1 Introduction ON AN ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS BY DYADIC POSITIVE OPERATORS ANDREI K. LERNER

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1 ON AN ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS BY DYADIC POSITIVE OPERATORS By ANDREI K. LERNER Abstract. Given a general dyadic grid D and a sparse family of cubes S = {Q } D, define a dyadic positive operator A D,S by A D,S f (x) =, f Q χ Q (x). Given a Banach function space X(R n ) and the maximal Calderón-Zygmund operator T, we show that T f X c(n, T ) sup A D,S f X. D,S This result is applied to weighted inequalities. In particular, it implies (i) the twoweight conecture by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the A 2 conecture ; (iii) an extension of certain mixed A p -A r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A 1 estimates (nown for T ) to the maximal Calderón-Zygmund operator T. 1 Introduction A Calderón-Zygmund operator in R n is an L 2 bounded integral operator with ernel K satisfying the following growth and smoothness conditions: (i) K (x, y) c/ x y n for all x = y; (ii) there exists 0 < δ 1 such that by K (x, y) K (x, y) + K (y, x) K (y, x ) c x x δ x y n+δ, whenever x x < x y /2. Given a Calderón-Zygmund operator T, define its maximal truncated version T f (x) = sup 0<ε<ν ε< y <ν K (x, y) f (y)dy. By a general dyadic grid D we mean a collection of cubes with the following properties: JOURNAL D ANALYSE MATHÉMATIQUE, Vol. 121 (2013) DOI /s

2 142 ANDREI K. LERNER (i) for any Q D its sidelength l Q is of the form 2, Z; (ii) Q R {Q, R, } for any Q, R D; (iii) the cubes of a fixed sidelength 2 form a partition of R n. We say that {Q } D is a sparse family of dyadic cubes if (i) the cubes Q are disoint in, with fixed; (ii) if = Q, then +1 ; (iii) +1 Q 1 2 Q. Given a dyadic grid D and a sparse family S = {Q } D, consider a dyadic positive operator A defined by A f (x) = A D,S f (x) =, f Q χ Q (x) (we use the standard notation f Q = 1 Q Q f ). Our main result is the following. Theorem 1.1. Let X be a Banach function space over R n equipped with Lebesgue measure. Then, for any appropriate f, T f X c(t, n) sup A D,S f X, D,S where the supremum is taen over arbitrary dyadic grids D and sparse families S D. We consider several applications of this result in the case when X is the weighted Lebesgue space, X = L p (u) (by a weight we mean a non-negative locally integrable function). Operators similar to A were used in [4, 14, 19] to deal with several classical transforms represented in terms of the Haar shift operators of bounded complexity (for example, the Hilbert, Riesz and Beurling transforms). Now, by Theorem 1.1, we have that the results obtained by this approach hold for arbitrary Calderón- Zygmund operators. In particular, we mention the wor [4] by D. Cruz-Uribe, J. Martell and C. Pérez where a very simple proof of both the two-weight and A 2 conectures for A (and hence for the above mentioned classical operators) was found. Now we see that this proof automatically extendeds to any Calderón- Zygmund operator; in particular, this yields the two-weight conecture due to D. Cruz-Uribe and C. Pérez in full generality. Moreover, the approach to A from [4] allows us to get a rather general sufficient condition for the two-weighted boundedness of T. First we observe that the two-weighted estimates for dyadic positive operators (and in particular for A) have been recently characterized by M. Lacey, E. Sawyer,

