) × [1 , ∞ ), is defined as(6) ri µ ( x, r, t ) := µB ( x, tr ) µB ( x, r ) , and the maximal relative increment function , as(7) mri µ ( r, t ) := sup x ∈ supp ( µ ) µB ( x, tr ) µB ( x, r ) . When µ is understood we will simply write ri and mri .This notation allows one to unify different conditions that have been considered regardingthe boundedness of maximal operators. For instance, on supp ( µ ) the doubling conditionsimply means that there is a constant C ≥ r > mri µ ( r, ≤ C ,and the d − -microdoubling condition, that for all r > mri µ ( r, d − ) ≤ C . Note that by τ -smoothness, the complement of the support of µ has µ -measure zero, so the relative incrementfunction is defined for almost every x . Example 3.5.
The interest of considering values of t > C ( µ ) K K (cid:16) log K log K (cid:17) of formula (10) below.This bound is related to the centered maximal operator when the supremum is restrictedto radii R between r and T r , T >
1. The constant K depends on T , as it must satisfy mri µ ( r, T ) ≤ K . For Lebesgue measure on R d with the ℓ ∞ -norm, C ( λ d ) = 1. If we set T = 2, then we can take K = 2 d , while K = d d for T = d , a choice which yields bounds oforder O ( d log d ). A 1 /d -microdoubling constant is K = e ( R d has the approximate midpointproperty, and in fact it is a geodesic space, so microdoubling is the same as microblossomingin this case) and K := max { K , e } = e .Returning to Example 3.2, by a rescaling argument it is clear that the situation for( Z d , ℓ ∞ , µ ) cannot be much worse than for ( R d , ℓ ∞ , λ d ), and in fact it is easy to see thatthe same argument of Stein and Str¨omberg (which will be presented in greater generality tein Str¨omberg covering theorem 7 below) yields the O ( d log d ) bounds. Now suppose we modify the measure so that at onesingle point it is much smaller. Clearly, this will have little impact in the weak type (1,1)bounds, since for d >> x ∈ Z d , and r >
1, the measure of B ( x, r ) will be changed bylittle or not at all, while for r ≤
1, balls with distinct centers do not intersect. For def-initeness, set ν = µ on Z d \ { } , and ν { } = d − d . Then the doubling constant, and the( t, K )-microdoubling constant, for any t >
0, is at least d d (3 d − ≤ K = K , much largerthan the corresponding constants for µ . However, the local comparability constant is stillvery close to 1, since intersecting balls of the same radius must contain at least 3 d pointseach, and a 1 /d -microblossoming constant can be taken to be very close to e . Setting T = d ,we get K ≤ d d (2 d + 1) d , so log K in this case is comparable to the constant obtained when T = 2. Remark 3.6.
One might define (
T, K )-macroblossoming, with
T >
1, by analogy withDefinition 3.1. However, since B ( x, T r ) ⊂ Blu ( x, r, T r ), assuming directly that mri µ ( r, T ) ≤ K is not stronger than ( T, K )-macroblossoming,4.
The Stein-Str¨omberg covering theorem
Next, we present the Stein-Str¨omberg argument using the terminology of blossoms. Notethat the next theorem does not require X to be geometrically doubling.Given an ordered sequence of sets A , A , . . . , we denote by D , D , . . . its sequence ofdisjointifications, that is D = A , and D n +1 = A n +1 \ ∪ n A i . We shall avoid reorderings andrelabelings of collections of balls, as this may lead to confusion regarding the meaning of D j .The unfortunate downside of this choice is an inflation of subindices. Theorem 4.1. Stein-Str¨omberg covering theorem for sparse radii.
