Weighted inequalities and vector-valued Calderón-Zygmund operators on non-homogeneous spaces

Publicacions Matem`atiques, Vol. 44 (2000), 613–640 WEIGHTED INEQUALITIES AND VECTOR-VALUED CALDER´ ON-ZYGMUND OPERATORS ON NON-HOMOGENEOUS SPACES J. Garc´ ıa-Cuerva and J. M. Martell Abstract Recently, F. Nazarov, S. Treil and A. Volberg (and independently X. Tolsa) have extended the classical theory of Calder´on-Zygmund operators to the context of a “non-homogeneous” space (X,d,µ), where, in particular, the measure µmay be non-doubling. In the present work we study weighted inequalities for these operators. Specifically, for 1 <p<∞, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted Lpinequality holds. We deal with this problem by developing a vector-valued theory for Calder´on-Zygmund operators on non-homogeneous spaces which is interesting in its own right. For the case of the Cauchy integral operator, which is the most important example, we even prove that the conditions for the weights are also necessary. 1. Introduction Let µbe a Borel measure in the complex plane. The Cauchy integral operator is defined as Cf(z)=Cµf(z)=C f(ξ) z−ξdµ(ξ),for µ-a.e. z∈C\supp f. It is natural to wonder whether this operator is bounded on L2(µ), on Lp(µ)orevenbetweenL1(µ) and L1,∞(µ). Besides, since the previous definition makes no sense for points in the support of the function, 2000 Mathematics Subject Classification. 42B20, 30E20. Key words. Non-doubling measures, Calder´on-Zygmund operators, vector-valued inequalities, weights, Cauchy integral. Both authors are partially supported by DGES Spain, under Grant PB97-0030. We would like to thank J. L. Torrea for many comments and suggestions. 614 J. Garc´ ıa-Cuerva, J. M. Martell another question is to find conditions on µin order to ensure the existence of the principal values on these spaces. For example, when µ is the one-dimensional Hausdorff measure over a Lipschitz curve, the boundedness was proved in [Cal] for small Lipschitz constant and the full result was obtained in [CMM]. Another approach to this problem is the T(b) theorem proved in [DJS], (see also [Da1]). For measures over rectifiable sets and the relation with analytic capacity see [Ch1], [Ch2], [Mur] and the references given there. See also the recent survey [Da2]. The answer for general measures has been obtained by Tolsa in [To1], [To2]. In the first work, it is established the equivalence of the uniform boundedness of the truncated Cauchy integrals in L2(µ) and some geometric conditions on the measure µ, namely: µhas linear growth —which means that the measure of each ball is controlled by a constat times the radius— and it satisfies certain local curvature condition (see [To1], [Mel], [MV], [MMV]). In the second reference, the author obtains that the boundedness in L2(µ) implies the existence of principal values. Besides, those measures, for which the existence of principal values holds, are completely characterized. In[NTV1]aT(1) theorem is proved for Calder´on-Zygmund operators in Cwith a measure such that µ(Q)≤(Q) for all squares Q⊂C, where (Q) stands for the side length of Q. They prove that Tis continuous in L2(µ) if and only if Tand its adjoint are bounded over characteristic functions of squares. (Actually, as it is pointed out in [NTV1], similar results work for “n-dimensional” measures in Rd,d≥nand Calder´onZygmund operators with “n-dimensional” kernels K.) In particular for the Cauchy integral, this result has been also obtained by [Ver]. In[NTV2] a generalization of this setting is given. They deal with nonhomogeneous spaces which are metric spaces endowed with a positive measure in such a way that the measure of a ball is controlled by the radius to the power n, where n>0 is a fixed real number. In these spaces (where the measure is not assumed to satisfy any doubling condition) from the L2(µ) boundedness, the authors manage to obtain weak and strong type estimates for Calder´on-Zygmund operators and for the maximal operators associated with them. The main example is the Cauchy integral where the metric space is Cand n=1. A non-homogeneous space (X,d) will be a separable metric space endowed with a non-negative “n-dimensional” Borel measure µ, that is, µ(B(x, r)) ≤rn,for all x∈X,r>0, where B(x, r)={y∈X:d(x, y)≤r}and nis a fixed positive number (not necessarily an integer). Two-Weight Theory on Non-Homogeneous Spaces 615 Definition 1.1. A bounded linear operator Ton L2(µ) is said to be a Calder´on-Zygmund operator with “n-dimensional” kernel Kif for every f∈L2(µ), Tf(x)=X K(x, y)f(y)dµ(y),for µ-almost every x∈X\supp f, where, for some A>0, K:X×X−→ Csatisfies (i) |K(x, y)|≤ A d(x, y)n,for all x=y; (ii) and the following two conditions hold: d(x,y)≥2d(x,x)|K(x, y)−K(x,y)|dµ(y)≤A, d(x,y)≥2d(x,x)|K(y,x)−K(y,x)|dµ(y)≤A. Note that this class of operators is slightly larger than those considered in [NTV2] where pointwise estimates for the kernel are used rather than integral estimates. Observe that if we take some measure µin C such that the Cauchy integral is bounded in L2(µ), µwill have linear growth (e.g. [To1]), that is, µis “1-dimensional”. In this case, the Cauchy integral is a Calder´on-Zygmund operator with “1-dimensional” kernel K(z,ξ)= 1 z−ξ. The aim of this paper is to obtain some weighted inequalities for these operators. If 1 <p<∞, consider the following two-weight inequality for T:X|Tf(x)|pu(x)dµ(x)≤X|f(x)|pv(x)dµ(x),(1) for f∈Lp(v)=Lp(vdµ) and where u,vare µ-a.e. positive functions. These inequalities in Rdwith the same weight in both sides have been recently studied by [OP]. They have obtained some results about Muckenhoupt weights and weighted inequalities for Calder´on-Zygmund operators. However, we are interested in a different type of inequalities, namely, we shall be concerned with the following problem: Find conditions on 0≤v<∞µ-a.e. (resp. u>0 µ-a.e.) such that (1) is satisfied by some u>0µ-a.e. (resp. 0≤v<∞µ-a.e.). 616 J. Garc´ ıa-Cuerva, J. M. Martell As we can see in Chapter VI of [GR] and in Chapters II, V of [Ste] this question is closely related to obtaining vector-valued inequalities for T. We shall use this connection to get an answer for this problem, that is, we shall develop a vector-valued theory for these operators to obtain the necessary vector-valued inequalities. Some references about classical vector-valued theory are [BCP], [RRT] and [GR]. For these operators the relevant classes of weights will be, as usual, Dpand Zp,1<p<∞, which are defined as follows: Dp=0≤w<∞µ-a.e. : X w(x)1−p(1 + d(x, x0))−np dµ(x)<∞ Zp=w>0µ-a.e. : X w(x)(1+d(x, x0))−npdµ(x)<∞, for some x0∈X. Note that these classes of weights do not depend on the point x0and that this definition becomes simpler for spaces with finite diameter (see Section 3). The concrete result is Theorem. Take p,1<p<∞.Ifu∈Zp(resp. v∈Dp), then there exists some weight 0<v<∞µ-a.e. (resp. 0<u<∞µ-a.e.) such that (1) holds. Moreover, v(resp. u) can be found in such a way that vα∈Zp(resp. uα∈Dp), provided that 