3 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 143 and I. Uriarte-Tuero [16]; a necessary and sufficient condition is expressed in terms of Sawyer-type testing conditions. We mention here a different simple characterization which is partially based on an idea used in [4] to deal with A. Its advantage is that it avoids the use of the notions of D and S. On the other hand, it requires the bi(sub)linear maximal operator defined by M( f, g)(x) = sup Q x ( 1 f Q Q )( 1 Q Q ) g, where the supremum is taen over arbitrary cubes Q containing the point x. Theorem 1.2. Let 1 < p < and let u,v be arbitrary weights. Then the equivalence sup A D,S Lp (v) L p (u) M L p (v) L p (u 1 p ) L 1 D,S holds with the corresponding constants depending only on n. In order to give a general formulation of the two-weighted Mucenhoupt type sufficient condition for T, we again invoe the Banach function space X. Given a cube Q, define the X-average of f over Q and the maximal operator M X by f X,Q = ( fχ Q )(l Q ) X, M X f (x) = sup f X,Q. Q x This operator was introduced and studied by C. Pérez [23, 24]. By X we denote the associate space to X. Theorems 1.1 and 1.2 easily imply the following. Theorem 1.3. Let 1 < p <, and let X and Y be Banach function spaces such that M X and M Y are bounded on L p and L p, respectively. Then (1.1) T Lp (v) L p (u) c sup u 1/p X,Q v 1/p Y,Q. Q Assume that X = L p. Then M X = M L p. The operator M Lp is not bounded on L p, since this would be equivalent to the boundedness of M on L 1. Similarly, if Y = L p, then M Y = M L p is not bounded on L p. It is natural that in the case X = L p, Y = L p, the condition of the theorem is not satisfied since in this case the finiteness of the right-hand side of (1.1) means that a couple (u,v) satisfies the A p Mucenhoupt condition. But it is well nown that (u,v) A p is not sufficient even for the two-weighted boundedness of the Hardy-Littlewood maximal operator M [25]. On the other hand, taing the X and Y averages on the right-hand side of (1.1) a bit larger than the L p and L p averages, the corresponding operators M X

4 144 ANDREI K. LERNER and M Y will be a bit smaller than M L p and M Lp, and we obtain their boundedness on L p and L p, respectively, and therefore a sufficient two-weighted condition. A typical situation occurs when X = L A is the Orlicz space defined by means of the Young function A, equipped with the Luxemburg norm. In this case, X = LĀ (with the equivalence of norms), where Ā is the Young function complementary to A. The boundedness of M L A on L p was characterized by C. Pérez [24]; a necessary and sufficient condition is the B p condition, which says that for some c > 0, c A(t) dt t p <. t Hence, if X = L A and Y = L B, the boundedness of M X and M Y on L p and L p is equivalent to that Ā B p and B B p. In this case, Theorem 1.3 yields the two-weight conecture by D. Cruz-Uribe and C. Pérez mentioned above (we use the notation f L A,Q = f A,Q ). Conecture 1.4. Given p, 1 < p <, let A and B be two Young functions such that Ā B p and B B p. If the pair of weights (u,v) satisfies sup u 1/p A,Q v 1/p B,Q <, Q then T f Lp (u) c f Lp (v). For a complete history of this conecture and partial results we refer to a recent boo [3]. Under certain restrictions on A and B, the conecture was proved in [2]. By means of the local mean oscillation decomposition the conecture was proved for any T in the case p > n in [17]. After that, using the same decomposition and the operator A, the conecture was proved for the Hilbert, Riesz and Beurling transforms in [4]. In the recent paper [21], the conecture was completely proved in the case p = 2 by means of the Bellman function method. Also, in [5], the conecture was proved for the so-called log bumps and for certain log log bumps. Further of applications of Theorem 1.1 are given in Section 2 below. We turn now to the main ingredients used in the proof of this theorem. A representation of T in terms of the Haar shift operatorss = S m, D obtained by T. Hytönen [7] (see also [8, 13]), and its maximal truncated corollary proved in [10]. A recent estimate by T. Hytönen and M. Lacey [9], where they used the local mean oscillation decomposition from [17] to bound S m, D by the sum κ+1 i =1 A i, where κ = max(, m, 1) is the complexity ofs m, D, and the operators