Let ( X, d, µ ) be a metric measure space, where µ satisfies a C ( µ ) local comparability condition, and let R := { r n : n ∈ Z } be a T -lacunary sequence of radii, i.e., r n > and r n +1 /r n ≥ T > .Suppose there exists a t > such that T t ≥ and µ t -microblossoms boundedly with constant K . Let { B ( x i , s i ) : s i ∈ R, ≤ i ≤ M } be a finite collection of balls with positive measure,ordered by non-increasing radii. Set U := ∪ Mi =1 B ( x i , ts i ) . Then there exists a subcollection { B ( x i , s i ) , . . . , B ( x i N , s i N ) } , such that, denoting by D i j the disjointifications of the reducedballs B ( x i j , ts i j ) , (8) µU ≤ ( K + 1) µ ∪ Nj =1 B ( x i j , ts i j ) , and (9) N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) ≤ C ( µ ) K + 1 . Proof.
We use the Stein-Str¨omberg selection algorithm. Let B ( x i , s i ) = B ( x , s ) and sup-pose that the balls B ( x i , s i ) , . . . , B ( x i k , s i k ) have already been selected. If k X j =1 µD i j µB ( x i j , s i j ) Bl ( x ij ,s ij ,ts ij ) ( x i k +1 ) ≤ , J. M. Aldaz accept B ( x i k +1 , s i k +1 ) := B ( x i k +1 , s i k +1 ) as the next ball in the subcollection. Otherwise,reject it. Repeat till we run out of balls. Let C be the collection of all rejected balls. Then µ a.e., ∪C < N X j =1 µD i j µB ( x i j , s i j ) Blu ( x ij ,s ij ,ts ij ) . Integrating both sides and using microblossoming we conclude that µ ∪ C ≤ K P Ni µD i j = Kµ ∪ Nj =1 B ( x i j , ts i j ), whence µU ≤ ( K + 1) µ ∪ Nj =1 B ( x i j , ts i j ).Next we show that N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) ≤ C ( µ ) K + 1 . Suppose P Nj =1 µD ij µB ( x ij ,s ij ) B ( x ij ,s ij ) ( z ) >
0. Let { B ( x i k , s i k ) , . . . , B ( x i kn , s i kn ) } be the collec-tion of all balls containing z (keeping the original ordering by decreasing radii). Then each B ( x i kj , s i kj ) has radius either equal to or (substantially) larger than s i kn . We separate the con-tributions of these balls into two sums. Suppose B ( x i k , s i k ) , . . . , B ( x i km , s i km ) all have radiilarger than s i kn , while B ( x i km +1 , s i km +1 ) , . . . , B ( x i kn , s i kn ) have radii equal to s i kn . Now for1 ≤ j ≤ m , by T lacunarity and the fact that T t ≥
1, we have s i kn ≤ ts i kj , so z ∈ B ( x i kj , s i kj )implies that x i kn ∈ Bl ( x i kj , s i kj , ts i kj ), whence m X j =1 µD i kj µB ( x i kj , s i kj ) Bl ( x ikj ,s ikj ,ts ikj ) ( x i kn ) ≤ , and thus m X j =1 µD i kj µB ( x i kj , s i kj ) B ( x ikj ,s ikj ) ( z ) ≤ . Next, note that the sets D i km +1 , . . . , D i kn are all disjoint and contained in Bl ( z, s i kn , ts i kn ).By microblossoming and local comparability, for j = m + 1 , . . . , n we have µ ∪ nj = m +1 D i kj ≤ µBl ( z, s i kn , ts i kn ) ≤ KµB ( z, s i kn ) ≤ K C ( µ ) µB ( x i kj , s i kn ) . It follows that n X j = m +1 µD i kj µB ( x i kj , s i kn ) B ( x ikj ,s ikn ) ( z ) ≤ C ( µ ) µBl ( z, s i kn , ts i kn ) µB ( z, s i kn ) ≤ C ( µ ) K. (cid:3) Denote by M R the centered Hardy-Littlewood maximal operator, with the additional re-striction that the supremum is taken over radii belonging to the subset R ⊂ (0 , ∞ ) (cf. [NaTa,p. 735]). We mention that under the hypotheses of the next corollary, it is not known whetherthe centered Hardy-Littlewood maximal operator M (with no restriction on the radii) is ofweak type (1,1). tein Str¨omberg covering theorem 9 Corollary 4.2.
