Lipschitz spaces and Calderón-Zygmund operators associated to non-doubling measures

Publ. Mat. 49 (2005), 285–296 LIPSCHITZ SPACES AND CALDER ´ ON-ZYGMUND OPERATORS ASSOCIATED TO NON-DOUBLING MEASURES Jos´ e Garc´ ıa-Cuerva and A. Eduardo Gatto Abstract In the setting of a metric measure space (X, d, µ) with an n-dimensional Radon measure µ, we give a necessary and sufficient condition for the boundedness of Calder´on-Zygmund operators associated to the measure µon Lipschitz spaces on the support of µ. Also, for the Euclidean space Rdwith an arbitrary Radon measure µ, we give several characterizations of Lipschitz spaces on the support of µ,Lip(α, µ), in terms of mean oscillations involving µ. This allows us to view the “regular” BMO space of X. Tolsa as a limit case for α→0 of the spaces Lip(α, µ). 1. Introduction The present paper is devoted, on the one hand, to study the invariance of Lipschitz spaces under Calder´on-Zygmund operators associated to an n-dimensional Radon measure µ. We do that in Section 2, in the context of a metric measure space (X, d, µ) with an n-dimensional measure, that is a measure satisfying condition (2.1). This allows, in particular, non-doubling measures. A second aim is to show that, for any Radon measure in the Euclidean space Rd, the Lipschitz spaces can be characterized by a host of integral oscillation conditions similar to the regular BMO condition introduced by Tolsa. This shows that the regular BMO space of Tolsa is a limit case of the natural Lipschitz spaces associated to the measure. The study of Calder´on-Zygmund operators associated to an n-dimensional Radon measure was carried out, in the Lebesgue spaces, by 2000 Mathematics Subject Classification. 42B20, 26B35, 47B38, 47G10. Key words. Calder´on-Zygmund theory, singular integrals, Lipschitz spaces, BMO, non-doubling measures. Supported in part by DGES, Spain, under grant BFM2001-0189. It is a pleasure for the second author, to thank the members of the Mathematics Department of Universidad Aut´onoma de Madrid for their friendly hospitality. 286 J. Garc´ ıa-Cuerva, A. E. Gatto Nazarov, Treil and Volberg (see [NTV1], [NTV2]) and also by Tolsa (see [To1], [To2]). Further results, dealing with BMO and H1and providing boundedness criteria in the spirit of the T(1) or T(b) theorems, were obtained as well (see [NTV3], [MMNO], [To3]). In [GG] we have also studied, on metric spaces, the theory of fractional integral operators associated to an n-dimensional Radon measure µon Lebesgue spaces and Lipschitz spaces. In a previous version of this paper we were only considering n-dimensional Radon measures, which are the ones we are mainly interested on and the only ones for which we can prove the boundedness of Calder´onZygmund operators. However, Xavier Tolsa made the observation, that we gratefully acknowledge, that Theorem 3.3 and its proof were valid for a general Radon measure, since the n-dimensional nature of the measure, was never used. We also want to thank Professor Peter Constantin and the referee for appropriate questions about Theorem 2.5 that helped us to obtain the present statement. 