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arXiv:0908.0643v2 [math.CA] 9 Mar 2010

BEHAVIOR OF WEAK TYPE BOUNDS FOR HIGH DIMENSIONAL

MAXIMAL OPERATORS DEFINED BY CERTAIN RADIAL MEASURES

J. M. ALDAZ AND J. P´

EREZ L ´

AZARO

Abstract. As shown in [A1], the lowest constants appearing in the weak type (1,1) in-

equalities satisﬁed by the centered Hardy-Littlewood maximal operator associated to certain

ﬁnite radial measures, grow exponentially fast with the dimension. Here we extend this result

to a wider class of radial measures and to some values of p > 1. Furthermore, we improve

the previously known bounds for p= 1. Roughly speaking, whenever p∈(1,1.03], if µis

deﬁned by a radial, radially decreasing density satisfying some mild growth conditions, then

the best constants cp,d,µ in the weak type (p, p) inequalities satisfy cp,d,µ ≥1.005dfor all d

suﬃciently large. We also show that exponential increase of the best constants occurs for

certain families of doubling measures, and for arbitrarily high values of p.

1. Introduction

Given a Borel measure µon Rdand a locally integrable function g, the Hardy-Littlewood

maximal operator Mµis given by

(1) Mµg(x) := sup

{r>0:µ(B(x,r))>0}

1

µ(B(x, r)) ZB(x,r)|g|dµ,

where B(x, r) denotes the euclidean closed ball of radius r > 0 centered at x. As is well

known, Mµis a positive, sublinear operator, acting on the cone of positive, locally integrable

functions (Mµis deﬁned by using |g|rather than g). The Hardy-Littlewood maximal operator

admits many variants: Instead of averaging |g|over balls centered at x(the centered operator)

as in (1), it is possible to consider all balls containing x(the uncentered operator) or average

over convex bodies more general than euclidean balls (and even over more general sets). And

as part of the current eﬀort to develop a calculus on metric spaces, the Hardy-Littlewood

maximal operator has been studied in settings far more general than Rd. Here we work

with the centered operator deﬁned using euclidean balls in Rd, associated to certain radial

measures µgiven by µ(A) := RAf(kyk2)dλd(y), where f: (0,∞)→[0,∞) is nonincreasing

(possibly unbounded) and not zero almost everywhere, and f(t)td−1∈L1

loc((0,∞), dt). We

emphasize that the function fdeﬁning µis allowed to vary with the dimension d. Additional

hypotheses, regarding the growth at 0 of fand its decay at ∞, are given below.

The Hardy-Littlewood maximal operator is an often used tool in Real and Harmonic

Analysis, mainly (but not exclusively) due to the fact that while |g| ≤ Mµga.e., Mµgis

2000 Mathematical Subject Classiﬁcation. 42B25.

The authors were partially supported by Grant MTM2009-12740-C03-03 of the D.G.I. of Spain.

1

2 J. M. Aldaz and J. P´erez-L´azaro

not too large (in an Lpsense) since it satisﬁes the following strong type (p, p) inequality:

kMµgkp≤Cpkgkpfor 1 < p ≤ ∞. For p= 1, Mµsatisﬁes instead the weak type (1,1)

inequality supα>0αµ({Mµg≥α})≤c1kgk1. Another aspect of the maximal operator that is

receiving increasing attention, but not touched upon here, is that of its regularity properties,

cf. for instance [AlPe1], [AlPe2], [AlPe3] and the references contained therein. When µ=λd,

the d-dimensional Lebesgue measure, we often simplify notation, by writing Mrather than

Mλdand dx instead of dλd(x).

Considerable eﬀorts have gone into determining how changing the dimension of Rdmodiﬁes

the best constants appearing in the weak and strong type inequalities. When p=∞, we can

take Cp= 1 in every dimension d, since averages never exceed a supremum. Quite remarkably,

E. M. Stein showed that for M, there exist bounds for Cpthat are independent of d([St1],

[St2], [StSt], see also [St3]). Stein’s result was generalized to the maximal function deﬁned

using an arbitrary norm by J. Bourgain ([Bou1], [Bou2], [Bou3]) and A. Carbery ([Ca])

when p > 3/2. For ℓqballs, 1 ≤q < ∞, D. M¨uller [Mu] showed that uniform bounds

again hold for every p > 1 (given 1 ≤q < ∞, the ℓqballs are deﬁned using the norm

kxkq:= (xq

1+xq

2+···+xq

d)1/q). It is still an open question whether the maximal operator

associated to cubes and Lebesgue measure is uniformly bounded for 1 < p ≤3/2.

When p= 1, the maximal operator is (typically) unbounded, so one considers weak type

(1,1) inequalities instead. In [StSt] E. M. Stein and J. O. Str¨omberg proved that the smallest

constants in the weak type (1,1) inequality satisﬁed by Mgrow at most like O(d) for euclidean

balls, and at most like O(dlog d) for more general balls. They also asked if uniform bounds

could be found, a question still open for euclidean balls. But for cubes the answer is negative,

cf. [A2]. In [Au], G. Aubrun reﬁned the result from [A2] by showing that c1,d ≥Θ(log1−εd),

where Θ denotes the exact order and ε > 0 is arbitrary. A very signiﬁcant extension of

the Stein and Str¨omberg’s O(dlog d) result, beyond the euclidean setting, has recently been

obtained by A. Naor and T. Tao, cf. [NaTa].