5 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 145 A i are defined by A i f (x) = A D,S,i f (x) =, f (Q ) (i)χ Q (x) (here Q (i) denotes the i-th ancestor of Q, that is, the unique dyadic cube containing Q and such that l Q (i) = 2 i l Q ). Observe that the idea to bounds m, D by operatorsa i goes bac to [4]. But a crucial point is the linear dependence on κ in [9], while it was exponential in [4]. The ey idea in [9] was that A i can be viewed as a Haar shift operator of complexity i, but with a positive ernel. This fact allowed to simplify certain arguments used when dealing with general Haar shift operators. Our novel point in this paper is that one can use again the local mean oscillation decomposition to bounda i. More precisely, we consider the formal adoint of A i given by A i f (x) = 1 f 2 in Q χ (Q ) (i)(x)., We show that given a finite sparse family S 1, there is a sparse family S 2 such that for a.e. x, (1.2) A S 1,i f (x) c(n)i ( M f (x) +A S2 f (x) ). Combining this estimate with the above mentioned ingredients leads easily to Theorem 1.1. Observe that the local mean oscillation decomposition proved in [17] states that (1.3) f (x) m f (Q 0 ) 4M #,d 1/4;Q 0 f (x) + 4, ω 1 2 n+2 ( f ; (Q ) (1) )χ Q (x) (see Section 4 below for the definitions of the obects involved here). This estimate would allow us to get (1.2) with A 1 instead ofa = A 0 on the right-hand side, and, as a result, we would get Theorem 1.1 witha 1. This is not actually important from the point of view of main applications. But in order to arrive at a smaller operator A, we use the following variant of (1.3) proved in Theorem 4.5 below: f (x) m f (Q 0 ) 4M #,d 1 f (x) n+2 ;Q 0, ω 1 2 n+2 ( f ; Q )χ Q (x). The main difference from (1.3) is that the oscillations here are taen over the cubes Q. The paper is organized as follows. In Section 2, we give some other applications of Theorem 1.1. Section 3 contains basic facts concerning the Haar shift

6 146 ANDREI K. LERNER operators. In Section 4, we prove the above mentioned version of the local mean oscillation decomposition. Theorem 1.1 is proved in Section 5. Finally, in Section 6, we prove Theorems 1.2 and 1.3. Throughout the paper we will use the following notation. Given a sparse family {Q }, set E = Q \ +1. Observe that the sets E are pairwise disoint and Q 2 E. In the case when the argument does not depend on a particular grid D and a sparse family S D we drop the subscripts D and S, and we assume that D is the standard dyadic grid. 2 Applications 2.1 The A 2 conecture. Given a weight w, define its A p characteristic by ( )( ) 1 1 p 1 w Ap sup A p (w; Q) = wdx w 1 p 1 dx. Q Q Q Q Q The A 2 conecture states that for a Calderón-Zygmund operator T, ( ) (2.1) T L p (w) c(t, p, n) w max 1, 1 p 1 A p (1 < p < ). Note that by extrapolation it suffices to get this result in the case p = 2 (this explains the name of the conecture). In its full generality, this conecture was recently settled by T. Hytönen [7] (see also [8, 13]). Soon after that, it was shown in [10] that (2.1) holds for T as well. The proof of (2.1) is based on the representation of T in terms of the Haar shift operatorss m, D. After that, the proof reduces to showing (2.1) for S m, D in place of T, with the corresponding constant depending linearly (or polynomially) on the complexity. Over the past year, several different proofs of the latter step have appeared (see, e.g., [15, 26]). It is clear that (2.1) follows immediately from Theorem 1.1 combined with the estimate (2.2) A L 2 (w) c(n) w A2 proved in [4]. The proof of (2.2) is quite elementary; we give it here for the sae of the completeness. Let M d w be the dyadic weighted maximal operator; we use the fact that it is bounded on L p (w) with bound independent of w. Assuming that