2. Calder´on-Zygmund operators In this section, (X, d, µ) will be a metric measure space (that is, dis a distance on Xand µis a Borel measure on X), such that, for every ball B(x, r) = {y∈X:d(x, y)< r}, x ∈X, r > 0, we have (2.1) µ(B(x, r)) ≤Crn, where nis some fixed positive real number and Cis independent of x and r. We shall also refer to condition (2.1) by saying that the measure µis n-dimensional. Whenever we refer to “the ball B”, we shall understand that we have chosen for it a fixed center and a fixed radius. That way, it makes sense to say that if Bis a ball and kis a positive real number, we shall denote by kB the ball having the same center as Band radius ktimes that of B. From now on, we shall assume that µ(X) = ∞. We shall use below two basic lemmas from Section 2 of [GG], which allow us to bound the integrals against an n-dimensional measure of potential kernels on balls or complements of balls. For completeness, we group here in a single statement without proofs the two lemmas and a third property of the measure which will also be important in the sequel. Calder´ on-Zygmund Operators 287 Lemma 2.1. Let µbe an n-dimensional measure on (X, d, µ),γ > 0 and r > 0. Then ZB(x,r) 1 d(x, y)n−γdµ(y)≤Crγ a) ZX\B(x,r) 1 d(x, y)n+γdµ(y)≤Cr−γandb) Zr/2≤d(x,y)<r 1 d(x, y)ndµ(y)≤C,c) where, in all cases, Cis a constant independent of r. Definition 2.2. Given α, 0 <α<1, we shall say that a function defined on the support of µ,f: supp(µ)→Cis a Lipschitz function of order α when (2.2) |f(x)−f(y)| ≤ Cd(x, y)αfor every x, y ∈supp(µ) and the smallest constant in inequality (2.2) will be denoted kfkLip(α,µ). The linear space of Lebesgue classes of Lipschitz functions of order α, modulo constants, becomes a Banach space with the norm k kLip(α,µ); it will be denoted Lip(α, µ). Note also that if a function defined µ-almost everywhere on Xsatisfies (2.2) µ-a.e., then it coincides µ-a.e. with a Lipschitz function on the support of µ. Next, we define the class of singular kernels that we consider in this paper. Definition 2.3. A singular kernel on a metric measure space (X, d, µ) with µ n-dimensional, will be a measurable function K(x, y) on X×X\ {x=y}satisfying the following conditions: (1) |K(x, y)| ≤ A1 d(x,y)n. (2) |K(x1, y)−K(x2, y)| ≤ A2d(x1,x2)δ d(x1,y)n+δfor 2d(x1, x2)≤d(x1, y), where δ, 0< δ ≤1 is a regularity constant specific to the kernel. (3) lim ε→0Rε<d(x,y)<1K(x, y) dµ(y) exists for µ-almost every point x. With Kwe associate the truncated kernels Kε(x, y) = K(x, y)χ{d(x,y)>ε}(x, y). Finally, through the truncated kernels, we are in a position to introduce the singular integral operators that will be our object of study in this article. 288 J. Garc´ ıa-Cuerva, A. E. Gatto Definition 2.4. For f∈ Lip(α, µ), 0 < α < δ ≤1, we define e Tεf(x) = ZX (Kε(x, y)−K1(x0, y)) f(y) dµ(y), where x0is a fixed point in Xand then we also define e Tf(x) = lim ε→0e Tεf(x). Note that it follows from the properties of K(x, y) and Lemma 2.1 that the limit exists µ-almost everywhere. Indeed, for ε < 1 e Tεf(x) = Zd(x,y)<1 Kε(x, y) (f(y)−f(x)) dµ(y) + Zε<d(x,y)<1 K(x, y) dµ(y)!f(x) +ZX (K1(x, y)−K1(x0, y)) f(y) dµ(y), where the first and third integrals are absolutely convergent and the second term converges by property (3) of Definition 2.3. Theorem 2.5. Let Kbe a singular kernel as above and let e Tbe the corresponding singular integral operator. Let 0< α < δ ≤1. Then e T is a bounded operator on Lip(α, µ)if and only if there are constants B1 and B2such that (a) e T(1)(x) = B1µ-a.e. and (b) Zr<d(x,y)<R K(x, y) dµ(y)≤B2,for all 0< r < R and µ-a.e. x. Proof: We shall show first that conditions (a) and (b) are sufficient. Except for a set of µ-measure zero that depends on K(x, y) and µ, we have e Tf(x1)−e Tf(x2) = lim ε→0ne Tεf(x1)−e Tεf(x2)o = lim ε→0ZX (Kε(x1, y)−Kε(x2, y)) f(y) dµ(y). The same computation with f= 1 and condition (a) imply that lim ε→0ZX (Kε(x1, y)−Kε(x2, y)) dµ(y) = 0. Calder´ on-Zygmund Operators 289 Therefore e Tf(x1)−e Tf(x2) = lim ε→0ZX (Kε(x1, y)−Kε(x2, y)) (f(y)−f(x1)) dµ(y). Now let r=d(x1, x2) and take ε < r. After splitting the