The weak type (1,1) case for integrable radial densities deﬁned via bounded decreasing

functions was studied in [A1]. It was shown there that the best constants c1,d satisfy c1,d ≥

Θ (1) 2/√3d/6, in strong contrast with the linear O(d) bounds known for M. This suggests

that for these measures and suﬃciently small values of p > 1, lack of uniform bounds in d

should also hold. We show here that this is indeed the case, and for a wider class of measures

than those considered in [A1]. We shall remove the assumption of boundedness on densities

and the assumption of ﬁniteness on measures, replacing these hypotheses with milder growth

conditions on the relative size of balls centered at the origin (a possibility suggested in [A1,

Remark 2.6]). Instead of working directly with norm (or strong type) inequalities when p > 1,

we shall consider the weak type (p, p) inequalities. This allows us to treat the cases p= 1 and

p > 1 simultaneously. Needless to say, lower bounds for weak type constants immediately

imply the same bounds for strong type constants. In Theorem 3.4 we show that if balls

centered at zero grow suﬃciently fast for some given radius, and this growth experiences a

certain rate of decay at inﬁnity (cf. the theorem for the exact technical conditions) then there

is exponential increase of the best constants cp,d in the weak type (p, p) inequalities, for every

Behavior of weak type bounds 3

p∈[1, p0), where p0≈1.0378. The proof follows the lines of [A1], but replacing the Dirac

delta δ0with χB(0,v)for some suitably chosen radius v, and using a better ball decomposition.

This allows us to improve the bound on c1,d from [A1] to c1,d ≥Θ (1) 2/551/6dfor p= 1,

even though we are considering the characteristic function of a ball, rather than the more

eﬃcient δ0. Of course, Dirac Deltas cannot be used when p > 1, since the only reasonable

deﬁnition of kδkpfor p > 1 is kδkp=∞.

Exponential dependency on the dimension of the best constants also holds for certain

collections of doubling measures and arbitrarily high values of p, cf. Theorem 3.12 below.

Thus, Stein result on uniform Lpbounds for Mdoes not extend to arbitrary doubling measures

on Rd, even though the class of doubling measures represents a natural generalization of λd.

To highlight the diﬀerence between λdand the measures considered here, we point out that

when Macts on radial, radially decreasing Lpfunctions, the best weak type (p, p) constants

cp,d equal 1 in every dimension, see [AlPe4, Theorem 2.6] (actually, the result is stated there

for c1,d, but c1,d ≥cp,d , cf. (5) below and the explanations afterwards), while the best strong

type constants satisfy Cp,d ≤21/qq1/p , where q=p/(p−1), see [AlPe4, Corollary 2.7].

Professor Fernando Soria informs us that he and Alberto Criado have also extended the

results from [A1] to some values of p > 1, cf. [Cr]; we mention that where [Cr] and this paper

overlap, the results presented here are more general and give better bounds.

2. Notation and background results.

The restriction of µto a measurable set Ais denoted by µ|A; that is, µ|A(B) = µ(A∩B).

We always assume that µ(Rd)>0 and µ(B(x, r)) <∞, i.e., measures are nontrivial and

locally ﬁnite. The maximal function of a locally ﬁnite measure νis deﬁned by

(2) Mµν(x) := sup

{r>0:µ(B(x,r))>0}

ν(B(x, r))

µ(B(x, r)).

Note that formula (1) is simply (2) in the special case ν << µ. Our choice of closed balls in

(1) and (2) is mere convenience; using open balls instead does not change the value of the

maximal operator at x, since each closed ball is a countable intersection of open balls. The

boundary of B(x, r) is the sphere S(x, r). Sometimes we use Bd(x, r) and Sd−1(x, r) to make

their dimensions explicit. If x= 0 and r= 1, we use the abbreviations Bdand Sd−1. Balls are

deﬁned using the ℓ2or euclidean distance kxk2:= px2

1+···+x2

d. The Lebesgue measure on

Rdis denoted by λd, and area measure on a d−1 sphere, by σd−1. Sometimes it is convenient

to use normalized versions of these measures, so balls and spheres have total mass 1; we use

Nas a subscript to denote these normalizations. Thus, λd

N(Bd) = 1 and σd−1

N(Sd−1) = 1.

Regarding the relationships between diﬀerent constants, let us recall that by the Besicovitch

Covering Theorem, for every locally ﬁnite Borel measure µon Rd, and every pwith 1 ≤p < ∞,

the maximal operator satisﬁes the following weak type (p, p) inequality:

(3) µ({Mµg≥α})≤ckgkp

αp

,

4 J. M. Aldaz and J. P´erez-L´azaro

where c=c(p, d, µ) depends neither on g∈Lp(Rd, µ) nor on α > 0. The constant ccan also

be taken to be independent of µand of p. Set q:= p/(p−1). Using the quantitative version

of the Besicovitch Covering Theorem given in [Su, p. 227], we have

(4) µ({Mµg≥α})≤(2.641 + o(1))dkgk1

α.

Thus, if g∈Lp(Rd, µ), then |g|p∈L1(Rd, µ), and it follows from Jensen’s inequality that

(5) µ({Mµg≥α}) = µ({(Mµg)p≥αp})≤µ({Mµ|g|p≥αp})≤(2.641 + o(1))dkgkp

p

αp.

Letting cp,d,µ be the best constant cin (3), we have cp,d,µ ≤(2.641 + o(1))d/p. This bound

is uniform in µ, and setting p= 1 in the exponent d/p, it can be made uniform in palso.