0<α<1. Once we have obtained sufficient conditions on the weights in order to ensure that (1) holds, we shall study how sharp are these classes, that is, we shall prove that for a particular example these conditions are also necessary. In [GR] this problem is treated for classical Calder´onZygmund operators in Rn. There the Riesz transforms are used to show that those classes of weights are necessary. In our setting this rˆole will be played by the Cauchy integral. Take a measure µfor which the Cauchy integral is bounded in L2(µ), see [To1]. Now, the weighted inequality is C|Cf(z)|pu(z)dµ(z)≤C(u, v)C|f(z)|pv(z)dµ(z),(2) for any f∈Lp(vdµ). We devote Section 4 to get the following theorem, which is, essentially, the converse of the previous one. Theorem. Take p,1<p<∞. Given 0<u<∞µ-a.e. (resp. 0<v< ∞µ-a.e.), if there exists some weight 0<v<∞µ-a.e. (resp. 0<u< ∞µ-a.e.) such that (2) holds, then u∈Zp(resp. v∈Dp). Two-Weight Theory on Non-Homogeneous Spaces 617 The plan of the paper is the following. Section 2 contains a vectorvalued version of the main theorem in [NTV2], which shall be proved in three steps. In Subsections 2.1 and 2.2 we shall obtain the weak type (1,1) estimate, whereas the strong inequalities are considered in Subsection 2.3. An immediate consequence is given in Subsection 2.4, indeed the vector-valued inequalities obtained there will be the main tool for solving the problem we are concerned with. Sections 3, 4 are devoted to this problem: the first one for general operators and the second one for the particular case of the Cauchy integral, where the necessity is proved. 2. The vector-valued theorem Throughout this section we shall consider vector-valued operators, that is, operators which take their values in Banach spaces. Let A,Bbe a couple of Banach spaces. L(A,B) will denote the set of bounded linear operators from Ato B. We shall say that K:X×X−→ L(A,B) is a (vector-valued) “n-dimensional” Calder´on-Zygmund kernel if, for some A>0, it verifies (i) K(x, y)L(A,B)≤A d(x, y)n,for all x=y; (ii) and the following two conditions hold: d(x,y)≥2d(x,x)K(x, y)−K(x,y)L(A,B)dµ(y)≤A, d(x,y)≥2d(x,x)K(y,x)−K(y,x)L(A,B)dµ(y)≤A. Definition 2.1. Let Tbe a linear operator mapping boundedly L2 A(µ) into L2 B(µ), such that, for any f∈L2 A(µ), Tf(x)=X K(x, y)f(y)dµ(y),for µ-a.e. x∈X\supp f, where Kis an “n-dimensional” Calder´on-Zygmund kernel. Then we shall say that Tis a vector-valued Calder´on-Zygmund operator. For r>0, the truncated operators are defined as follows Trf(x)=X\B(x,r) K(x, y)f(y)dµ(y), and we can consider the maximal operator associated with T, Tf(x) = sup r>0Trf(x)B. 618 J. Garc´ ıa-Cuerva, J. M. Martell For 1 ≤p≤∞, it is well known that Lp A∗(µ)⊂(Lp A(µ))∗.If the Banach space Ais reflexive equality holds, however it fails in general. When we deal with a reflexive Banach space A, we can define T∗, the adjoint of T, which turns out to be a vector-valued Calder´onZygmund operator that maps boundedly L2 B∗(µ)intoL2 A∗(µ). The kernel is  K(x, y)=K(y,x)∗∈L(B∗,A∗) (the adjoint operator of K(y,x)). Besides, T∗L2 B∗(µ)→L2 A∗(µ)≤TL2 A(µ)→L2 B(µ). Since K(y,x)∗L(B∗,A∗)≤ K(y,x)L(A,B)and K(y1,x 1)∗−K(y2,x 2)∗L(B∗,A∗)≤K(y1,x 1)−K(y2,x 2)L(A,B),  Kwill be an “n-dimensional Calder´on-Zygmund kernel with the same constant A. Let M(X) be the space of all complex-valued Borel measures on X. The space A⊗M(X) will consist of all finite linear combinations of elements of the form aη with a∈Aand η∈M(X). For one of these elements we define by convenience T(aη)(x)=X