integral in the limit above as the sum of the integral over B(x1,3r) and the integral over X\B(x1,3r), we can write e Tf(x1)−e Tf(x2) = lim ε→0ZB(x1,3r) Kε(x1, y) (f(y)−f(x1)) dµ(y) −lim ε→0ZB(x1,3r) Kε(x2, y) (f(y)−f(x2)) dµ(y) −(f(x2)−f(x1)) lim ε→0ZB(x2,2r) Kε(x2, y) dµ(y) −(f(x2)−f(x1)) lim ε→0ZB(x1,3r)\B(x2,2r) Kε(x2, y) dµ(y) + lim ε→0ZX\B(x1,3r) (Kε(x1, y)−Kε(x2, y)) (f(y)−f(x1)) dµ(y) =I1−I2−I3−I4+I5. Observe now that, by combining condition (1) of Definition 2.3 together with the Lipschitz condition for fand using part a) of Lemma 2.1, the integral in I1converges absolutely and |I1| ≤ C1kfkLip(α,µ)rα. Then, after realizing that B(x1,3r)⊂B(x2,4r), we can use the same argument to see that the integral in I2also converges absolutely and, we have |I2| ≤ C2kfkLip(α,µ)rα. To control I3we use condition (b) to obtain |I3| ≤ B2kfkLip(α,µ)rα. Next, we can use part c) of Lemma 2.1 to prove that the integral in I4converges absolutely and we have |I4| ≤ C4kfkLip(α,µ)rα. Finally, using condition (2) in Definition 2.3, and part b) of Lemma 2.1, we see that the integral in I5converges absolutely and |I5| ≤ C5kfkLip(α,µ)rα. This completes the proof of the sufficiency. Next we shall show that the conditions are necessary. First of all, we observe that condition (a) follows from the fact that k1kLip(α,µ)= 0. To prove (b) consider first r=d(x, x2), x, x2∈supp(µ) such that the limit in condition (3) of Definition 2.3 exists both for xand x2. Let 290 J. Garc´ ıa-Cuerva, A. E. Gatto f(x) = d(x, x2)α. From the decomposition we made above in the proof of the sufficiency, we can write e T(f)(x)−e T(f)(x2)−I1+I2+I4−I5=−I3 =d(x, x2)αlim ε→0Zd(x2,y)<2r Kεdµ(y). Since the left hand side is less than or equal to B d(x, x2)αwith a constant Bindependent of x, we obtain lim ε→0Zd(x2,y)<2r Kε(x2, y) dµ(y)≤B with r=d(x, x2), µ-a.e. in supp(µ). Since µis n-dimensional, we also have (2.3) lim ε→0Zd(x2,y)<r Kε(x2, y) dµ(y)≤B0 with r=d(x, x2), µ-a.e. in supp(µ). Now, it is easy to see that condition (2.3) holds for all r > 0. Indeed, let r > 0 and assume that d(x2, x)6=r, for x∈supp(µ) except for a fixed set Eof measure 0. If there is no ¯x∈supp(µ)\Ewith r/2≤d(x2,¯x)< r, then Rr/2≤d(x2,y)<r Kε(x2, y) dµ(y)=0; if there is ¯xsatisfying d(x2,¯x)=s with r/2≤s < r, we have lim ε→0Zd(x2,y)<r Kε(x2, y) dµ(y)≤lim ε→0Zd(x2,y)<s Kε(x2, y) dµ(y) +Zr/2≤d(x2,y)<r |Kε(x2, y)|dµ(y) ≤B+C. Finally, we can get condition (b). Let 0 < r < R. We have Zr<d(x2,y)<R Kε(x2, y) dµ(y) = lim ε→0Zε<d(x2,y)<R K(x2, y) dµ(y) −lim ε→0Zε<d(x2,y)≤r K(x2, y) dµ(y), which implies (b) with constant B2independent of x2,rand R. Calder´ on-Zygmund Operators 291 Remark 2.6.Conditions (1) and (2) in Definition 2.3 are traditionally called “standard estimates” in the literature on singular integrals. Condition (3) is also classical for singular integrals of principal value type. Therefore, we can say that Theorem 2.5 is valid for all standard singular integrals of principal value type. Condition (b) in Theorem 2.5 is a weak cancellation condition that is also present in the classical literature on singular integrals on Euclidean spaces with Lebesgue measure dating back to Calder´on and Zygmund. It can be seen, for instance, in [BCP]. It has also appeared recently in the context of general measures in the work of P. Mattila. See, for example [M2], where it is used, for the Riesz kernels, to derive some local rectifiability properties of the measure. 