Replacing (2.641 + o(1))d/p by c1,d,µ in the right hand side of (5), we also obtain cp,d,µ ≤

(c1,d,µ)1/p ≤c1,d,µ. Let Cp,d,µ be the lowest constant in

(6) kMµgkLp(Rd,µ)≤Cp,d,µkgkLp(Rd,µ).

It is an immediate consequence of Chebyshev’s inequality that cp,d,µ ≤Cp,d,µ, since

(7) αpµ({(Mµg)p≥αp})≤ kMµgkp

p≤Cp

p,d,µkgkp

p.

When pis small, lower bounds for cp,d,µ are quite often not just formally stronger, but

substantially stronger than lower bounds for Cp,d,µ, since it is well-known that for many

measures C1,d,µ =∞and limp→1Cp,d,µ =∞, while cp,d,µ ≤c1,d,µ ≤(2.641 + o(1))d.

Let d >> 1, and consider Lebesgue measure restricted to the unit ball. Most of its mass

is concentrated near Sd−1(0,1), since volume scales like Rd, so the ball “looks” very much

like the sphere. The main idea in [A1] and here is to realize that this is a rather general

phenomenon: Rotationally invariant measures with a certain decay at inﬁnity, will often be

very similar in a certain region to area on some sphere Sd−1(0, R1). Hence, the size of balls

in that region can be estimated by intersecting them with Sd−1(0, R1) and then using the

area of the spherical caps resulting from such intersections. Given a unit vector v∈Rdand

s∈[0,1), the sspherical cap about vis the set C(s, v) := {θ∈Sd−1:hθ, vi ≥ s}. Spherical

caps are just geodesic balls BSd−1(x, r) in Sd−1. For spheres other than Sd−1, spherical caps

are deﬁned in an entirely analogous way. If v=e1= (1,0,...,0) and s= 2−1, then

(8) BSd−1(e1, π/3) = C(2−1, e1) = Sd−1∩B(e1,1).

More generally, given any angle r∈(0, π/2), writing s= cos rand t= sin r, we have

(9) BSd−1(e1, r) = C(s, e1)⊂B(se1, t).

The following lemma shows that σd−1

N(C(s, e1)) = td/Θ(√d), where Θ stands for exact order

(i.e., g= Θ(h) if and only if g=O(h) and h=O(g)); the special case r=π/3 is used

in the proof of [A1, Theorem 2.3]. We recall the following results on volumes and areas: i)

λd(Bd) = πd/2

Γ(1+d/2) ; ii) σd−1(Sd−1) = dλd(Bd); iii) σd−1(BSd−1(x, r)) = σd−2(Sd−2)Rr

0sind−2tdt

(cf. for instance [Gra, (A.11) pg. 259]).

Behavior of weak type bounds 5

Lemma 2.1. Let r∈(0, π/2), let σd−1

Nbe normalized area on the sphere Sd−1(0, R), and

let s= cos r,t= sin r, so with this notation, σd−1

N(BSd−1(0,R)(Re1, Rr)) = σd−1

N(C(Rs, Re1)).

Then

(10) td−1

√2πd ≤σd−1

N(C(Rs, Re1)) ≤td−1

s√2πd r1 + 1

d.

Proof. Observe ﬁrst that the relative size of caps depends neither on the center of the ball

nor on the radius. In particular, since we are dealing with normalized area, we may assume

that R= 1. We use the following Gamma function estimate (an immediate consequence of

the log-convexity of Γ on (0,∞), cf. Exercise 5, pg. 216 of [Web]):

(11) d

21/2

≤Γ(1 + d/2)

Γ(1/2 + d/2) ≤d+ 1

21/2

.

From i), ii), iii), (11) and the fact that cos u≥son [0, r], we get:

(12) σd−1

N(C(s, e1)) ≤σd−2(Sd−2)

sσd−1(Sd−1)Zr

0

sind−2ucos udu

(13) = 1

sd

λd−1(Bd−1)

λd(Bd)td−1≤td−1

s√2πd r1 + 1

d.

Likewise, since cos u≤1,

(14) σd−1

N(C(s, e1)) ≥σd−2(Sd−2)

σd−1(Sd−1)Zr

0

sind−2ucos udu =1

d

λd−1(Bd−1)

λd(Bd)td−1≥td−1

√2πd .

3. Weak type (p, p)bounds for rotationally invariant measures

Fix d∈N\ {0}, and let f: (0,∞)→[0,∞) be a nonincreasing (possibly unbounded)

function, not zero almost everywhere, such that f(t)td−1∈L1

loc((0,∞), dt). Then the function

fdeﬁnes a locally integrable, rotationally invariant (or radial) measure µon Rdvia

(15) µ(A) := ZA

f(kyk2)dλd(y).

Observe that the local integrability of f(t)td−1is assumed for a ﬁxed d, not for all values

of dsimultaneously. Note also that fcan depend on d. When A=B(0, R), integration in

polar coordinates yields the well known expression µ(B(0, R)) = σd−1(Sd−1)RR

0f(t)td−1dt.

Since (unlike [A1]) ﬁniteness of measures and boundedness of densities are not assumed in

the present paper, we need to impose some conditions on the rate of growth of balls centered

at zero. To this end, we deﬁne, for all u∈(0,1] and all R > 0 such that µ(B(0, uR)) >0,

(16) hu(R) := µ(B(0, R))

µ(B(0, uR)).