Let ( X, d, µ ) be a metric measure space, where µ satisfies a C ( µ ) local com-parability condition, and let R := { r n : n ∈ Z } be a T -lacunary sequence of radii. Sup-pose there exists a t > with T t ≥ such that µ ( t, K ) -microblossoms boundedly. Then k M R k L − L , ∞ ≤ ( K + 1) ( C ( µ ) K + 1) . The proof is standard. We present it to keep track of the constants.
Proof.
Fix ε >
0, let a >
0, and let f ∈ L ( µ ). For each x ∈ { M R f > a } select B ( x, r )with r ∈ R , such that aµB ( x, r ) < R B ( x,r ) | f | . Then the collection of “small” balls { B ( x, tr ) : x ∈ { M R f > a }} is a cover of { M R f > a } . By the τ -smoothness of µ , there is a finitesubcollection { B ( x i , ts i ) : s i ∈ R, ≤ i ≤ M } of balls with positive measure, ordered bynon-increasing radii, such that(1 − ε ) µ { M R f > a } ≤ (1 − ε ) µ ∪ { B ( x, tr ) : x ∈ { M R f > a }} < µ ∪ Mi =1 B ( x i , ts i ) . Next, let { B ( x i , s i ) , . . . , B ( x i N , s i N ) } be the subcollection given by the Stein-Str¨ombergcovering theorem for sparse radii. Then we have µ ∪ Mi =1 B ( x i , ts i ) ≤ ( K + 1) µ ∪ Nj =1 B ( x i j , ts i j ) = ( K + 1) N X j =1 µD i j = ( K + 1) N X j =1 µD i j µB ( x i j , s i j ) Z B ( x ij ,s ij ) ≤ ( K + 1) 1 a Z | f | N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) ≤ ( K + 1) ( C ( µ ) K + 1) 1 a Z | f | . (cid:3) In the specific case of d -dimensional Lebesgue measure λ d , C ( λ d ) = 1. Choosing t = 1 /d and T = d , K above can be taken to be e , so the constant obtained is ( e + 1) , whichis worse than the constant ( e + 1)( e + 1) yielded by the Stein-Str¨omberg argument. Thisdiscrepancy is due to the fact that our definition of microblossoming uses the uncenteredblossom instead of the blossom, so from the assumption µ ( Blu ( x, r, tr )) ≤ KµB ( x, r ) we getthe same bound µ ( Bl ( x, r, tr )) ≤ KµB ( x, r ) for the potentially smaller centered blossom. Ofcourse, we could strengthen the definition, using blossoms, to obtain the same constant asin the Stein-Str¨omberg proof, but in the case of Lebesgue measure we prefer to consider itseparately, using different values of ( t, K ) to lower the known bounds. We do this in the nextsection.While Corollary 4.2 follows from the proof of the Stein-Str¨omberg covering theorem, it wasnot stated there but in [MeSo, Lemma 4] for Lebesgue measure, and in the microdoublingcase, in [NaTa, Corollary 1.2]. A source of interest for this result comes from the fact thatunder ( t, K )-microblossoming, the maximal operator defined by a (1 + t )-lacunary set of radii R is controlled by the sum of N maximal operators with lacunarity 1 /t , where N is the leastinteger such that (1 + t ) N ≥ /t . Thus, the bound k M R k L − L , ∞ ≤ N ( K + 1) ( C ( µ ) K + 1)follows. Under the additional assumption of ( t, K / )-microdoubling, the maximal operator defined by taking suprema of radii in [ a, (1 + t ) a ) is controlled by K / times the averagingoperator of radius (1 + t ) a . Putting these estimates together, and using the better bound for µBl ( x, r, tr ) ≤ K / µB ( x, R ), the following result due to Naor and Tao (cf. [NaTa, Corollary1.2]) is obtained. Of course, in this case µ is doubling and X , geometrically doubling. Corollary 4.3.
