ACCRETIVE SYSTEM T b-THEOREMS ON NONHOMOGENEOUS SPACES

We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderón-Zygmund operator on L2(μ). We do not assume any kind of doubling condition on the measure μ, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L. Thus we answer positively a question of Christ as to whether the L-assumption can be replaced by a BMO assumption. We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry. In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calderón-Zygmund operators with respect to very bad measures. 0. Introduction: Main objects and results In what follows, the symbol K (x, y) stands for a Calderón-Zygmund kernel defined for x, y ∈ Rn of order d: |K (x, y)| ≤ C1 |x − y|d , |K (x1, y)− K (x2, y)| + |K (y, x1)− K (y, x2)| ≤ C2|x1 − x2| |x1 − y|d+α for a positive α and any points x1, x2, y satisfying |x1 − x2| ≤ (1/2)|x1 − y|. We always assume that μ is adapted to K via the Ahlfors condition: μ ( B(x, r) ) ≤ C3r d . However, we assume no estimate from below. DUKE MATHEMATICAL JOURNAL Vol. 113, No. 2, c © 2002 Received 25 May 2000. Revision received 9 July 2001. 2000 Mathematics Subject Classification. Primary 47B37; Secondary 30E20. Authors’ work supported by National Science Foundation grant number DMS 9970395 and United States–Israel Binational Science Foundation grant number 00030. 259 260 NAZAROV, TREIL, and VOLBERG By a Calderón-Zygmund operator with kernel K we mean any bounded operator on L2(μ) such that T f (x) = ∫ K (x, y) f (y) dμ(y) (0.1) for all x outside the support of f dμ. Sometimes we use the phrase an operator with Calderón-Zygmund kernel. By that we mean any operator with this property for f ’s from its domain of definition (which can consist of, say, all bounded functions with compact support, or all smooth functions, or anything like that). Usually we are dealing with boundedness of an operator with Calderón-Zygmund kernel on L2(μ) (then it is bounded on all L p(μ), 1 < p <∞; see [24]). Very often the kernel K is continuous (and the measure has a compact support); thus (0.1) makes sense for any x and defines a Calderón-Zygmund operator. But we are not interested in this obvious L2-boundedness. Our goal is to obtain the bound that depends only on C1,C2,C3, α from the definition of K and maybe on some other parameters, but not on the norm of K as a continuous function. Notice also that kernel K does not define T uniquely. In fact, we have the following. THEOREM 0.1 Let T1, T2 be Calderón-Zygmund operators with kernel K . Then T1 f − T2 f = m · f , where m is a bounded μ measurable function. One can find an easy proof in [27, Chap. 1]. The next definition differs essentially from the classical one. Given a 3 > 1, we introduce the space BMO3, BMO3(μ) := { f ∈ Lloc(μ) : ‖ f ‖ 2 ∗ := sup Q 1 μ(3Q) ∫ Q | f − 〈 f 〉Q | 2 dμ <∞ } , where Q stands for an arbitrary cube in Rn . Here 〈 f 〉Q := (1/μ(Q)) ∫ Q f dμ. We remark that the space BMO3(μ) depends essentially on 3 (see [25]). However, in what follows, any 3 > 1 works. These doubling BMO spaces turn out to be natural when one works with nondoubling measures. They are obviously larger than the usual BMO space. The main players in what follows are accretive systems. A system of functions {bS} is called an accretive system if there exists δ > 0 such that for any cube S in Rn ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES 261 there exists bS from the system having the properties ‖bS‖∞ ≤ 1, ∣∣∣ ∫ S bS dμ ∣∣∣ ≥ δ · μ(S). As always, T is a Calderón-Zygmund operator. We denote the set of CalderónZygmund parameters of the kernel by z. To formulate our results, let us introduce three sets of assumptions on T and an accretive system. Each set of assumptions is weaker than the previous one. By an almost cube, we mean a rectangle in Rn with sides parallel to the axis and with ratio of any two sides between 1/2 and 2. In all three sets of assumptions, we are talking about