7 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 147 f, g 0, by Hölder s inequality, we have (A f )gdx = f Q R n g Q Q, 2 ( A 2 (w; Q 1 )( 1 ) w 1 (Q, ) f Q w(q ) 2 w A2 Mw d 1( fw)m w(gw d 1 )dx, E 2 w A2 R n M d w 1( fw)m d w (gw 1 )dx Q ) g E which yields (2.2) by duality. 2 w A2 M d w 1( fw) L 2 (w 1 ) M d w(gw 1 ) L 2 (w) c w A2 f L2 (w) g L2 (w 1 ), To summarize, a proof of the A 2 conecture can be based on te following ingredients. A representation of T in terms of S m, D [7, 8, 13]. The local mean oscillation decomposition bound of S m, D A i [9]. The local mean oscillation decomposition bound of A i by A. The L 2 (w) bound of A [4]. by the operators 2.2 Mixed A p -A estimates. Given a weight w, define its A characteristic by 1 w A = sup M(wχ Q ). Q R n w(q) Q Note that w A c(p, n) w Ap for any p > 1. M. Lacey [14] showed that for classical singular integrals, (2.1) can be improved to ( (2.3) T Lp (w) c(t, p, n) w 1/p A p max ( w A ) 1/p, ( σ A ) 1/p), where σ = w 1 p. Also, it was conectured in [14] that this estimate holds for any Calderón-Zygmund operator. Soon afterwards, the conecture was proved in [9]; the proof was based on the analysis of the operators A i. On the other hand, the proof in [14] was based on showing (2.3) for A in place of T. Hence, by Theorem 1.1, we see that this proof actually yields (2.3) in the general case.

8 148 ANDREI K. LERNER 2.3 Mixed A p -A r estimates. Given a weight w, define its mixed A p -A r characteristic by w (Ap ) α (A r ) β = sup Q R n A p (w; Q) α A r (w; Q) β, where α,β 0. In [19], it was proved that for any 2 p r <, (2.4) A L p (w) c(p, r, n) w (Ap ) 1 p 1 (A r ) 1 1 p 1. By duality, it follows from this that for any 1 < p < 2 and r > p, (2.5) A L p (w) c(p, r, n) σ (Ap ) 1 p 1 (A r ) 1 1 From this, estimates (2.4) and (2.5) were obtained in [19] for classical singular integrals in place of A. Now, by Theorem 1.1, we have that they hold for any Calderón-Zygmund operator T (and T ). Note that the difference between these estimates and (2.3) is that in the mixed A p -A r characteristic only one supremum is involved, while the right-hand side of (2.3) involves two independent suprema. It was shown in [19] by simple examples that the right-hand sides in (2.4) and (2.3) are incomparable. In [9], a new conecture was posed about the L p (w) bound for T implying the estimates of both types. By Theorem 1.1, we have that it suffices to prove this conecture for A. However, even for this simple operator, the new conecture is still not clear. p Sharp A 1 estimates. Recall that w is an A 1 weight if there exists c > 0 such that Mw(x) cw(x) a.e.; the smallest possible c here is denoted by w A1. It was proved in [20] that for any w A 1, (2.6) T f L p (w) c(n, T )pp w A1 f L p (w) (1 < p < ) and (2.7) T f L 1, (w) c(n, T ) w A1 log(1 + w A1 ) f L 1 (w). The so-called wea Mucenhoupt-Wheeden conecture says that (2.7) holds with linear dependence on w A1. However, this was recently disproved in [22], which raises the conecture that the L log L dependence on w A1 in (2.7) is best possible. Very recently, both estimates (2.6) and (2.7) have been improved by T. Hytönen and C. Pérez [12] as follows: (2.8) T f Lp (w) c(n, T )pp w 1/p A 1 w 1/p A f Lp (w) (1 < p < )