Let ( X, d, µ ) be a metric measure space, where µ satisfies a C ( µ ) localcomparability condition and is ( t, K / ) -microdoubling. If N is the least integer such that (1 + t ) N ≥ /t , then k M k L − L , ∞ ≤ N K / ( K + 1) ( C ( µ ) K / + 1) . This shows that the Stein-Str¨omberg covering theorem for sparse radii in metric spacessuffices to prove the Naor-Tao bounds, but no greater generality is achieved in either the spacesor the measures, since microdoubling is used in the last step. A second approach, which yieldsa slightly more general version of the result and gives better constants, consists in going backto the original Stein-Str¨omberg argument. Recall that when defining ( t, K )-microblossoming,we set 0 < t < K ≥
1. In the proof of the next result K := max { K , e } is used todetermine the size of the steps. For convenience we take K ≥ e , but e is just one possiblechoice. Note that the condition on mri ( r, T ) below entails that µ is doubling on its support,and hence supp ( µ ) is geometrically doubling. Theorem 4.4. Stein-Str¨omberg covering theorem for bounded radii.
Let ( X, d, µ ) be a metric measure space such that µ satisfies a C ( µ ) local comparability condition, and is ( t, K ) -microblossoming. Set K = max { K , e } . Let r > , and suppose there exists a T > such that K := mri ( r, T ) < ∞ . Let { B ( x i , s i ) : r ≤ s i < T r, ≤ i ≤ M } be a finite col-lection of balls with positive measure, given in any order, and let D = B ( x , ts ) , . . . , D M = B ( x M , ts M ) \ ∪ M − B ( x i , ts i ) be the disjointifications of the t -reduced balls. Then (10) M X i =1 µD i µB ( x i , s i ) B ( x i ,s i ) ≤ C ( µ ) K K (cid:18) K log K (cid:19) . Since the big d log d part in the estimates for the maximal operator (in R d with Lebesguemeasure) comes from this case, which does not require any particular ordering nor any choiceof balls, it is natural to enquire whether some additional selection process can lead to animprovement in the bounds. In general metric spaces this cannot be done, by [NaTa, Theorem1.4], but it might be possible in R d . However, I have not been able to find such a new selectionargument.In the statement above, T is not assumed to be close to 1, and in fact it could be muchlarger than 2 (recall Example 3.5). From the viewpoint of the proof, the difference between T >> t -microdoubling lies in the fact that the size of the steps willvary depending on the growth of balls, rather than having increments given by the constantfactor 1 + t at every step. But the total number of steps will be determined by K and K ,not by whether the factors are all equal to 1 + t or not. tein Str¨omberg covering theorem 11 Proof.
Suppose M X i =1 µD i µB ( x i , s i ) B ( x i ,s i ) ( y ) > . Let s = min { s i : 1 ≤ i ≤ M and y ∈ B ( x i , s i ) } . Then r ≤ s < T r . Select h = sup { h > µB ( y, (1 + h ) s ) ≤ KµB cl ( y, s ) and (1 + h ) s ≤ T r } . This is always possible since lim h ↓ µB ( y, (1 + h ) s ) = µB cl ( y, s ). Now either (1 + h ) s = T r ,in which case the process finishes in one step, and then it could happen that µB cl ( y, (1 + h ) s ) < KµB cl ( y, s ), or (1 + h ) s < T r , in which case µB ( y, (1 + h ) s ) ≤ KµB cl ( y, s ) ≤ µB cl ( y, (1 + h ) s ) (the last inequality must hold, since otherwise we would be able to selecta larger value for h ).If h , . . . , h m have been chosen, let h m +1 := sup { h > µB ( y, s (1 + h )Π mi =1 (1 + h i ) ≤ KµB cl ( y, s Π mi =1 (1 + h i )) and s (1 + h )Π mi =1 (1 + h i ) ≤ T