the action of T on bounded functions. In the first two sets of assumptions, T acts on bounded functions with compact support. In the third set of assumptions, we are working with T b, where b is, generally, a bounded function but not with compact support. In this case, the meaning of T b is not readily clear even if T is assumed to be bounded in L2(μ). How to interpret condition T b ∈ BMO3. To interpret this condition in our situation, we assume only that T maps bounded functions with compact support into L2(μ). In particular, this is so if T is a priori bounded in L2(μ). Fix a cube Q. We define T b as a functional on L0(Q, μ), where L 2 0(Q, μ) := { f ∈ L2(Q, μ) : ∫ Q f dμ = 0}. Fix a function a from L 2 0(Q, μ), and let R = 3Q with 3 > 1. Let b1 := bχR, b2 := b − b1. We have (T b)(a) := (T b, a) := (T b1, a)L2(μ) + (b 2, T a). (0.2) Notice that this definition does not depend on 3 > 1. Notice also that the last term is an absolutely convergent integral: (b2, T a) = ∫ Q ∫ Rn\R [K (x, y)− K (x0, y)]a(x)b(y) dμ(x) dμ(y). (0.3) We introduce T bχRn\R := ∫ Rn\R[K (x, y)− K (x0, y)]b(y) dμ(y). We say that T b ∈ BMO3 if the thus-defined functional satisfies |(T b)(a)| ≤ Cμ(3Q)‖a‖L2(μ) with the same finite C for all Q. The absolute convergence of the last integral is standard (see, e.g., [25, Lem. 2.1]. Here are our three sets of assumptions. 262 NAZAROV, TREIL, and VOLBERG L-system of accretive functions supported on almost cubes. For every almost cube Q, there is a function bQ with the following properties: supp bQ ⊂ Q, |〈bQ〉Q | := ∣∣∣ 1 μ(Q) ∫ Q bQ dμ ∣∣∣ ≥ δ, ‖bQ‖∞ ≤ 1, ‖T (bQ)‖∞ ≤ B. BMO3-system of accretive functions supported on almost cubes. We have the following: supp bQ ⊂ Q, |〈bQ〉Q | := ∣∣∣ 1 μ(Q) ∫ Q bQ dμ ∣∣∣ ≥ δ, ‖bQ‖∞ ≤ 1, ‖T QBMO23 ≤ B. BMO3-system of accretive functions assigned to almost cubes. We call the attention of the reader to the fact that in this set of assumptions we do not assume that bQ are supported on Q: |〈bQ〉Q | := ∣∣∣ 1 μ(Q) ∫ Q bQdμ ∣∣∣ ≥ δ, ‖bQ‖∞ ≤ 1, ‖T QBMO23 ≤ B. Our main results are the following. THEOREM 0.2 Let T be a Calderón-Zygmund operator with Calderón-Zygmund parameters z and norm ‖T ‖. Then there exists an L-system of accretive functions supported on almost cubes {bS} such that ‖T bS‖∞ ≤ B(z, ‖T ‖) < ∞ and such that the constant of accretivity δ ≥ c(z, ‖T ‖) > 0. Conversely (notice that we need accretive functions only on cubes rather than on almost cubes), we have the following. ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES 263 THEOREM 0.3 Let T be a Calderón-Zygmund operator with Calderón-Zygmund parameters z and norm ‖T ‖. Suppose that for T there exists an L-system of accretive functions supported on cubes {bS}. Suppose that for T ∗ there exists an L-system of accretive functions supported on cubes {bS}. Let B := sup{‖T b 1 S‖∞, ‖T bS‖∞}. Then ‖T ‖ ≤ A(z, δ, B) <∞. These results are the generalization of the results of Christ [1] to nonhomogeneous spaces. Theorem 0.4 allows one to replace an L-assumption with a BMO assumption, answering a question of Christ which seemed to be open even in the homogeneous situation. THEOREM 0.4 Let T be a Calderón-Zygmund operator with Calderón-Zygmund parameters z and norm ‖T ‖. Suppose also that T has an antisymmetric kernel. Suppose that there exists a BMO3-system of accretive functions assigned to cubes {bS} such that B := sup ‖T SBMO23 . Then ‖T ‖ ≤ A(z, δ, B) <∞. The “classical” T b-theorem for Calderón-Zygmund operators with antisymmetric kernels on nonhomogeneous spaces (see [25]) corresponds to the case bS = b with b being weakly accretive. Notice that in all the theorems above there is no assumption of weak boundedness. But weak boundedness is usually essential in the theory. The explanation is simple: in Theorem 0.4 antisymmetry is a sort of weak boundedness. In Theorem 0.3 the existence of a local L accretive system turns out to be such a powerful assumption that it allows us to