9 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 149 and (2.9) T f L 1, (w) c(n, T ) w A1 log(1 + w A ) f L 1 (w). Note that (2.9) follows from (2.8) by means of the Calderón-Zygmund method. Inequality (2.8) was deduced in [12] from a sharp version of the reverse Hölder inequality, along with the estimate (2.10) T f L p (w) c(n, T )pp ( 1 ) 1 1/pr f L r 1 p (M r w) (1 < r < 2) proved in [20] (here M r w = M(w r ) 1/r ). The method of proof of (2.10) leaves open the question whether this inequality (and so (2.8) and (2.9)) holds for the maximal Calderón-Zygmund operator T as well. Theorem 1.1 yields a positive answer to this question. Theorem 2.1. Inequalities (2.8) and (2.9) remain true for T in place of T. It follows from the discussion above that it suffices to prove (2.10) for T. The rest of the argument is exactly the same as in [12]. Next, by Theorem 1.1, it suffices to prove (2.10) fora. This can be done in a variety of ways. For example, it was shown in [17] that (2.10) follows from T f wdx c(n, T ) (M f ) δ M ( (M f ) 1 δ w ) dx (0 < δ 1). R n R n Exactly as in [17], we have that this inequality with A in place of T would imply (2.10) for A. But this is almost trivial: (A f )wdx = f Q R n w Q Q 2 ( f Q ) δ ((M f ) 1 δ w) Q E,, 2 (M f ) δ M((M f ) 1 δ w)dx, E 2 (M f ) δ M ( (M f ) 1 δ w ) dx. R n 3 Haar shift operators We recall briefly the main definitions concerning Haar shift operators. For more details we refer to [7, 10, 13]. Definition 3.1. We say that h Q is a Haar function on a cube Q D if (i) h Q is a function supported on Q and constant on the children of Q;

10 150 ANDREI K. LERNER (ii) h Q = 0. We say that h Q is a generalized Haar function if it is a linear combination of a Haar function on Q and χ Q (in other words, only condition (i) above is satisfied). Definition 3.2. Given a general dyadic grid D, (m, ) Z 2 +, and Q D, set S Q f (x) = Q,Q D,Q,Q Ql(Q )=2 m l(q),l(q )=2 l(q) f, h Q Q h Q Q Q (x), where h Q Q is a (generalized) Haar function on Q, and h Q Q is one on Q such that h Q Q L hq Q L 1. We say that S is a (generalized) Haar shift operator of complexity type (m, ) if S f (x) = S m, D f (x) = Q DS Q f (x). The number κ = max(m,, 1) is called the complexity of S. Also, by definition, the L 2 boundedness of the generalized Haar shift operator is assumed (for the usual Haar shift this follows automatically from its properties). Definition 3.3. Given a generalized Haar shift S, define its associated maximal truncations by S f (x) = sup S ε,v f (x), 0<ε v< where S ε,v f (x) = S Q f (x). Q D:ε l Q v The importance of these obects follows from the following result proved by T. Hytönen [7] and simplified in [13]. Theorem 3.4. Let T be a Calderón-Zygmund operator which satisfies the standard estimates with δ (0, 1]. Then for all bounded and compactly supported functions f and g, g, T f = c(t, n)e D,m =0 2 (m+)δ/2 g,s m, D f, where E D is the expectation with respect to a probability measure on the space of all general dyadic grids.

11 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 151 By means of Theorem 3.4, the following estimate was deduced in [10]. Proposition 3.5. We have the pointwise bound T f (x) c(t, n) (M f (x) +E D,m =0 ) 2 (m+)δ/2 (S m, D ) f (x). 4 A local mean oscillation decomposition Definition 4.1. The non-increasing rearrangement of a measurable function f on R n is defined by f (t) = inf{α > 0 : {x R n : f (x) < α} < t} (0 < t < ). Definition 4.2. Given a measurable function f onr n and a cube Q, the local mean oscillation of f on Q is defined by ( ) ( ) ω λ ( f ; Q) = inf ( f c)χq λ Q c R (0 < λ < 1). Definition 4.3. By a median value of f over Q we mean a (possibly nonunique) real number m f (Q) such that max ( {x Q : f (x) > m f (Q)}, {x Q : f (x) < m f (Q)} ) Q /2. It is easy to see that the set of all median values of f is either one point or a closed interval. In the latter case, we assume for the definiteness that m f (Q) is the maximal median value. Observe that it follows from the definitions that (4.1) m f (Q) ( fχ Q ) ( Q /2). This estimate implies (4.2) (( f m f (Q))χ Q ) (λ Q ) 2ω λ ( f ; Q) (0 < λ 1/2). We also mention (cf. [6, Lemma 2.2]) that (4.3) lim Q 0,Q x m f (Q) = f (x) for a.e. x R n. Given a cube Q 0, denote by D(Q 0 ) the set of all dyadic cubes with respect to Q 0. The dyadic local sharp maximal function M #,d λ;q 0 f is defined by M #,d λ;q 0 f (x) = sup ω λ ( f ; Q ). x Q D(Q 0 ) The following theorem was proved in [17].