r } . Since µB ( y, T r ) < ∞ , the process stops after a finite number of steps, so there is an N ≥ s Π N +1 i =1 (1 + h i ) = T r and µB cl ( y, s Π Ni =1 (1 + h i )) ≤ µB ( y, T r ) ≤ KµB cl ( y, s Π Ni =1 (1 + h i )) . To estimate N , note that since r ≤ s , µB ( y, T r ) ≤ K µB ( y, s ) ≤ K K µB cl ( y, (1 + h ) s ) ≤ · · · ≤ K K N µB cl ( y, s Π Ni =1 (1 + h i )) ≤ K K N µB ( y, T r ) . Hence K N ≤ K and thus N ≤ log K / log K .The remaining part of the argument is a variant of what was done in Stein-Str¨ombergfor sparse radii, when considering the contribution of balls with the same radius as thesmallest ball. Here we arrange the balls containing y into N + 2 “scales” (instead of just one)depending on whether their radii R are equal to s , or s Π mi =1 (1 + h i ) < R ≤ s Π m +1 i =1 (1 + h i ), or s Π Ni =1 (1 + h i ) < R ≤ T r .For the first scale, consider all balls B ( x i , , s ) , . . . , B ( x i ,k , s ) containing y . Since for1 ≤ j ≤ k , x i ,j ∈ B ( y, s ), it follows that the disjoint sets D i ,j are all contained in Bl ( y, s, ts ) . By microblossoming and local comparability we have, for j = 1 , . . . , k , µ ∪ k j =1 D i ,j ≤ µBl ( y, s, ts ) ≤ K µB ( y, s ) ≤ K C ( µ ) µB ( x i ,j , s ) , so k X j =1 µD i ,j µB ( x i ,j , s i ,j ) B ( x i ,j ,s i ,j ) ( y ) ≤ C ( µ ) µBl ( y, s, ts ) µB ( y, s ) ≤ C ( µ ) K . The contributions of all the other scales are estimated in the same way as the second one,which is presented next. Again, consider all balls B ( x i , , s i , ) , . . . , B ( x i ,k , s i ,k ) containing y and with radii s i ,j in the interval ( s, (1 + h ) s ]. Then all the sets D i ,j are contained in Bl ( y, (1 + h ) s, t (1 + h ) s ) . Using microblossoming, the choice of h , and the local comparability of µ , for j = 1 , . . . , k we have(11) µ ∪ k j =1 D i ,j ≤ µBl ( y, (1 + h ) s, t (1 + h ) s ) ≤ K µB ( y, (1 + h ) s ) ≤ K KµB cl ( y, s ) ≤ K K C ( µ ) µB ( x i ,j , s i ,j ) , so k X j =1 µD i ,j µB ( x i ,j , s i ,j ) B ( x i ,j ,s i ,j ) ( y ) ≤ C ( µ ) µBl ( y, (1 + h ) s, t (1 + h ) s ) µB cl ( y, s ) ≤ C ( µ ) K K. Adding up over the N + 2 scales we get (10). (cid:3) Next we put together the two parts of the Stein-Str¨omberg covering theorem. This helpsto see why the original argument gives better bounds than domination by several sparseoperators.
Theorem 4.5. Stein-Str¨omberg covering theorem.
Let ( X, d, µ ) be a metric measurespace, where µ satisfies a C ( µ ) local comparability condition, and is ( t, K ) -microblossoming.Set K = max { K , e } , and suppose K := sup r> mri ( r, /t ) < ∞ . Let { B ( x i , s i ) : s i ∈ R, ≤ i ≤ M } be a finite collection of balls with positive measure, ordered by non-increasing radii,and let U := ∪ Mi =1 B ( x i , ts i ) . Then there exists a subcollection { B ( x i , s i ) , . . . , B ( x i N , s i N ) } , such that, denoting by D i = B ( x i , ts i ) , . . . , D i N = B ( x i N , ts i N ) \ ∪ N − B ( x i j , ts i j ) , we have (12) µU ≤ ( K + 1) µ ∪ Nj =1 B ( x i j , ts i j ) , and (13) N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) ≤ C ( µ ) K K (cid:18) K log K (cid:19) . Proof.
The selection process is the same as in the proof of Theorem 4.1, yielding the desiredsubcollection, with (12) being the same as (8). As for the right hand side of (13) the 1 comesfrom the contribution of balls with very large radii, as in (9), while C ( µ ) K K (cid:16) log K log K (cid:17) isthe bound from (10). (cid:3) The same argument given for Corollary 4.2 now yields
Corollary 4.6.
