An example on the maximal function associated to a nondoubling measure

Publ. Mat. 49 (2005), 453–458 AN EXAMPLE ON THE MAXIMAL FUNCTION ASSOCIATED TO A NONDOUBLING MEASURE J. M. Aldaz Abstract We show that there is a measure µ, defined on the hyperbolic plane and with polynomial growth, such that the centered maximal operator associated to µdoes not satisfy weak type (1,1) bounds. 1. Introduction and main result Let Xbe a metric space and let µbe a Borel measure defined on X. In a paper [NTV] that has exerted considerable influence in later developments, F. Nazarov, S. Treil, and A. Volberg showed that a good deal of the theory of Calder´on Zygmund operators still holds if one replaces the doubling condition on the measure µby the following polynomial growth condition: There exist constants c,dsuch that for every x∈X and every radius r > 0, µ(B(x, r)) ≤crd. Polynomial growth is a natural assumption in this area: In the euclidean case X=Rd, G. David showed it is needed for the L2boundedness of singular integral operators that, like the Hilbert transform, are associated to kernels Ksatisfying |K(x, y)| ≥ C|x−y|−d(cf. [Pa, Comment 1, p. 60]). The “Calder´on Zygmund philosophy” consists in controlling singular integral operators via the maximal function, which for doubling measures is of weak type (1,1). In this context it is natural to ask whether the assumption µ(B(x, r)) ≤crdcan replace the doubling hypothesis and still yield weak type (1,1) bounds. F. Nazarov, S. Treil, and A. Volberg bypassed this issue in [NTV], resorting instead to a modified maximal function ˜ M, where one does not take the usual average but divides by the measure of the ball with the same center and triple radius: ˜ Mf(x) := supr>0(µ(B(x, 3r)))−1RB(x,r)|f|dµ. This modified maximal 2000 Mathematics Subject Classification. 42B25. Key words. Maximal function, hyperbolic plane, nondoubling measure with polynomial growth. Partially supported by Grant BFM2003-06335-C03-03 of the D.G.I. of Spain. 454 J. M. Aldaz function is of weak type (1,1), but in principle does not control anything. Polynomial growth is then used to show that there exist “large doubling balls” centered at every point, making it possible to utilize ˜ Min order to find bounds for other operators. A positive answer to the question whether the hypothesis µ(B(x, r)) ≤ crdis sufficient to ensure that the weak type (1,1) of the centered maximal function would have allowed a development of the theory more in parallel with the classical case. The negative answer we obtain shows that such reduction is not possible. In the example we present the metric space Xis just the hyperbolic plane H, with a suitably defined Borel measure. More precisely, we prove the following Theorem. There exists a Borel measure µon the hyperbolic plane H and a constant c > 0such that for every w∈Hand every radius s > 0, µ(B(w, s)) ≤cs, and the centered maximal function associated to µis not of weak type (1,1). Let us make a comment on terminology: In [NTV] and some later papers, a measure satisfying µ(B(x, r)) ≤crdis called d-dimensional. But the preceding condition does not really give a genuine notion of dimension: some measures can have uncountably many “dimensions” (for instance, planar Lebesgue measure on the unit disc), while other perfectly good measures have none (for example, the doubling measure µ(A) := RAx2dx on R, which under any reasonable definition ought to be regarded as one-dimensional). So we think it is more precise to speak of polynomial growth, as is done, for instance, in [Pa]. This research was carried out while visiting the Universidad Aut´onoma de Madrid. I am indebted to the Department of Mathematics, and specially to Prof. Jos´e Garc´ıa-Cuerva, for the invitation, and also for several useful conversations. 