K(x, y)adη(y),x∈X\supp η. As in [NTV2], we consider the following version of the Hardy-Littlewood maximal function:  Mf(x) = sup r>0 1 µ(B(x, 3r)) B(x,r)|f|dµ. This maximal function is bounded in Lp(µ), 1 <p≤∞, and acts continuously from L1(µ)toL1,∞(µ). We shall need the following result, which is a kind of boundedness of an “atom” away from its support. Lemma 2.2. For η=J i=1 aiηi∈A⊗M(X)with supp η⊂B(x, ρ) and η(X)=X dη = J  i=1 aiX dηi= J  i=1 aiηi(X)=0, we have X\B(x,2ρ)Tη(y)Bdµ(y)≤A J  i=1 aiAηi, where Ais the constant in the definition of the kernel. Two-Weight Theory on Non-Homogeneous Spaces 619 Proof: The proof is standard. By using the properties of ηand condition (ii), we can write X\B(x,2ρ)Tη(y)Bdµ(y) =X\B(x,2ρ) J  i=1 B(x,ρ) (K(y,x)−K(y,x))aidηi(x)B dµ(y) ≤ J  i=1 aiA B(x,ρ)d(x,y)≥2d(x,x) K(y,x)−K(y,x)L(A,B)dµ(y)d|ηi|(x) ≤A J  i=1 aiAηi. Remark 2.3.Just as before, the following can be proved: if η= J i=1 aiηi+fdµ ∈A⊗M(X)+L1 A(µ) with supp η⊂B(x, ρ) and η(X)=J i=1 aiηi(X)+Xfdµ= 0, we also obtain X\B(x,2ρ)Tη(y)Bdµ(y)≤AJ  i=1 aiAηi+fL1 A(µ). 2.1. Weak type inequality for elementary measures. An elementary measure will be an element of A⊗M(X), where the measures involved are unit point masses, namely ν= N  i=1 αiδxi∈A⊗M(X). Theorem 2.4. For an elementary measure as above, the following inequality holds TνL1,∞ B(µ)≤C N  i=1 αiA, where Conly depends on the dimension n, the constant Ain the definition of the kernel Kand the norm TL2 A(µ)→L2 B(µ). 620 J. Garc´ ıa-Cuerva, J. M. Martell Observe that here, there is no problem with the definition of Tν because the sum is finite and Tν(x)= N  i=1 T(αiδxi)(x)= N  i=1 K(x, xi)αi makes sense everywhere except at finitely many points. Proof: We shall follow the proof of [NTV2, Theorem 5.1] paying special attention to those details that differ from the scalar case. We can assume that N i=1 αiA= 1 and prove that TνL1,∞ B(µ)≤C. Fix some t>0, and suppose that µ(X)>1 t. Following the “scalar” case proof —with αiAinstead of αi— we are able to find some Borel sets E1,...,E N such that B(xi,ρ i)\ i−1  =1 E⊂Ei⊂B(xi,ρ i)\ i−1  =1 Eand µ(Ei)=αiA t, where B(xi,ρ i)={y∈X:d(xi,y)<ρ i}. It is clear that the sets Ei are pairwise disjoint, if we put E=iEi,  i B(xi,ρ i)⊂E⊂ i B(xi,ρ i) and µ(E)=1 t. Define σ= i χX\B(xi,2ρi)Tαi αiA χEi, and Tν−tσ = i ϕi= iT(αiδxi)−tχ X\B(xi,2ρi)Tαi αiA χEi. Since B(xi,ρ i)⊂E,wehave X\EϕiBdµ ≤X\B(xi,2ρi)Tαiδxi−tαi αiA χEidµB dµ +B(xi,2ρi)\B(xi,ρi)T(αiδxi)Bdµ ≤2AαiA+2 nAαiA, where we have used Lemma 2.2 for the first term and condition (i) of the kernel for the second. Thus X\ETν−tσBdµ ≤ N  i=1 X\EϕiBdµ ≤2n+1A N  i=1 αiA=2 n+1A, Two-Weight Theory on Non-Homogeneous Spaces 621 and µ{x∈X:(Tν−tσ)(x)B>2n+1At}≤2 t, since µ(E)=1 t. Then, it might be enough to find some big constant A0such that µ{σB>A 0}≤2 t.(3) In this case, µ{x∈X:Tν(x)B>(2n+1A+A0)t}≤4 t. In order to finish we only have to observe that the above inequality is obvious when µ(X)≤1 t. Then, if we take C=4(2 n+1A+A0), we have just obtained TνL1,∞ B(µ)≤C. Let us show how can we get (3) in this vector-valued framework. First, we prove this inequality under the assumption that Ais a reflexive Banach space. For a fixed A0, to be chosen later, suppose that µ{σB> A0}>2 t. Then, there exists a Borel set F,F⊂{σB>A 0}, such that µ(F)=1 t.Thusσχ F∈L1 B(µ), because Xσχ FBdµ ≤µ(F)1/2σL2 B(µ)≤TL2 A(µ)→L2 B(µ) 1 t N  i=1 αi1/2 A<∞. Since L1 B(µ) is isometrically contained in (L∞ B∗(µ))∗, the Hahn-Banach theorem implies the existence of some β∈L∞ B∗(µ), βL∞ B∗(µ)= 1, such that β,σχF=σχ F(L∞ B∗(µ))∗=Fσ(x)Bdµ(x)>A 0µ(F)=A0 t.