forget about weak boundedness. This effect was found earlier by Christ in [1] for the homogeneous situation. In general, nonhomogeneous harmonic analysis has received considerable attention recently. We refer the reader to [5], [6], [22]–[25], and [29]–[31]. Some applications One of the corollaries of Theorems 0.2 and 0.3 is the following result, in which we use the notions of analytic capacity γ and the Cauchy integral operator. Recall that for a measure μ on the complex plane C (the canonical value of) the Cauchy integral While preparing the galley proofs, the authors found out about the preprint of P. Auscher, S. Hoffman, C. Muscalu, T. Tao, and C. Thiele, “Carleson measures, extrapolation, and T (b) theorems,” which contains close results about the local T (b) theorems (but in a homogeneous setting) along with interesting connections with the theory of phase space analysis from the point of view of wave packets on tiles. 264 NAZAROV, TREIL, and VOLBERG operator Tμ is defined by its bilinear form: (Tμφ,ψ) := ∫ ∫ φ(z)ψ(ζ )− φ(ζ )ψ(z) ζ − z dμ(ζ ) dμ(z) for φ,ψ ∈ C 0 . The analytic capacity γ of a compact subset K of the plane is γ (K ) := sup { lim z→∞ |z f (z)| : f is bounded and analytic in C \ K , ‖ f ‖∞ ≤ 1, f (∞) = 0 } . By the symbol H 1 we denote the 1-Hausdorff measure. It is well known that γ (K ) ≤ H 1(K ). THEOREM 0.5 Let E be a compact subset of C, 0 < H 1(E) < ∞. Then the Cauchy integral operator T with respect to H 1|E is bounded on L2(H 1|E) if and only if there exists δ > 0 such that δ ·H 1(S ∩ E) ≤ γ (S ∩ E) for any square S. This result has been proved by T. Murai [21] under the extra assumption that E is Ahlfors regular, namely, when there exist 0 < c1, c2 <∞ such that c1l(S) ≤ H 1(S ∩ E) ≤ c2l(S) for every square S centered on E , where l(S) denotes the length of the side of S. The meaning of Theorem 0.5 is that the Cauchy integral operator acts boundedly on L2 with respect to H 1|E if and only if the analytic capacity on portions of E is equivalent to the measure. One can deduce from Theorem 0.5 another proof of G. David’s celebrated characterization of Ahlfors-David curves (originally obtained in [4]). We refer the reader to the preprint [26], where this new proof is explained together with the above mentioned generalization of Murai’s theorem. The proof has a sinful feature: there is absolutely no geometry involved in it. But sometimes this may become useful. The proof also illustrates that the assumptions of our accretive system theorems are actually verifiable in practice. Finally, let us remark that we think that accretive system T b-theorems are the most flexible tools for dealing with Calderón-Zygmund operators. ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES 265 Plan In Section 1 we list the basic results on Calderón-Zygmund operators on nonhomogeneous spaces. We use them in what follows. Then we prove Theorem 0.2 in Section 2. In Section 3 we start to prove Theorems 0.3 and 0.4. Section 7 finishes the proofs. In the appendix we prove some technicalities needed in the estimate of the diagonal part of T . Sections 4 and 5 are the most difficult and involved. They use the strictly technical results from the appendix (surgery by means of accretive systems). 1. Basic facts on Calderón-Zygmund operators on nonhomogeneous spaces We use the notation T ∈ C Z(K ) to say that T is a Calderón-Zygmund operator with kernel K . We also need the notion of the cut-off of T : Tε f (x) := ∫ y:|y−x |≥ε K (x, y) f (y) dμ(y), T ε f (x) := ∫ K (x, y)ψ ( |x − y| ε ) f (y) dμ(y), where ψ is a C function, which vanishes on B(0, 1/2) and equals 1 on Rn \B(0, 1). Notice that T ε are operators with Calderón-Zygmund kernels, while Tε are not. The right maximal operator is now M̃ f (x) := sup r 1 μ(B(x, 3r)) ∫ B(x,r) | f (y)| dμ(y). Obviously, ∣∣(Tε − T ε )( f )(x)∣∣ ≤ AM̃ f (x). (1.1) We also introduce the singular maximal function T ] f (x) := sup ε |Tε f (x)|. The