6 J. M. Aldaz and J. P´erez-L´azaro

In the extreme case µ=δ0,hu(R) = 1 always, and for every gwith g(0) <∞, we have

Mµg=g=g(0) a.e. with respect to δ0. Thus, for all p≥1 and all d≥1, cp,d,δ0=Cp,d,δ0= 1.

Of course, in this case there is no relationship between δ0and d. For the measures considered

in [A1, Theorem 2.3], limR→0hu(R) = u−dand limR→∞ hu(R) = 1; we present this fact, which

appears within the proof of [A1, Theorem 2.3], as part of the next proposition.

Proposition 3.1. Fix d∈N\ {0}. Let f: (0,∞)→[0,∞)be a nonincreasing function with

f > 0on some interval (0, a)and f(t)td−1∈L1

loc(0,∞). If µis the measure deﬁned by (15),

then for every u∈(0,1) and every R > 0we have hu(R)≤u−d. If additionally fis bounded,

then for every u∈(0,1),supR>0hu(R) = limR→0hu(R) = u−d. Regardless of whether fis

bounded or not, if µis ﬁnite, then for every u∈(0,1) we have limR→∞ hu(R) = 1.

Proof. The fact that supR>0hu(R)≤u−dis obvious since fis nonincreasing, so the case

where fis constant yields the largest possible growth, and then we just have a multiple of

Lebesgue measure. Or, more formally:

(17) µ(B(0, R))

µ(B(0, uR)) =µ(B(0, uR)) + µ(B(0, R)\B(0, uR))

µ(B(0, uR))

(18) = 1 + σd−1(Sd−1)RR

uR f(t)td−1dt

σd−1(Sd−1)RuR

0f(t)td−1dt ≤1 + f(uR)RR

uR td−1dt

f(uR)RuR

0td−1dt =u−d.

Suppose next that in addition to being nonincreasing, fis bounded. Then the aver-

ages 1

λd(B(0,R)) RB(0,R)f(kxk2)dx are bounded and nonincreasing with respect to R. Thus,

limR→01

λd(B(0,R)) RB(0,R)f(kxk2)dx =Lexists, and

(19) lim

R→0

µ(B(0, R))

µ(B(0, uR)) = lim

R→0RB(0,R)f(kxk2)dx

RB(0,uR)f(kxk2)dx = lim

R→0

Lλd(B(0, R))

Lλd(B(0, uR)) =u−d.

The last assertion about ﬁnite measures is obvious.

Remark 3.2. The condition limR→0hu(R) = u−dcan be satisﬁed by unbounded densities

with a mild singularity a 0. Consider, for instance, f(x) = |log(x)χ(0,1](x)|, for every d≥1.

Lemma 3.3. Let µbe a rotationally invariant measure on Rd, let 1≤p < ∞, and let

q:= p/(p−1). For 0< R and 0< v < 1, write H:= √R2+v2R2. If the pair (v, R)is such

that µ(B(0, vR)) >0, then

(20) cp,d,µ ≥µ(B(0, vR))1/qµ(B(0, R))1/p

2µ(B(Re1, H)) .

If additionally there exist T, t0>0and v0∈(0,1) such that sup{R>0:v0R≥T}hv0(R)≤v−t0d

0,

then

(21) cp,d,µ ≥sup

{R>0:v0R≥T}

vt0d/q

0µ(B(0, R))

2µ(B(Re1, H)) .

Behavior of weak type bounds 7

Proof. Note that

(22) MµχB(0,vR)(Re1)≥kχB(0,vR)k1

2µ(B(Re1, H)) =: α.

By rotational invariance of µ, we have B(0, R)⊂ {MµχB(0,vR)≥α}. And since χB(0,vR)=

χp

B(0,vR), it follows that kχB(0,vR)k1=kχB(0,vR)kp

p. Using (3) we see that

(23) cp,d,µ ≥α{MµχB(0,vR)≥α}1/p

kχB(0,vR)kp≥kχB(0,vR)k1

2µ(B(Re1, H))

µ(B(0, R))1/p

kχB(0,vR)kp

(24) = µ(B(0, vR))1/q µ(B(0, R))1/p

2µ(B(Re1, H)) .

Specializing to v=v0and using the hypothesis on Tand t0we obtain

(25) cp,d,µ ≥vt0d/q

0

µ(B(0, R))

2µ(B(Re1, H))

for every R > 0 such that v0R≥T.

The preceding Lemma is more general than needed in the present paper, since we are not

assuming that µis of the form given by (15); this greater generality will be useful in future

work. If µis given by (15), then by Proposition 3.1, the condition on hv(R) is satisﬁed for

some t0≤1, all v∈(0,1), and all T > 0. So the Lemma is applicable and furthermore, any

v0∈(0,1) can be used (in the next Theorem we take v0= 1/2). The idea of the proof is to

choose R1so µ(B(R1e1, H)) is exponentially small (in d) when compared with µ(B(0, R1)),

and then to adjust qin (21) so v−1/q

0is suﬃciently close to 1. This yields exponential growth

of the constants for p > 1 small enough. Recall that Cp,d,µ denotes the best constant in the

strong type (p, p) inequalities. We emphasize that in the next result, we can have diﬀerent

functions fassociated to diﬀerent dimensions d.

Theorem 3.4. Fix d∈N\{0}, and set u=p2/3. Let f: (0,∞)→[0,∞)be a nonincreasing

function and let µbe the radial measure deﬁned via (15). Assume µsatisﬁes

(26) sup

R>0

hu(R)≥u−(6 log 2−log 55

3 log 3−3 log 2 )d=64

55d

6

≥lim sup

R→∞

hu(R).