(4) On the other hand, βχ F∈L2 B∗(µ) with βχ FL2 B∗(µ)≤t−1/2and we can use the adjoint operator to obtain β,σχF=Xσ(x)χF(x),β(x)dµ(x) = N  i=1 Xαi αiA χEi(x),T∗(βχ F\B(xi,2ρi))(x)dµ(x) ≤ N  i=1 X χEi(x)T∗(βχ F\B(xi,2ρi))(x)A∗dµ(x). For every x∈Ei⊂B(xi,ρ i), by condition (i) of the kernel, T∗(βχ F\B(xi,2ρi))(x)−T∗(βχ F\B(x,ρi))(x)A∗ ≤B(xi,2ρi)\B(x,ρi) K(x, y)L(B∗,A∗)β(y)B∗dµ(y)≤2nA. 628 J. Garc´ ıa-Cuerva, J. M. Martell vector-valued theory, and we can use the self-improvement result (Theorem 2.9) in order to obtain this sequence-valued extension. Corollary 2.10. Let Tbe an operator as above and take q,1<q<∞. Then (i) µ    x:   j|Tfj(x)|q   1 q >λ     ≤C λX   j|fj(x)|q   1 q dµ(x). (ii)    j|Tfj|q   1 qLp(µ) ≤C   j|fj|q   1 qLp(µ) ,if1<p<∞. Remark 2.11.These vector-valued results will be further used in [GM] to obtain similar estimates for the maximal operator associated with T, which, under the appropriate conditions, fits into this vector-valued theory. In particular, we shall prove the previous inequalities for the supremum of the truncated Cauchy integrals. By means of them, weighted inequalities for this maximal operator will be obtained and we shall be able to study the existence of principal values in weighted Lebesgue spaces. 3. Vector-valued inequalities and weights The relation between weighted inequalities and vector-valued inequalities was discovered by J. L. Rubio de Francia in [R] and it can be also found in Chapter VI of [GR]. The two-weight problem for an operator Tconsists in finding all pairs (u, v) of positive functions for which the inequality X|Tf(x)|pu(x)dµ(x)≤C(u, v)X|f(x)|pv(x)dµ(x),(5) (f∈Lp(v)) holds true. We are going to consider the following weak variant of this general problem: Find conditions on 0≤v<∞µ-a.e. (resp. u>0 µ-a.e.) such that (5) is satisfied by some u>0µ-a.e. (resp. 0≤v<∞µ-a.e.). To start, we need the following result, proved in [FT], which establishes the concrete relationship between vector-valued inequalities and weights. This theorem is closely related to those contained in [GR, pp. 549–554]. Two-Weight Theory on Non-Homogeneous Spaces 629 Theorem 3.1. Let (Y,dν)be a measure space; F,GBanach spaces, and {Ak}k∈Za sequence of pairwise disjoint measurable subsets of Y such that Y=kAk. Consider 0<s<p<∞and Ta sublinear operator which satisfies the following vector-valued inequality    jTfjp G   1 pLs(Ak,d ν) ≤Ck   jfjp F   1 p ,k∈Z,(6) where, for every k∈Z,Ckonly depends on F,G,pand s. Then, there exists a positive function u(x)on Ysuch that YTf(x)p Gu(x)dν(x)1 p ≤CfF where Cdepends on F,G,pand s. Moreover, given a sequence of positive numbers {ak}k∈Zwith kap k<∞, and σ=p s,u(x)can be found in such a way that u−1χAkLσ−1(Ak,dµ)≤(a−1 kCk)p. In our context (Y,dν)=(X,dµ) which is a σ-finite measure space. Then, a simple argument shows that the weight ucan be also taken so that u<∞a.e. Given 1 <p<∞and some x0∈X, remember the definition of the classes of weights in X: Dp=0≤w<∞µ-a.e. : X w(x)1−p(1 + d(x, x0))−np dµ(x)<∞ Zp=w>0µ-a.e. : X w(x)(1 + d(x, x0))−npdµ(x)<∞. Note that these classes of weights do not depend on the point x0. Remark 3.2.In the case that the diameter of the space is finite, (or equivalently, the distance is bounded), there exists Rlarge enough such that X⊂B(x0,R) and so µ(X)≤Rn<∞. Thus, the previous classes can be given by the equivalent definition: Dp=0≤w<∞µ-a.e. : X w(x)1−pdµ(x)<∞ Zp=w>0µ-a.e. : X w(x)dµ(x)<∞. 