12 152 ANDREI K. LERNER Theorem 4.4. Let f be a measurable function on R n and let Q 0 be a fixed cube. Then there exists a (possibly empty) sparse family of cubes Q D(Q 0 ) such that for a.e. x Q 0, f (x) m f (Q 0 ) 4M #,d 1/4;Q 0 f (x) + 4, ω 1 2 n+2 ( f ; (Q ) (1) )χ Q (x). Here we prove a similar result with the local mean oscillations taen over the cubes Q instead of (Q )(1). Theorem 4.5. Let f be a measurable function on R n and let Q 0 be a fixed cube. Then there exists a (possibly empty) sparse family of cubes Q D(Q 0 ) such that for a.e. x Q 0, f (x) m f (Q 0 ) 4M #,d 2 n 2 ;Q 0 f (x) + 2, The ey element of the proof is the following. ω 2 n 2( f ; Q )χ Q (x). Lemma 4.6. There exists a (possibly empty) collection of pairwise disoint cubes {Q 1 } D(Q 0) such that Q1 1 2 Q 0 and for a.e. x Q 0, (4.4) f m f (Q 0 ) = g 1 + α,1 χ Q 1 + ( f m f (Q 1 ))χ Q 1, where g 1 2M #,d 2 n 2 ;Q 0 f for a.e. x Q 0 \ Q1 and the numbers α,1 satisfy a,1 2ω 2 n 2( f ; Q 0 ). After this lemma is established, the proof of Theorem 4.5 follows exactly the same lines as the proof of Theorem 4.4. Therefore, we only outline briefly the main details. Proof of Theorem 4.5. Iterating (4.4) for each Q 1 and for every subsequent cube, we get that for a.e. x Q 0, (4.5) f m f (Q 0 ) = g + α,1 χ Q 1 + =2 i:qi 1 = where = Q, and the family {Q } is sparse. Moreover, :Q Q 1 i g 2M #,d 2 n 2 ;Q 0 f and α (i), 2ω 2 n 2( f ; Q 1 i ). α (i), χ Q, The first sum in (4.5) is bounded by 2ω 2 n 2( f ; Q 0 ) 2M #,d 2 n 2 ;Q 0 f. Further, :Q Q 1 i α (i), χ Q 2ω 2 n 2( f ; Q 1 i )χ Q 1. i

13 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 153 Hence the second sum in (4.5) is bounded by 2 2 i ω 2 n 2( f ; Q 1 i )χ Q 1. i Combining the obtained estimates completes the proof. Proof of Lemma 4.6. Set f 1 (x) = f (x) m f (Q 0 ) and E 1 = {x Q 0 : f 1 (x) > ( f 1 χ Q0 ) (λ n Q 0 )}, where λ n = 1/2 n+2. If E 1 = 0, then by (4.2) we trivially have f (x) m f (Q 0 ) 2ω λn ( f ; Q 0 ) 2M #,d λ n ;Q 0 f (x) for a.e. x Q 0. Assume therefore that E 1 > 0. Let m Q0 f 1 (x) = sup max x Q D(Q 0 ) Q i :Q (1) i =Q m f1 (Q i ) (the maximum is taen over 2 n dyadic children of Q). Consider the set 1 = {x Q 0 : m Q0 f 1 (x) > ( f 1 χ Q0 ) (λ n Q 0 )}. By (4.3), m Q0 f 1 (x) f 1 (x) a.e., and hence 1 E 1 > 0. We can write 1 = Q 1, where Q1 are pairwise disoint cubes from D(Q 0 ) with the property that they are maximal such that (4.6) max Q i :Q (1) i =Q 1 m f1 (Q i ) > ( f 1 χ Q0 ) (λ n Q 0 ). In particular, this means that each Q 1 satisfies m f1 (Q 1 ) ( f 1 χ Q0 ) (λ n Q 0 ) 2ω λn ( f ; Q 0 ). Since m f1 (Q 1 ) = m f (Q 1 ) m f (Q 0 ), we have f m f (Q 0 ) = f 1 χ Q0 \ 1 + m f1 (Q 1 )χ Q 1 + ( f m f (Q 1 ))χ Q 1, which proves (4.4) with g 1 = f 1 χ Q0 \ 1 and α,1 = m f1 (Q 1 ). By the properties established above, g 1 and α,1 satisfy the statement of the lemma. It remains to show that 1 Q 0 /2. If Q i is a child of Q, then by (4.1), m f (Q i ) ( fχ Qi ) ( Q i /2) ( fχ Q ) ( Q /2 n+1 ).

