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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.
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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