Under the assumptions and with the notation of the preceding result, thecentered maximal function satisfies the weak type (1,1) bound k M k L − L , ∞ ≤ ( K + 1) (cid:18) C ( µ ) K K (cid:18) K log K (cid:19)(cid:19) . tein Str¨omberg covering theorem 13 For Lebesgue measure on R d , with balls defined by an arbitrary norm and t = d − , thisis worse (by a factor of e ) than the bound (1 + e )(1 + o (1)) e d log d obtained by Stein andStr¨omberg.Regarding lower bounds, currently it is known that for the centered maximal functiondefined using ℓ ∞ -balls (cubes) the numbers k M k L − L , ∞ diverge to infinity (cf. [A]) at a rateat least O ( d / ) (cf. [IaSt]). No information is available for other balls. In particular, thequestion (asked by Stein and Str¨omberg) as to whether or not the constants k M k L − L , ∞ diverge to infinity with d , for euclidean balls, remains open.5. Sharpening the bounds for Lebesgue measure
Here we revisit the original case studied by Stein and Str¨omberg, Lebesgue measure λ d on R d , with metric (and hence, with maximal function) defined by an arbitrary norm. Since λ d is ( t, (1 + t ) d )-microdoubling for every t >
0, values of t = 1 /d can be used to obtainimprovements on the size of the constants. Theorem 5.1.
Consider R d with Lebesgue measure λ d and balls defined by an arbitrary norm.Let R := { r n : n ∈ Z } be a d -lacunary sequence of radii, and let M R be the corresponding(sparsified) Hardy-Littlewood maximal operator. Then k M R k L − L , ∞ ≤ ( e /d + 1)(1 + 2 e /d ) .Furthermore, if the maximal function is defined using the ℓ ∞ -norm, so balls are cubes withsides perpendicular to the coordinate axes, then k M R k L − L , ∞ ≤ . As we noted above, using the original argument from [StSt] one obtains k M R k L − L , ∞ ≤ ( e + 1)( e + 1). Proof.
Suppose, for simplicity in the writing, that r n +1 = dr n (the case r n +1 ≥ dr n is provenin the same way). We apply the Stein Str¨omberg selection process with t = 1 /d and mi-crodoubling constant K = (1+1 /d ) d < e /d . As before, given 0 ≤ f ∈ L and a >
0, we coverthe level set { M R f > a } almost completely, by a finite collection of “small” balls { B ( x i , ts i ) : s i ∈ R, ≤ i ≤ M } ordered by non-increasing radii, and such that aµB ( x i , s i ) < R B ( x i ,s i ) f .From this collection we extract a subcollection { B ( x i , ts i ) , . . . , B ( x i N , ts i N ) } satisfying µ ∪ Mi =1 B ( x i , ts i ) ≤ ( e /d + 1) µ ∪ Nj =1 B ( x i j , ts i j ) = ( e /d + 1) N X j =1 µD i j . Next, we obtain the bound N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) ≤ e /d + 1 , by considering z such that P Nj =1 µD ij µB ( x ij ,s ij ) B ( x ij ,s ij ) ( z ) >
0. Select the ball B with largestindex that contains z . Since B belongs to the subcollection obtained by the Stein-Str¨ombergmethod, all balls containing z and with radii ≥ d r ( B ) (where r ( B ) denotes the radius of B ), contribute at most 1 to the sum. Next we have to consider two more scales, all the balls with radius r ( B ), and all the balls with radius dr ( B ). By the usual argument (asin the proof of Theorem 4.1) each of these scales contributes at most e /d to the sum, so k M R k L − L , ∞ ≤ ( e /d + 1)(1 + 2 e /d ) follows. The result for cubes is obtained by letting d → ∞ , since in this case it is known that the weak type (1,1) norms increase with thedimension (cf. [AV, Theorem 2]). (cid:3) Theorem 5.2.