14 154 ANDREI K. LERNER Therefore, if (4.6) holds, ( f 1 χ Q0 ) (λ n Q 0 ) < ( f 1 χ Q 1 ) ( Q 1 /2 n+1 ). Hence {x Q 1 : f 1(x) > ( f 1 χ Q0 ) (λ n Q 0 )} Q 1 /2n+1, and thus 1 2 n+1 Q 1 {x Q 1 : f 1 (x) > ( f 1 χ Q0 ) (λ n Q 0 )} {x Q 0 : f 1 (x) > ( f 1 χ Q0 ) (λ n Q 0 )} λ n Q 0, which completes the proof. 5 Proof of Theorem 1.1 Taing into account Proposition 3.5, in order to prove Theorem 1.1, it suffices to show that (5.1) M f X c(n) A f X ( f 0) and (5.2) (S m, D ) f X c(n)κ 2 A f X. Here and below, A i f X is understood as sup D,S A D,S,i f X, where the supremum is taen over arbitrary dyadic grids D and sparse families S D. 5.1 Banach function spaces. For a general account of Banach function spaces we refer to [1, Ch. 1]. Here we mention only several facts which are used below. The associate space X consists of measurable functions f for which f X = sup f (x)g(x) dx <. g X 1 R n This definition implies the Hölder inequality (5.3) f (x)g(x) dx f X g X. R n Further [1, p. 13], (5.4) f X = sup f (x)g(x) dx. g X =1 R n

15 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 155 By Fatou s lemma [1, p. 5], if f n f a.e. and lim inf n f n X <, then f X and (5.5) f X lim inf n f n X. 5.2 Proof of (5.1). We use the well-nown principle saying that in order to estimate the usual maximal operator it suffices to estimate the dyadic one. This principle has several forms. We need one attributed in the literature to M. Christ and, independently, to J. Garnett and P. Jones. However, we have found it stated in a very clear form only in [12, proof of Th. 1.10]. Proposition 5.1. There are 2 n dyadic grids D α such that for any cube Q R n there exists a cube Q α D α such that Q Q α and l Qα 6l Q. It follows from this Proposition that 2n (5.6) M f (x) 6 n M D α f (x). α =1 By the Calderón-Zygmund decomposition, if {x : M d f (x) > 2 (n+1) } = Q, then the family {Q } is sparse and M d f (x) 2 n+1, f Q χ E (x) 2 n+1 A f (x). From this and from (5.6), we have 2n (5.7) M f (x) 2 12 n A Dα,S α f (x), wheres α D α depends on f. This implies (5.1) with c(n) = 2 24 n. 5.3 Proof of (5.2). We start with the following lemma of Hytönen and Lacey [9]. α =1 Lemma 5.2. If S has complexity κ, then for any dyadic Q ( ω λ (S f ; Q) c(λ, n) κ f Q + κ ) f Q (i). Observe that here dyadic means that Q D if S = S D. Combining Lemma 5.2 with Theorem 4.5, we get ( (S m, D ) f (x) m (S m, D ) f (Q 0) c(n) κm f (x) + κa f (x) + i =1 κ i =1 ) A i f (x).