630 J. Garc´ ıa-Cuerva, J. M. Martell If the support of the measure is a bounded set, we can restrict the whole space to this set, and we would be in the previous case. So, when we talk about spaces with finite diameter, we shall be concerned with both cases. We would like to apply the last theorem to our operators. In what follows Twill be a “scalar” Calder´on-Zygmund operator T, that is, an operator like those in Definition 1.1. Proposition 3.3. Take 0<s<1<p<∞and v∈Dp. Then, if the diameter of Xis equal to infinity, we have    j|Tfj|p   1 pLs(Sk,d µ) ≤Cs,p2kn s   jfjp Lp(vdµ)   1 p , for k=0,1,..., where S0={x:d(x, x0)≤1}and Sk={x:2 k−1<d(x, x0)≤2k}, for k=1,2,.... Otherwise,    j|Tfj|p   1 pLs(µ) ≤Cs,p    jfjp Lp(vdµ)   1 p . Proof: Let us see what happens in the first situation. Fix k≥0 and set Bk+1 =B(x0,2k+1). Every function fis split as f=f+f = fχ Bk+1 +fχ X\Bk+1 . Then, for x∈Skand y∈X\Bk+1 we observe that 2d(x, y)>d(y,x0) and thus |Tf(x)|≤X\Bk+1 A d(x, y)n|f(y)|dµ(y) ≤4nAX (1 + d(y,x0))−n|f(y)|v(y)1 pv(y)−1 pdµ(y) ≤4nAX|f(y)|pv(y)dµ(y)1 pX v(y)1−p (1 + d(y,x0))np dµ(y)1 p ≤CfLp(vdµ). Two-Weight Theory on Non-Homogeneous Spaces 631 Note that the last inequality holds because v∈Dp. Then, since µ(Sk)≤ µ(Bk)≤2kn, we prove    j|Tf j|p   1 pLs(Sk,dµ) ≤C2kn s   jfjp Lp(vdµ)   1 p . On the other hand, due to that fact that 0 <s<1, we can use Kolmogorov inequality (see [GR, p. 485]) and Corollary 2.10 to obtain    j|Tf j|p   1 pLs(Sk,dµ) ≤Csµ(Sk)1 s−1   j|Tf j|p   1 pL1,∞(Sk,dµ) ≤Cµ(Sk)1 s−1Bk+1    j|fj(x)|p   1 p v(x)1 pv(x)−1 pdµ(x) ≤Cµ(Sk)1 s−1  X j|fj(x)|pv(x)dµ(x)   1 pBk+1 v(x)−p pdµ(x)1 p =Cµ(Sk)1 s−1   jfjp Lp(vdµ)   1 pBk+1 v(x)1−pdµ(x)1 p . As 1 s−1>0, we observe µ(Sk)1 s−1≤µ(Bk)1 s−1≤(2kn)1 s−1. Furthermore, Bk+1 v(x)1−pdµ(x)1 p =Bk+1 v(x)1−p (1 + d(x, x0))np (1 + d(x, x0))np dµ(x)1 p ≤C2n2(k+1) n, since v∈Dp. Then,    j|Tf j|p   1 pLs(Sk,dµ) ≤C2kn s   jfjp Lp(vdµ)   1 p . Collecting these inequalities, we get the desired estimate. 632 J. Garc´ ıa-Cuerva, J. M. Martell When the space has finite diameter, it measure will be finite as well. Thus, we proceed like we did with the functions f j. Since 0 <s<1, we can apply Kolmogorov inequality (see [GR, p. 485]) and Corollary 2.10 to obtain    j|Tfj|p   1 pLs(µ) ≤Csµ(X)1 s−1   j|Tfj|p   1 pL1,∞(µ) ≤Cµ(X)1 s−1X   j|fj(x)|p   1 p v(x)1 pv(x)−1 pdµ(x) ≤Cµ(X)1 s−1  X j|fj(x)|pv(x)dµ(x)   1 pX v(x)−p pdµ(x)1 p ≤C   jfjp Lp(vdµ)   1 p , because Xhas finite measure and v∈Dp(which, in this case, means v1−p∈L1(µ)). Once we have the vector-valued inequalities we can use Theorem 3.1 to obtain weighted inequalities. Theorem 3.4. Take p,1<p<∞.Ifu∈Zp(resp. v∈Dp), then there exists some weight 0<v<∞µ-a.e. (resp. 0<u<∞µ-a.e.) such that (5) holds. Moreover, v(resp. u) can be found in such a way that vα∈Zp(resp. uα∈Dp), provided that 0<α<1. Proof: Assume that the case v∈Dpis proved. If u∈Zp, then u= u1−p∈Dp. Apply this assumption to the adjoint operator T∗(which is an operator with the same properties as T) in order to obtain some weight v,0<v<∞µ-a.e., such that X|T∗f(x)|pv(x)dµ(x)≤CX|f(x)|pu(x)dµ(x). Take vso that v=v1−p. Then, since 0 <v<∞µ-a.e., an standard argument yields that the last inequality implies (5). Furthermore, we can choose vsuch that vα∈Dp, provided that 0 <α<1. That is, we can find vin such a way that vα∈Zp. Two-Weight Theory on Non-Homogeneous Spaces 633 Let us prove the case v∈Dp. Fix 0 <α<1 and put q=1+α(p−1). Then 1 <q<p and we can find some s,0<s<1, such that σ=p s> q. When Xhas infinite diameter, we