Then for every psuch that 1≤p < 6 log 2

log 55 ≈1.0378, we have

(27) 55−1/621/p >1and Cp,d,µ ≥cp,d,µ ≥1

4 + Θ 1

√d21/p

551/6d

.

Proof. Assume that d≥2, and set H=pR2

1+v2R2

1, as in Lemma 3.3. Arguing as in [A1,

Theorem 2.3], we look for a radius R1such that B(R1e1, H) has very small measure compared

to B(0, R1). Fix 0 < ε < 1/10. Deﬁne A={R > 0 : hu(R)≥(1 −ε)(64/55)d/6}. By the

continuity in Rof huand the hypotheses in (26) Ais a nonempty closed set. If Ais unbounded,

8 J. M. Aldaz and J. P´erez-L´azaro

we choose R1∈Aso large that hu(R1)≥(1 −ε)(64/55)d/6and hu(u−1R1), hu(u−2R1)<

(1 + ε)(64/55)d/6. If Ais bounded, then R1:= max Aautomatically satisﬁes the preceding

conditions on hu(R1), hu(u−1R1), and hu(u−2R1). Set v= 1/2. Then H=R1√5/2. Write

T=pR2

1+H2= 3R1/2, and observe that T=u−2R1, so B(R1e1, H)∩ {x1≤R1} ⊂

B(0,3R1/2). Since the density of µis radially decreasing,

(28) µ(B(R1e1, H)∩ {x1≥R1})≤µ(B(R1e1, H)∩ {x1≤R1}).

Thus µ(B(R1e1, H)≤2µ(B(R1e1, H )∩ {x1≤R1}), so is enough to control this latter term.

To this end, we split B(R1e1, H)∩ {x1≤R1}into the following three pieces and estimate

the measure of each one: B(0, uR1)∩B(R1e1, H ), B(0, R1)c∩B(R1e1, H)∩ {x1≤R1}, and

(B(0, R1)\B(0,p2/3R1)) ∩B(R1e1, H). First we bound the part containing the origin:

(29) µ(B(0, uR1)∩B(R1e1, H)) ≤µ(B(0, uR1)) ≤µ(B(0, R1))

1−ε55

64d/6

.

The other two parts are contained inside certain cones, whose radial projections into the

unit sphere are spherical caps. So we apply Lemma 2.1. To control µ(B(0, R1)c∩B(R1e1, H)∩

{x1≤R1}), we deﬁne νon Sd−1(0,1) as the pushforward (via the radial projection map) of

µrestricted to B(0, u−2R1)\B(0, R1). Now νis a rotationally invariant measure on Sd−1,

so it must be a multiple mσd−1

Nof normalized area. Since ν(Sd−1) = m, we have m=

µ(B(0, u−2R1)\B(0, R1)) and thus ν=µ(B(0, u−2R1)\B(0, R1)) σd−1

N.We use symmetry

to ﬁnd the spherical cap Cdetermined by the intersection of S(0, R1) with B(R1e1, H),

restricting ourselves to the x1x2-plane. Simultaneously solving x2

1+x2

2=R2

1and (x1−R1)2+

x2

2=H2, we ﬁnd that the radial projection of Cinto Sd−1is C(3/8, e1). Now

(30) µ(B(0, R1)c∩B(R1e1, H))

(31) = µ(B(0, R1)c∩B(R1e1, H)∩ {x1≤R1}) + µ(B(R1e1, H)∩ {x1> R1}).

By Lemma 2.1 with cos r= 3/8 (so sin r=√55/8) and by the choice of R1,

(32) µ(B(0, R1)c∩B(R1e1, H)∩{x1≤R1})≤µ(B(0, R1))(1 +ε)264

55d/3

σd−1

N(C(3/8, e1))

(33) ≤µ(B(0, R1)) 55

64d/6

Θ1

√d.

Regarding the measure of (B(0, R1)\B(0,p2/3R1)) ∩B(R1e1, H), this set is contained in

the (positive) cone subtended by the cap Cresulting from the intersection of Sd−1(0,p2/3R1)

with B(R1e1, H). The said cone is formed by all rays starting at 0 and crossing C. Let

rbe the maximal angle between a vector in this cap and the x1-axis. We consider the

intersection of Cwith the x1x2-plane, in order to determine s:= cos rand t:= sin r. Solving

(x1−R1)2+x2

2=H2and x2

1+x2

2= (p2/3R1)2, we obtain t=√1077/(24√2). Projecting

Behavior of weak type bounds 9

radially µ|B(0,R1)to ν=µ(B(0, R1))σd−1

Non Sd−1, and likewise projecting radially Conto

C(s, e1), from Lemma 2.1 we obtain

(34) µ((B(0, R1)\B(0, uR1)) ∩B(R1e1, H)) ≤ν(C(s, e1)) ≤µ(B(0, R1))tdO1

√d.

The preceding estimates, together with t=√1077/(24√2) <(55/64)1/6, entail that

(35) µ(B(R1e1, H)∩ {x1≤R1})≤µ(B(0, R1)) 55

64d/61

1−ε+ Θ 1

√d,

and we already know from (28) that µ(B(R1e1, H)) is at most twice as large. Since by

Proposition 3.1, µ(B(0, R)) ≤v−dµ(B(0, vR)) for every v∈(0,1) and every R > 0, we can

apply Lemma 3.3 with t0= 1, R=R1and v0= 1/2. This yields

(36) cp,d,µ ≥21/p55−1/6d

4

1−ε+ Θ 1

√d.