next theorem is important in what follows (see [24] for the proof). THEOREM 1.1 If T ∈ C Z(K ), then any of the following equivalent assertions hold: (1) {Tε}ε>0 are uniformly bounded in L2(μ), (2) {T ε }ε>0 are uniformly bounded in L2(μ), (3) T ] is bounded in L2(μ). Let M0(R) stand for all complex measures with compact support in Rn . THEOREM 1.2 If T ∈ C Z(K ), then 266 NAZAROV, TREIL, and VOLBERG (1) T is a bounded operator from L1(μ) to L1,∞(μ); (2) if, in addition, the kernel K is continuous, then T is a bounded operator from M0(R) to L1,∞(μ) and its norm depends only on the Calderón-Zygmund parameters z and the norm of T in L2(μ). The next lemma is well known and widely used (see [1]). LEMMA 1.3 Let X be a compact Hausdorff space, and let T be a bounded linear operator from M(X ) to C(X ), where M(X ) is the space of complex measures on X . Also, assume that the adjoint operator T ∗ acts from M(X ) to C(X ), and assume that for a finite positive measure μ the following holds: μ{x ∈ X : |T ν(x)| > t} ≤ A‖ν‖ t , ∀ν ∈ M(X ). (1.2) Then for any Borel set E ⊂ X , 0 < μ(E), there exists a function h on E, 0 ≤ h ≤ 1, such that ∫ E h dμ ≥ μ(E) 2 , (1.3) ‖T (h dμ)‖∞ < 4 A. (1.4) THEOREM 1.4 Let T ∈ C Z(K ) with ‖T ‖ denoting its norm as an operator in L2(μ). Let δ ∈ (0, 1). Then there exist constants C1,C2 depending on δ and ‖T ‖ only such that T ] f (x) ≤ C1(M̃ |T f | δ)1/δ(x)+ C2(M̃ f )(x), ∀ f ∈ L 1(μ). (1.5) This is called Cotlar’s inequality. Notice that |T f | is summable by Theorem 1.2. This makes the right-hand side almost everywhere finite. 2. The proof of Theorem 0.2 We prove a slightly more general result. Our operator is assumed to be bounded in L2(E, dμ). (Here E denotes just the support of μ.) Then its norm ‖T ‖ bounds it as an operator in L2(E , μ) for any Borel subset E ′ of E . Thus Theorem 0.2 follows from the next theorem. THEOREM 2.1 Let μ be supported by a compact set E, and let an operator T with CalderónZygmund kernel be bounded in L2(μ). Then there exist δ > 0 and B < ∞ which ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES 267 depend only on ‖T ‖, and a function b such that ‖b‖∞ ≤ 1, (2.1) ∫ E b dμ ≥ δ μ(E), (2.2) ‖T (b dμ)‖∞ ≤ B. (2.3) Proof We deduce from Theorems 1.1 and 1.2 that (T ε ) ] : M(E)→ L1,∞(μ) (2.4) uniformly in ε. Obviously, T ε satisfies all the assumptions of Lemma 1.3. Applying Lemma 1.3, we obtain the family of functions {h} having the following properties: 0 ≤ h ≤ 1, ∫ E h dμ ≥ δμ(E), ‖T ε (h ε dμ)‖∞ ≤ B. Let us consider a decreasing sequence εn → 0 such that hn → h weakly in L2(μ) and Tεn → T0 in the weak operator topology. (We can do that because the Tε are uniformly bounded operators from a separable to a reflexive space.) It is clear that T0 ∈ C Z(K ). Let hn := hn , Tn := Tεn . LEMMA 2.2 The functions Tn hm , m ≥ n, are uniformly bounded. Proof We know that ‖T ε (h dμ)‖∞ ≤ B. Using (1.1) and the uniform boundedness of hm , we conclude that ‖Tm hm‖ ≤ B1. Similarly to (1.1), |Tδ f (x)− T2δ f (x)| ≤ A M̃ f (x). Thus if εn ∈ [εm, 2εm], then ‖Tn hm‖∞ ≤ B2. If εn > 2εm , then (T ψ εm )εn = Tεn . Thus in this case, ‖Tn hm‖∞ = ‖(T ψ εm )εn h m‖∞ ≤ ‖(T ψ εm ) hm‖∞. (2.5) By Cotlar’s inequality (see Th. 1.4), we get (T εm ) ] hm (x) ≤ C1(M̃ |T ψ εm hm |δ)1/δ(x)+ C2(M̃h εm )(x). (2.6) 268 NAZAROV, TREIL, and VOLBERG But both hm and T εm h εm are uniformly bounded. Thus (2.5) and (2.6) give us, finally, the uniform boundedness of Tn hm for all m ≥ n. To continue the proof of Theorem 2.1, let us choose {an k } mn k=0, a n k ≥ 0, ∑ k a n k = 1, in such a way that gn := ∑mn j=0 a n j hn+ j converges in L 2(μ)-norm to an h. Consider Tn gn . Then ‖Tn gn‖∞ ≤ ( mn ∑ j=0 aj ) sup m≥n ‖Tn hm‖∞ ≤ A <∞. In particular, let n j be a sequence such that Tn j gn j → f weakly in L 2(μ). (2.7) Clearly, f is a bounded function and ‖ f ‖∞ ≤ A := lim sup n→∞ ‖Tn gn‖∞ <∞. (2.8) Let us prove that T h is bounded. This finishes the proof of the theorem because ∫