2. Proof of the theorem We shall utilize the upper half plane model of the hyperbolic plane H. The following properties of this model, which can be found in geometry textbooks (see for instance, [McC, p. 237]), will be used in the proof. The hyperbolic balls B=Bh(p, s) are also euclidean balls B=Be(w, r), but centers and radii vary: Namely, if w= (a, b) is the euclidean center of Band r < b its euclidean radius, then the hyperbolic center of B is p= (a, √b2−r2), and the hyperbolic radius is s= log qb+r b−r. So as a topological space, our set Xis just {(x, y)∈R2:y > 0} with the usual topology, metrized by the hyperbolic distance. We define The Centered Maximal Operator 455 a Borel measure µon X=Has follows: let m1be the restriction to the upper half plane of the usual Gaussian probability on R2, let dm2:= χAdw, where dw stands for planar Lebesgue measure and A:= {(x, y)∈R2:x > 1, x−1> y > 0}, and finally, set µ:= m1+m2. First we show that Mµf(w) := sup s>0 1 µBh(w, s)ZBh(w,s)|f|dµ is not of weak type (1,1). By the usual approximation argument via convolutions, we may use a Dirac delta instead of a function. So consider δ(R+1/2,1), where R0. We will see that µMµδ(R+1/2,1)(w)>(R−1)3/2 3>1 2R, from which the result follows by letting R→ ∞. Since µ{(x, y)∈R2:R < x < R + 1,0< y < x−1} > m2{(x, y)∈R2:R < x < R + 1,0< y < x−1} = log 1 + 1 R>1 2R, it is enough to prove that {(x, y)∈R2:R < x < R + 1,0< y < x−1} ⊂Mµδ(R+1/2,1)(w)>(R−1)3/2 3. Fix (x, y) such that R < x < R + 1, 0 < y < x−1, and select r∈[1/2,1) so that the hyperbolic center of Be((x, 1), r) is (x, y). Since Be((x, 1), r) contains the point (R+ 1/2,1), Mµδ(R+1/2,1)((x, y)) ≥1 µBe((x, 1), r)≥1 µBe((R, 1),1). To estimate µBe((R, 1),1), note first that m1Be((R, 1),1) <Z∞ R−1 e−t2/2dt < Z∞ R−1 t R−1e−t2/2dt =e−(R−1)2/2 R−1. Suppose next that (x−R)2+ (y−1)2<1 and 0 < y < x−1. Solving for yin (x−R)2+ (y−1)2<1 we get (x−R)2 2<1−p1−(x−R)2< y < x−1<1 R−1. 456 J. M. Aldaz So {(x, y)∈R2: (x−R)2+ (y−1)2<1,0< y < x−1} ⊂((x, y)∈R2:R−r2 R−1<x<R+r2 R−1,0< y < (R−1)−1), whence m2Be((R, 1),1) <Z1 R−1 0ZR+√2 R−1 R−√2 R−1 dx dy =2√2 (R−1)3/2. Thus, by taking Rsufficiently large, µ Be((R, 1),1) <3 (R−1)3/2, and it follows that Mµis not of weak type (1,1). Next we prove that there is a constant c > 0 such that for all w∈H, µBh(w, s)≤cs. We find constants c1, c2>0 with m1Bh(w, s)≤c1s and m2Bh(w, s)≤c2sfor all w∈H. From now on we shall adhere to the convention whereby a constant cmay change its value from one line to the next. Recall that we use Be((a, b), r) = Bh((a, b0), s) to denote the same ball B⊂H, with respect to the euclidean metric in the first case and the hyperbolic metric in the second. Here s=1 2log 1 + 2r b−r. For small values of b, and therefore of r,scontrols r, so m1and m2can simply be replaced by planar Lebesgue measure to prove the polynomial growth. More precisely, suppose 0 < b ≤3. Then 0 <r<3 and there exists a c > 0 such that s≥cr. So for i= 1,2, miBh((a, b0), s) = miBe((a, b), r)≤cr ≤c0s. Suppose next that b > 3. Then m2Be((a, b), r) = 0 unless r > b −1. In this case we have m2Bh((a, b0), s) = m2Be((a, b), r)≤log(1+2r)≤log 1 + 2r b−r= 2s. Finally, to prove the polynomial growth of m1on {b > 3}we consider the following cases. If rand bare comparable, say, b/3≤r < b, everything is trivial, since m1Bh((a, b0), s) = m1Be((a, b), r)≤1/2 and s≥1 2log 2. The Centered Maximal Operator 457 So suppose 0 < r < b/3. Then m1Bh((a, b0), s) = m1Be((a, b), r)≤m1Be((0, b), r) ≤1 