Consider R d with Lebesgue measure λ d and balls defined by an arbitrary norm.If ε > , then k M k L − L , ∞ ≤ (2 + 3 ε ) d log d for all d = d ( ε ) sufficiently large. The bound from the proof of [StSt, Theorem 1] is k M k L − L , ∞ ≤ e ( e + 1)(1 + o (1)) d log d. Proof.
Fix ε ∈ (0 , d − − ε ) d = 1 + d − ε + O ( d − ε ), it follows that λ d is ( d − − ε , d − ε + O ( d − ε ))-microdoubling. Note that if a ball B contains the center of a second ball ofradius 1, and the latter ball is contained in (1 + d − − ε ) B , then the radius r B of B must satisfy r B ≥ d ε . Let L be any natural number such that (1 + d − − ε ) L ≥ d ε . Taking logarithms toestimate L , and using log(1 + x ) > x − x for x sufficiently close to 0, we see that it is enough,for the preceding inequality to hold, to choose L satisfying L ( d − − ε − d − − ε ) ≥ (1 + ε ) log d ,or, L ≥ (1 + o ( d − ))(1 + ε ) d ε log d . For the least such integer we will have L ≤ o ( d − ))(1 + ε ) d ε log d. Again we apply the Stein Str¨omberg selection process with t = d − − ε , covering a given levelset { M f > a } almost completely (up to a small δ >
0) by a finite collection of small balls { B ( x i , ts i ) : s i ∈ R, ≤ i ≤ k } ordered by non-increasing radii, and such that aµB ( x i , ts i ) < R B ( x i ,ts i ) | f | . Using the Stein Str¨omberg algorithm, we extract a subcollection { B ( x i , ts i ) , . . . , B ( x i N , ts i N ) } satisfying(14) (1 − δ ) µ { M f > a } ≤ (2 + d − ε + O ( d − ε )) N X j =1 µD i j , where the sets D i j denote the disjointifications determined by the above subcollection. Tosharpen the usual uniform bound for N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) , we use the fact that the sets D i are disjoint across different steps, and not just within thesame step. More precisely, let z satisfy(15) N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) ( z ) > . Select the ball B with largest index that contains z . Since B belongs to the subcollectionobtained by the Stein-Str¨omberg method, all balls containing z and with radii ≥ d ε r ( B ) tein Str¨omberg covering theorem 15 contribute at most 1 to the sum. Next we consider the first two scales, since for all the others,the argument is the same as for the second.Take all the balls with radii equal to r B . In order to bound (15) from above, we supposethat (1 + d − − ε ) B is completely filled up with the sets D i associated to balls with radii r B ,and hence, no D j associated to a ball with larger radius intersects (1 + d − − ε ) B . When weconsider the sum (15), but just for the balls with radius r B , we obtain the upper bound(1 + d − − ε ) d . For the second level, we consider all balls in the subcollection with radii in( r B , (1 + d − − ε ) r B ], and as before, we suppose that (1 + d − − ε ) B \ (1 + d − − ε ) B is completelyfilled up with the sets D j associated to these balls. The estimate we obtain for this second levelis (1 + d − − ε ) d − d − ε + O ( d − ε ). For balls with radii in ((1 + d − − ε ) k r B , (1 + d − − ε ) k +1 r B ],0 ≤ k < L , we use the same estimate. Adding up over all scales we obtain N X j =1 µD i j µB ( x i j , s i j ) B ( x ij ,s ij ) ( z ) ≤ d − ε + O ( d − ε )+(1 + ( d − ε + O ( d − ε ))(1 + o ( d − ))(1 + ε ) d ε log d ) ≤ (1 + O ( d − ε ))(1 + ε ) d log d. Multiplying this bound with the bound from (14) and adding an ε to absorb the big Ohterms, for d large enough we obtain k M k L − L , ∞ ≤ (2 + 3 ε ) d log d . (cid:3) References [A] J.M. Aldaz,
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Instituto de Ciencias Matem´aticas (CSIC-UAM-UC3M-UCM) and Departamento de Matem´aticas,Universidad Aut´onoma de Madrid, Cantoblanco 28049, Madrid, Spain.
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