Setting 21/p55−1/6= 1, we ﬁnd the solution p0= (6 log 2)/log 55 ≈1.03782. Observing that

cp,d,µ does not depend on our choice of ε, the result follows by letting ε→0.

Remark 3.5. For p≤1.03, 21/p 55−1/6>1.005. Thus, if dis “high”, cp,d,µ ≥1.005d. How

high must dbe can be explicitly determined from the proof, by keeping track of the constants

in Lemma 2.1, instead of writing Θ(1/√d). Note also that in the speciﬁc case p= 1, the

preceding theorem is more general and gives a better bound (since 55−1/62>(2/√3)1/6) than

[A1, Theorem 2.3], even though χB(0,R1/2) is a very poor choice when p= 1 (using δ0is much

more eﬃcient). We shall explore the case p= 1 in more detail elsewhere.

Remark 3.6. The hypotheses contained in (26) are selected so that all ﬁnite, radial, radially

decreasing measures with bounded densities are included, and still a concrete range for pis

obtained. Numerically, t0:= 6 log 2−log 55

3 log 3−3 log 2 ≈0.1246. Provided that the singularity at 0 is not

too strong, Theorem 3.4 also applies to measures with unbounded densities. In particular, it

applies to all measures deﬁned via (15), with ft(r) = r−td χ(0,1](r) and t∈(0,1−t0]. This

last condition comes from the fact that for these measures, hu(R) = u−(1−t)dwhen R≤1.

For inﬁnite measures, however, (26) can be rather restrictive. Deﬁne µt,d as in the preceding

remark but without truncation, i.e., using ft(r) = r−td . Then the theorem applies only to

t= 1 −t0. Observe, however, that values diﬀerent from t0and p2/3 could have been

used, with the same qualitative results. Thus, a simple way to obtain a theorem covering an

inﬁnite subfamily of the measures µt,d is to assume diﬀerent rates of growth for the sup and

the lim sup in (26). The proof of the next result is essentially identical to that of Theorem

3.4, so it will be omitted. We use u=p2/3 to be able to apply the same splitting of the ball

centered at R1e1, but other values are possible. Also, the upper bound given below for t1can

be modiﬁed, by suitably choosing a diﬀerent value for u. Recall that f: (0,∞)→[0,∞) is

nonincreasing and that µis deﬁned by fvia (15).

10 J. M. Aldaz and J. P´erez-L´azaro

Theorem 3.7. Fix d∈N\{0}, choose t0∈(0,1),t1∈(0,log(64/55)/log(9/4)), and set u=

p2/3. Then there exists a p0=p0(t0, t1)>1with the following property: For all p∈[1, p0)

we can ﬁnd a b(p, t0, t1)>1, such that for every measure µsatisfying supR>0hu(R)≥u−t0d

and lim supR→∞ hu(R)≤u−t1d,we have Cp,d,µ ≥cp,d,µ ≥Θ (1) b(p, t0, t1)d.

Remark 3.8. If t0< t1, then the preceding result covers all the measures µt,d deﬁned by

ft(r) = r−td such that t0≤1−t≤t1.

Returning to Theorem 3.4, it admits a simpler statement when fis bounded and f(x)xd−1∈

L1(0,∞), so µis ﬁnite. By Proposition 3.1, the conditions supR>0hu(R)≥u−t0dand

lim supR→∞ hu(R)≤u−t1dare then automatically satisﬁed for all t0, t1, u ∈(0,1).

Corollary 3.9. Fix d∈N\ {0}. Suppose fis bounded and f(x)xd−1∈L1(0,∞). If µis the

ﬁnite measure deﬁned via (15), then for every

(37) p∈1,6 log 2

log 55 we have Cp,d,µ ≥cp,d,µ ≥1

4 + Θ 1

√d21/p

551/6d

.

Example 3.10. When dealing with concrete families of measures it is possible to obtain

tighter bounds. We revisit the example from [A1, Remark 2.7], adapting the arguments given

there to p > 1. Let νd(A) := λd(A∩Bd) be Lebesgue measure restricted to the unit ball. We

apply Lemma 3.3 with R= 1 and v= 1/2, so H= 2−1√5, and Bd∩B(e1, H ) is the union

of two solid spherical caps, the largest of which is Bd∩ {x1≥3/8}(since the smaller sphere

has larger curvature). Solving x2

1+x2

2= 1 and (x1−1)2+x2

2=H2we obtain

νd(B(e1,2−1√5)) ≤2λd(Bd∩ {x1≥3/8}) = 2λd−1(Bd−1)Z1

3/8q1−x2

1d−1

dx1

≤16λd−1(Bd−1)

3Zπ/2

arcsin(3/8)

cosdtsin tdt =16

3(d+ 1) √55

8!d+1

λd−1(Bd−1).

By Lemma 3.3,

(38) cp,d,νd≥1

2d/q λd(Bd)

2λd−1(Bd−1)

3(d+ 1)

16 8

√55d+1

= Θ(√d)22+1/p

√55 d

.

Setting 22+1/p =√55 and solving for pwe obtain that cp,d grows exponentially fast with d

whenever

p < log 55

2 log 2 −2−1

≈1.1227.