use Theorem 3.1 with (Y,dν)= (X,dµ), F=Lp(vdµ), G=C,{Ak}k={Sk}∞ k=0 and Ck=C2kn s. The vector-valued inequality (6) is supplied by Proposition 3.3. Then, we know that there exists a weight usuch that (5) holds. Moreover, u can be taken in such a way that u−1Lσ−1(Sk,dµ)≤C(a−1 k2kn s)p, with ak>0 and kap k<∞. Therefore, X u(x)1−q (1 + d(x, x0))np dµ(x)= ∞  k=0 Sk u(x)1−q (1 + d(x, x0))np dµ(x) ≤2np  ∞  k=0 2−knp Sk u(x)1−σdµ(x)q−1 σ−1 µ(Sk) 1 (σ−1 q−1) ≤2np C ∞  k=0 a−p(q−1) k2 kn −p+p(q−1) s+1 (σ−1 q−1), where we have used H¨older’s inequality with exponent σ−1 q−1>1. Observe that −p+p(q−1) s+1 σ−1 q−1=q−p<0, so, we can choose ε>0 such that q−p+ε<0. Take the sequence {ak}k verifying a−p(q−1) k=2 knε. Then ∞  k=0 ap k= ∞  k=0 2−knε q−1<∞ and X u(x)1−q (1 + d(x, x0))np dµ(x)≤C ∞  k=0 2kn(q−p+ε)<∞. In order to finish it is enough to note that α=1−q 1−pand thus uα∈Dp. When the space has finite diameter, as well as before, we use Theorem 3.1 with (Y,dν)=(X,dµ), F=Lp(vdµ), G=C. In this case, we do not decompose the space, that is, we just take A0=Xand Ak=∅if k= 0. The vector-valued inequality (6) is provided by the second part of Proposition 3.3. Then, there exists a weight usuch that (5) holds. 634 J. Garc´ ıa-Cuerva, J. M. Martell Furthermore, ucan be taken in such a way that u−1Lσ−1(X,dµ)≤C. Since the measure of Xis finite and σ−1 q−1>1, we use H¨older’s inequality for this exponent to conclude X u(x)1−qdµ(x)≤X u(x)1−σdµ(x)q−1 σ−1 µ(X) 1 (σ−1 q−1) <∞. Observe that α=1−q 1−pand we have uα∈Dp. 4. Cauchy integral operator For a non-negative Borel measure µin the complex plane C, the Cauchy integral operator of a compactly supported function f∈Lp(µ), 1≤p≤∞, is defined as Cf(z)=Cµf(z)=C f(ξ) z−ξdµ(ξ),for µ-a.e. z∈C\supp f. Assume that µis such that the truncated Cauchy integrals are uniformly bounded in L2(µ). By [To1], µwill be in particular “1-dimensional”. In that case, we know that the existence of the principal value for compactly supported functions holds (see [To2]). Then, a bounded extension to the whole L2(µ) arises from these facts. Thus, we have a metric space C with the euclidean metric and µa “1-dimensional” measure for which the Cauchy integral operator is bounded in L2(µ). We observe that this operator falls into the theory developed by [NTV2]. The Cauchy integral operator is defined for compactly supported function in L2(µ) by means of its kernel K(z,ξ)= 1 z−ξ, that is clearly a “1-dimensional” Calder´onZygmund kernel. Then we can apply the results we have obtained to get vector-valued inequalities for C. By Corollary 2.10, the following result is established. Theorem 4.1. Under the above assumptions and for 1<p,q<∞we have (i) µ    z∈C:   j|Cfj(z)|q   1 q >λ     ≤C λC   j|fj(z)|q   1 q dµ(z). (ii)    j|Cfj|q   1 qLp(µ) ≤C   j|fj|q   1 qLp(µ) . Two-Weight Theory on Non-Homogeneous Spaces 635 In this framework, for 1 <p<∞, the classes of weights will be Dp=0≤w<∞µ-a.e. : C w(z)1−p(1 + |z|)−pdµ(z)<∞ Zp=w>0µ-a.e. : C w(z)(1 + |z|)−pdµ(z)<∞. If the measure has bounded support, these classes admit the equivalent definition given in Remark 3.2. In fact, several results will be easier when it happens. For w≥0 a.e. we denote w(A)=Aw(z)dµ(z), for any measurable set A⊂C. We would like to apply to this operator the results about weights we have proved. The point is that here we can obtain that these classes are sharp for this weak variant of the two-weight problem for the Cauchy integral operator: suppose that for some fixed 0 <u,v<∞µ-a.e., the following