2πZr −rZ∞ b−r e−x2+y2 2dy dx ≤1 2π e−(b−r)2 2 b−rZr −r e−x2 2dx ≤e−(b−r)2 2 b−rmin{1,2r}. Now if 0 < r < 1/2, then e−(b−r)2 2 b−r2r≤c2r b≤c01 2log 1 + 2r b≤c0s, while if 1/2≤r < b/3, then e−(b−r)2 2 b−r≤c0 b≤c00 1 2log 1 + 1 b≤c00s. Remark. One might ask for which locally finite Borel measures µon H is the centered maximal function Mµof weak type (1,1). Locally finite means that for every x∈Hthere is an open neighborhood of xwith finite measure. This implies that compact sets and balls have finite measure. If µhas compact support (and hence it is finite), then it follows from Besicovitch’s covering theorem that Mµis of weak type (1,1). But being finite, even in the presence of polynomial growth, is not enough to ensure the weak type of Mµ. The example we present above can be easily modified so that µ(H)<∞: Instead of using the set {(x, y)∈R2: x > 1, x−1>y>0}to define m2, take for instance A:= {(x, y)∈R2: x > 0, e−x> y > 0}, and argue as before. On the other hand, the centered maximal function associated to area in the hyperbolic plane is of weak type (1,1) (cf. [Str]), even though area does not satisfy any polynomial growth condition. So there seems to be no significant relationship between the weak type of the maximal operator and the polynomial growth of the underlying measure. To finish, we mention that while the doubling condition on the measure is sufficient to ensure the weak type of the maximal function, for H this is irrelevant: It follows from the remark in p. 67 of [CW] together with Example 3.5.2 of [Luu] that the hyperbolic plane admits no doubling measures (in particular, area is not doubling). The referee points out that one can relax the doubling condition on a measure µ, by requiring that it hold only at the points of its support, and 458 J. M. Aldaz this weaker version also entails the weak type of the maximal operator. With this definition there are plenty of “doubling measures” on H, so one might object to the assertion that the doubling condition on the measure is irrelevant for H. However, this weakened version of doubling ignores in an essential way the geometry of H. For instance, let µbe the linear Lebesgue measure on {(x, 1) : x∈R} ⊂ H. This measure is certainly doubling in the weaker sense; but I would be inclined to think that if one works with µone is doing analysis on the real line rather than on the hyperbolic plane. In fact, any such weakly doubling measure must live on a set with geometry substantially different from that of H. References [CW] R. R. Coifman and G. Weiss,“Analyse harmonique noncommutative sur certains espaces homog`enes. ´ Etude de certaines int`egrales singuli`eres”, Lecture Notes in Mathematics 242, Springer-Verlag, Berlin-New York, 1971. [Luu] J. Luukkainen, Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35(1) (1998), 23–76. [McC] J. McCleary,“Geometry from a differentiable viewpoint”, Cambridge University Press, Cambridge, 1994. [NTV] F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calder´on-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 9(1998), 463–487. [Pa] H. Pajot,“Analytic capacity, rectifiability, Menger curvature and the Cauchy integral”, Lecture Notes in Mathematics 1799, Springer-Verlag, Berlin, 2002. [Str] J.-O. Str¨ omberg, Weak type L1estimates for maximal functions on noncompact symmetric spaces, Ann. of Math. (2) 114(1) (1981), 115–126. Departamento de Matem´aticas y Computaci´on Universidad de La Rioja 26004 Logro˜no, La Rioja Spain E-mail address:[email protected] Primera versi´o rebuda el 25 de gener de 2005, darrera versi´o rebuda el 10 de mar¸c de 2005.