Remark 3.11. It is possible to present more involved arguments in Theorem 3.4 and in

Example 3.10, by trying to optimize in v∈(0,1] instead of simply using v= 1/2. But

this does not seem to signiﬁcantly improve the value of p. Speciﬁcally , using B(e1, H )

with H=√1 + v2in Example 3.10, the same steps followed above lead us to maximize

Behavior of weak type bounds 11

g(v, q) := 2v1/q

(3+2v2−v4)1/2.The particular choice v= 1/2 yielded the critical value p0≈1.1227

and its conjugate exponent q0≈9.1474. Now it is elementary to check that for all q≤9

and all v∈[0,1), g(v, q)<1.Thus, with the methods of the present paper we cannot

get exponential increase in Example 3.10 for any p≥9/8 = 1.125, which is very close to

p0≈1.1227. Even the general bound p0≈1.0378 from Theorem 3.4 is not far from 1.125.

We mention that although for small values of q, say, q≈10 it is better to consider as our

Lpfunction χB(0,v)with v≈1/2, in order to maximize g(v, q) as q→ ∞, we must let v→0.

Of course, at the endpoint value p= 1, the Dirac delta measure δ0is a better choice than all

the functions χB(0,v),v∈(0,1).

In general, good upper bounds for cp,d and Cp,d are easier to establish when µis doubling,

that is, when there exists an absolute constant Csuch that for all x∈Rdand all R > 0,

µ(B(x, 2R)) ≤Cµ(B(x, R)). The doubling condition captures the property of Lebesgue

measure that yields weak type bounds via covering lemmas of Vitali type. It might be

expected that arbitrary doubling measures would behave like Lebesgue measure, but in our

context this is not the case: There is a collection of doubling measures for which exponential

increase holds even when pis high. For all t∈(0,1), let µt,d be deﬁned on Rdby dµt,d :=

kxk−td

2dx, and consider the families Mt:= {µt,d :d∈N\ {0}}. It is well known that the

measures µt,d are indeed doubling, cf. for instance [St3, 2.7, p. 12]. For simplicity, instead

of Cp,d,µt,d and cp,d,µt,d we shall write Cp,d,t and cp,d,t to denote the best strong type and weak

type (p, p) constants for the measures in Mt.

Theorem 3.12. Fix p0∈[1,∞). Then there exist constants t0=t0(p0)∈(0,1) and b0=

b0(p0)>1such that for all p∈[1, p0]and all t∈[t0,1), we have Cp,d,t ≥cp,d,t ≥b(1−t)d

0/6.

Proof. Assume that d≥2. We use χB(0,1/2) as our Lpfunction, taking R= 1 and H= 2−1√5

in Lemma 3.3. Given p0, we select a ﬁxed c∈(1,21/p0), so 21/p0/c > 1, and split B(e1, H ) into

the three sets B(0, c/2) ∩B(e1, H), (B(0,1) \B(0, c/2)) ∩B(e1, H ), and B(0,1)c∩B(e1, H).

We shall select t0=t0(c)<1 so close to 1 that for all t∈[t0,1) the dominant term will be

(39) µt,d(B(0, c/2) ∩B(e1, H )) ≤µt,d(B(0, c/2)) = σd−1(Sd−1)c

2(1−t)d

(1 −t)d,

where the last equality is obtained by integrating in polar coordinates. Solving x2

1+x2

2=c2/4

and (x1−1)2+x2

2= 5/4, we ﬁnd that the middle section (B(0,1) \B(0, c/2)) ∩B(e1, H ) is

contained in the cone subtended by the cap C((c2−1)/8, ce1/2), or, if we radially project this

cap into the unit sphere, by C((c2−1)/(4c), e1). The radius of the smallest ball containing

this cap is x2(c) = (4c)−1√18c2−c4−1. Since x2(1) = 1 and x2is strictly decreasing on

[1,2] (as can be seen from its geometric meaning or by diﬀerentiating) x2(c)<1 on (1,2]. By

Lemma 2.1, σd−1

N(C((c2−1)/(4c), e1)) = x2(c)d/Θ(√d) and

(40) µt,d ((B(0,1) \B(0, c/2)) ∩B(e1, H )) ≤x2(c)1/(1−t)(1−t)d

Θ(√d)

σd−1(Sd−1)

(1 −t)d.

12 J. M. Aldaz and J. P´erez-L´azaro

Solving x1

1+x2

2= 1 and (x1−1)2+x2

2= 5/4 shows that B(0,1)c∩B(e1, H) is contained

in the cone subtended by the cap C(3/8, e1). The distance from the boundary of this cap to

the x1axis is x2=√55/8. Thus, Lemma 2.1, together with integration in polar coordinates

from 1 to 1 + 2−1√5 in the radial variable and over the said cap in the angular variable, yield

(41) µt,d(B(0,1)c∩B(e1, H )) ≤

√55

8!1/(1−t)

1 + 2−1√5

(1−t)d

σd−1(Sd−1)

(1 −t)dΘ(√d).

Select d0=d0(c) such that if d≥d0, the expressions 1/Θ(√d) in (40) and (41) coming from

Lemma 2.1 are bounded above by 1. As t→1, both x2(c)1/(1−t)→0 and √55/81/(1−t)→0,

so by choosing t0close enough to 1, we can make the term in (39) larger than those in (40)

and (41) for every d≥d0and all t∈[t0,1). Hence,

(42) µt,d(B(e1, H )) ≤3µt,d(B(0, c/2)) = 3σd−1(Sd−1)c(1−t)d

(1 −t)d2(1−t)d.