two-weight inequality holds C|Cf(z)|pu(z)dµ(z)≤C(u, v)C|f(z)|pv(z)dµ(z),(7) for any f∈Lp(vdµ). We are going to prove that, in this case, the weights belong to the given classes. If z=z1+iz 2,ξ=ξ1+iξ 2and fis a real-valued function, for µ-a.e. z∈C\supp f, we observe Cf(z) = Re(Cf(z)) + iIm(Cf(z)) =C z1−ξ1 |z−ξ|2f(ξ)dµ(ξ)−iC z2−ξ2 |z−ξ|2f(ξ)dµ(ξ). Lemma 4.2. Assume that (7) holds. Then for any z∈supp µthere exits a radius rz>0, such that, u(B(z,rz)) <∞. Proof: Fix z0=z0 1+iz0 2∈supp µ, then µ(B(z0,r)) >0 for all r>0. For z=z1+iz 2, we write |z|∞= max{|z1|,|z2|} and F1={z∈C:|z−z0|∞=z1−z0 1},F 2={z∈C:|z−z0|∞=z2−z0 2}, F3={z∈C:|z−z0|∞=z0 1−z1},F 4={z∈C:|z−z0|∞=z0 2−z2}. Set Bk=B(z0,2−k) and Sk=Bk\Bk+1. Then, there is some k0≥0 such that Sk0has positive measure (otherwise µ(B0) = 0). Assume for instance that µ(Sk0 F1)>0 (in the other cases we proceed in a similar way). Thus, there will exist A⊂Sk0 F1so that µ(A)>0 and v(A)<∞.Forz∈Bk0+2 and ξ∈A, we have |z−ξ|≤5·2−k0−2. Since 636 J. Garc´ ıa-Cuerva, J. M. Martell ξ∈A⊂F1, 2−k0−1≤|ξ−z0|≤√2 max{|ξ1−z0 1|,|ξ2−z0 2|} =√2(ξ1−z0 1). Besides, z1−z0 1≥−|z−z0|≥−2−k0−2and ξ1−z1=ξ1−z0 1+z0 1−z1≥1 √2|ξ−z0|−2−k0−2≥(√2−1)2−k0−2. Therefore, for z∈Bk0+2,ξ∈A ξ1−z1 |z−ξ|2≥(√2−1)2−k0−2 (5 ·2−k0−2)2=√2−1 25 2k0+2 =Ck0. Then, if z∈Bk0+2, −Re(C(χA)(z)) = A ξ1−z1 |z−ξ|2dµ(ξ)≥Ck0µ(A)=C>0. So, for the left hand side of (7) we have C|C(χA)(z)|pu(z)dµ(z)≥Bk0+2 (−Re(C(χA)(z)))pu(z)dµ(z) ≥CpBk0+2 u(z)dµ(z). Use this estimate and (7), with f=χA∈Lp(v), to obtain u(Bk0+2)< ∞. Then, by taking rz0=2 −k0−2the proof is finished. Lemma 4.3. Assume that (7) holds, then there exists R>0such that C\B(0,R) u(z) (1 + |z|)pdµ(z)<∞. Proof: For j=1,...,4, set Ejby putting z0= 0 in the definition of Fj. Then, it might be enough to find some Rj>0, for each j, such that, Ej\B(0,Rj) u(z) (1 + |z|)pdµ(z)<∞. We shall only do it for j= 1 and the other cases can be performed in the same manner. We can assume that µ(E1)>0 (otherwise there is nothing to prove). If E supp µis a bounded set, the estimate is trivial by choosing R1large enough. In the other case, there exists R1such that µ(B(0,R 1/2) E1)>0. Take A⊂B(0,R 1/2) E1with µ(A)>0 and v(A)<∞. Then, for z∈E1\B(0,R 1) and ξ∈A,|z|>2|ξ|and |z−ξ|≤|z|+|ξ|≤3 2|z|. Moreover, since both points belong to E1, |z|=!z2 1+z2 2≤√2 max{|z1|,|z2|} =√2z1,ξ 1=|ξ1|≤|ξ|<1 2|z|, Two-Weight Theory on Non-Homogeneous Spaces 637 and hence z1−ξ1≥1 √2|z|−1 2|z|=√2−1 2|z|. Then, z1−ξ1 |z−ξ|2≥2(√2−1) 9 1 |z|≥2(√2−1) 9 1 1+|z|, and for z∈E1\B(0,R 1), Re(C(χA)(z)) = A z1−ξ1 |z−ξ|2dµ(ξ)≥2(√2−1) 9 1 1+|z|µ(A) =C 1+|z|>0. Therefore, for the left hand side of (7) we get C|C(χA)(z)|pu(z)dµ(z)≥E1\B(0,R1) (Re(C(χA)(z)))pu(z)dµ(z) ≥CpE1\B(0,R1) u(z) (1 + |z|)pdµ(z). Since v(A)<∞, (7) can be used. Then the right hand side of this inequality is finite and the proof is finished. Now, we are able to prove the following result, which, together with Theorem 3.4, gives us necessary and sufficient conditions on the weights in order to solve, for the Cauchy integral operator, the weak variant of the two-weight problem we are dealing with. Theorem 4.4. Take p,1<p<∞. Given 0<u<∞µ-a.e. (resp. 0< v<∞µ-a.e.), if there exists some weight 0<v<∞µ-a.e. (resp. 0< u<∞µ-a.e.) such that (7) holds, then u∈Zp(resp. v∈Dp). Proof: We shall use the previous lemmas. Fix 0 <u,v<∞µ-a.e. such that (7) holds. By taking the radius R>0 supplied by Lemma 4.3, we only have to see what happens on the ball. Lemma 4.2 and a compactness argument lead to B(0,R) u(z) (1 + |z|)pdµ(z)<∞.