3. Characterization of Lipschitz spaces All throughout this section, µwill be a fixed Radon measure on Rd. This is all that we need for the results we prove in this section, in particular for Theorem 3.3. Only for Remark 3.8 we will need to assume that the measure is n-dimensional. From now on, all balls that we consider will be centered at points in the support of µ. In order to prove the main theorem of this section we will need the following known definition and lemma (see [To3]). Definition 3.1. Let βbe a fixed constant. A ball Bis called β-doubling if µ(2B)≤βµ(B). Lemma 3.2. Let f∈L1 loc(µ). If β > 2d, then, for almost every xwith respect to µ, there exists a sequence of β-doubling balls Bj=B(x, rj) with rj→0, such that lim j→∞ 1 µ(Bj)ZBj f(y) dµ(y) = f(x). Proof: We will show that for almost every xwith respect to µthere is aβ-doubling ball centered at xwith radius as small as we wish. This fact, combined with the differentiation theorem, completes the proof of the lemma. We know that for almost every xwith respect to µ (3.1) lim r→0 µ(B(x, r)) rd>0 292 J. Garc´ ıa-Cuerva, A. E. Gatto (differentiation of µwith respect to Lebesgue measure, see [M1]). Now for xsatisfying (3.1) take B=B(x, r) and assume that none of the balls 2−kB,k≥1, is β-doubling. Then it easy to see that µ(B)> βkµ(2−kB) for all k≥1. Therefore µ(2−kB) (2−kr)d<2d βkµ(B) rd. Note that, since β > 2d, the right hand side tends to zero for k→ ∞, which is a contradiction. Now we can state and prove the main result of this section. Theorem 3.3. For a function f∈L1 loc(µ), the conditions I, II, and III below, are equivalent (I)There exist some constant C1and a collection of numbers fB, one for each ball B, such that these two properties hold: For any ball B with radius r (3.2) 1 µ(2B)ZB |f(x)−fB|dµ(x)≤C1rα, and for any ball Usuch that B⊂Uand radius(U)≤2r, (3.3) |fB−fU| ≤ C1rα. (II)There is a constant C2such that (3.4) |f(x)−f(y)| ≤ C2|x−y|α for µ-almost every xand yin the support of µ. (III)For any given p,1≤p≤ ∞, there is a constant C(p), such that for every ball Bof radius r, we have (3.5) 1 µ(B)ZB |f(x)−mB(f)|pdµ(x)1/p ≤C(p)rα, where mB(f) = 1 µ(B)RBf(y) dµ(y)and also for any ball Usuch that B⊂Uand radius(U)≤2r, (3.6) |mB(f)−mU(f)| ≤ C(p)rα. In addition, the quantities: inf C1,inf C2, and inf C(p)with a fixed p are equivalent. Proof: (I) ⇒(II). Consider xas in the lemma and let Bj=B(x, rj), j≥1, a sequence of β-doubling balls with rj→0. We will show first that (3.2) implies lim j→∞ fBj=f(x). Calder´ on-Zygmund Operators 293 It suffices to observe that mBj(f)−fBj≤1 µ(Bj)ZBjf(y)−fBjdµ(y) ≤µ(2Bj) µ(Bj) 1 µ(2Bj)ZBjf(y)−fBjdµ(y)≤βC1rα j. Next, let xand ybe two points as in the lemma. Take B=B(x, r) any ball with r≤ |x−y|and let U=B(x, 2|x−y|). Now define Bk= B(x, 2kr), for 0 ≤k≤¯ k, where ¯ kis the first integer such that 2¯ kr≥ |x−y|. Then |fB−fU| ≤ ¯ k−1 X k=0 fBk−fBk+1 +fB¯ k−fU ≤C1 ¯ k X k=0 2krα≤C0C1|x−y|α, where C0is independent of xand B. A similar argument can be made for the point ywith any ball B0= B(y, s) such that s≤ |x−y|and V=B(y, 3|x−y|). Therefore |fB−fB0| ≤ |fB−fU|+|fU−fV|+|fV−fB0| ≤ C00C1|x−y|α. Finally, take two sequences of β-doubling balls Bj=B(x, rj) and B0 j= B(y, sj) with rj→0 and sj→0. We have |f(x)−f(y)|= lim j→∞ fBj−fB0 j≤C00C1|x−y|α. (II) ⇒(III). It is immediate. Note also that (II) ⇒(I) is immediate as well. (III) ⇒(I). Define first fB=mB(f). Then (3.3) is exactly (3.6). In addition, the left hand side of (3.2) is less than or equal to the left hand side of (3.5). This concludes the proof of the theorem. Remark 3.4.Theorem 3.3 is also true if the number 2 in condition (I) is replaced by any fixed ρ > 1. In that case, the proof uses (ρ, β)-doubling balls, that is, balls satisfying µ(ρB)≤βµ(B). However this extension is not needed in our paper.