By Lemma 3.3 we obtain cp,d,t ≥1

61

2(1−t)d/q 2

c(1−t)d=1

621/p

c(1−t)d≥1

621/p0

c(1−t)dfor

every p∈[1, p0]. Finally, by the choice of c, the inequality b0:= min{61/d0, c−121/p0}>1

holds, so we have exponential increase of the best constants in d(the term 61/d0has been

included to account for small values of d).

Example 3.13. Deﬁne µvon Rdas the sum of area measure on Splus Lebesgue on Bd(0, v).

If v= 1, then the arguments used in this paper apply and we do get exponential growth

of cp,d for suﬃciently small values of p > 1. However, suppose we let v→0; taking χB(0,r)

as our Lpfunction, we see that having v < r < 1 oﬀers no advantage over 0 < r ≤v, so

r→0 as v→0. This forces us to let q=q(r)→ ∞ in Lemma 3.3, and we do not obtain a

uniform value of pfor this family. While the measures µvare not absolutely continuous, by

taking fv=χ[0,v]+χ[1−1/d,1] we observe the same phenomenon for densities. Thus, additional

hypotheses are needed in order to go beyond radially decreasing densities.

References

[A1] Aldaz, J. M. Dimension dependency of the weak type (1,1) bounds for maximal functions associated

to ﬁnite radial measures. Bull. Lond. Math. Soc. 39 (2007) 203–208. Available at the Math. ArXiv.

[A2] Aldaz, J. M. The weak type (1,1) bounds for the maximal function associated to cubes grow to inﬁnity

with the dimension. Available at the Math. ArXiv.

[AlPe1] Aldaz, J.M., P´erez L´azaro, J. Functions of bounded variation, the derivative of the one dimensional

maximal function, and applications to inequalities, Trans. Amer. Math. Soc. 359 (5) (2007), 2443–

2461. Available at the Math. ArXiv.

[AlPe2] Aldaz, J. M.; P´erez L´azaro, J. Boundedness and unboundedness results for some maximal operators

on functions of bounded variation. J. Math. An. Appl. Volume 337, Issue 1, (2008) 130–143. Available

at the Math. ArXiv.

[AlPe3] Aldaz, J. M.; P´erez L´azaro, J. Regularity of the Hardy-Littlewood maximal operator on block de-

creasing functions. Studia Math. 194 (3) (2009) 253–277. Available at the Math. ArXiv.

Behavior of weak type bounds 13

[AlPe4] Aldaz, J. M.; P´erez L´azaro, J. The best constant for the centered maximal operator on radial func-

tions. To appear, Math. Ineq. Appl.; available at the Math. ArXiv.

[Au] Aubrun, G. Maximal inequality for high-dimensional cubes. Conﬂuentes Mathematici, Volume 1,

Issue 2, (2009) pp. 169–179, DOI No: 10.1142/S1793744209000067. Available at the Math. ArXiv.

[Bou1] Bourgain, J. On high-dimensional maximal functions associated to convex bodies. Amer. J. Math.

108 (1986), no. 6, 1467–1476.

[Bou2] Bourgain, J. On the Lp-bounds for maximal functions associated to convex bodies in Rn. Israel J.

Math. 54 (1986), no. 3, 257–265.

[Bou3] Bourgain, J. On dimension free maximal inequalities for convex symmetric bodies in Rn. Geometrical

aspects of functional analysis (1985/86), 168–176, LNM, 1267, Springer, Berlin, 1987.

[Ca] Carbery, A. An almost-orthogonality principle with applications to maximal functions associated to

convex bodies. Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 269–273.

[Cr] Criado, A. On the lack of dimension free estimates in Lp for maximal functions associated to radial

measures. Available at the Math. ArXiv.

[Gra] Gray, A. Tubes (Addison-Wesley Publishing Company, 1990).

[Mu] M¨uller, D. A geometric bound for maximal functions associated to convex bodies. Paciﬁc J. Math.

142 (1990), no. 2, 297–312.

[NaTa] Naor, A.; Tao, T. Random martingales and localization of maximal inequalities. To appear in J.

Func. Anal. Available at the Math. ArXiv.

[St1] Stein, E. M. The development of square functions in the work of A. Zygmund. Bull. Amer. Math.

Soc. (N.S.) 7 (1982), no. 2, 359–376.

[St2] Stein, E. M. Three variations on the theme of maximal functions. Recent progress in Fourier analysis

(El Escorial, 1983), 229–244, North-Holland Math. Stud., 111, North-Holland, Amsterdam, 1985.

[St3] Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals.

Princeton University Press, Princeton, NJ, 1993.

[StSt] Stein, E. M.; Str¨omberg, J. O. Behavior of maximal functions in Rnfor large n.Ark. Mat. 21

(1983), no. 2, 259–269.

[Su] Sullivan, John M. Sphere packings give an explicit bound for the Besicovitch covering theorem, J.

Geom. Anal. 4 (1994), no. 2, 219–231.

[Web] Webster, R. J. Convexity (Oxford University Press, 1997).

Departamento de Matem´

aticas, Universidad Aut´

onoma de Madrid, Cantoblanco 28049,

Madrid, Spain.

E-mail address:jesus.munarriz@uam.es

Departamento de Matem´

aticas y Computaci´

on, Universidad de La Rioja, 26004 Logro˜

no,

La Rioja, Spain.

E-mail address:javier.perezl@unirioja.es