16 156 ANDREI K. LERNER Assuming that f is bounded and with compact support, we have by (4.1) that m (S m, D ) f (Q) 0 as Q expands unboundedly. Therefore, the previous inequality combined with Fatou s lemma (5.5) implies (S m, D ) f X c(n)κ ( ) κ M f X + A f X + A i f X. From this and from (5.1), it follows that in order to prove (5.2) it suffices to show that (5.8) A i f X c(n)i A f X. Exactly as above, one can assume that A i is defined by means of the standard dyadic grid. Also, since we shall deal below only with A i and M, one can assume that f 0. Consider the formal adoint of A i : A i f (x) = 1 2 in, f Q χ (Q ) (i)(x). Our goal is to show that the operator A i is of wea type (1, 1) with the bound depending linearly on i. This will be done using the classical Calderón-Zygmund argument. Hence we start with the L 2 boundedness of A i. In the proof below, we use the well-nown fact that M d L p p. Proposition 5.3. For any i N, A i f L 2 = A i f L 2 8 f L 2. Proof. Similarly to the proof of (2.2), we have (A i f )gdx = f (Q R n ) (i)g Q Q 2,, 2 (M d f )(M d g)dx. R n E i =1 (M d f )(M d g)dx From this, using Hölder s inequality, the L 2 boundedness of M d and duality, we get the L 2 bound for A i. Lemma 5.4. For any i N, A i f L 1, ci f L 1, where c is an absolute constant (for i big enough, one can tae c = 5).

17 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 157 Proof. Let = {x : M d f (x) > α} = l Q l, where Q l are maximal pairwise disoint dyadic cubes such that f Ql > α. Set also b l = ( f f Ql )χ Ql, b = l b l and g = f b. We have (5.9) {x : A i f (x) > α} + {x : A i g(x) > α/2} Further, 1 α f L 1 ; and, by the L 2 boundedness of A i, + {x c : A i b(x) > α/2}. {x : A i g(x) > α/2} 4 α 2 A i g 2 L 2 c α 2 g 2 L 2 c α g L 1 c α f L 1. It remains therefore to estimate the term in (5.9). For x c, consider A i b(x) = 1 2 in (b l ) Q χ (Q ) (i)(x). l, The second sum is taen over those cubes Q for which Q Q l =. If Q l Q, then (b l ) Q = 0. Therefore, one can assume that Q Q l. On the other hand, if (Q )(i) c =, then Q l (Q )(i). Hence, for x c, we have A i b(x) = 1 2 in l, :Q Q l (Q )(i) (b l ) Q χ (Q ) (i)(x). The latter sum is nontrivial if i 2. In this case, the family of all dyadic cubes Q for which Q Q l Q (i) can be decomposed into i 1 families of disoint cubes of equal length. Therefore,, :Q Q l (Q )(i) χ Q (i 1)χ Ql. From this we get {x c : A i b(x) > α/2} 2 α A i b L 1 ( c ) 2 b l dx α The proof is complete. l, :Q Q l (Q )(i) 4(i 1) f α L 1. Q 2(i 1) α l Q l b l dx

18 158 ANDREI K. LERNER Lemma 5.5. Let i N. For any dyadic cube Q, ω λn (A i f ; Q) c(n)i f Q. Hence Proof. For x Q, 1 f 2 in Q χ (Q ) (i)(x) = 1 2 in, :Q (Q )(i) A i f (x) c χ Q (x) = 1 2 in, :(Q )(i) Q From this and from Lemma 5.4, we obtain, :Q (Q )(i) f Q c. f Q χ (Q ) (i)(x) A i ( fχ Q )(x). inf c ((A i f c)χ Q) (λ n Q ) (A i ( fχ Q)) (λ n Q ) c(n)i f Q, which completes the proof. We are now ready to prove (5.8). By the standard limiting argument, one can assume that the sum defining A i is finite. Then m A i f (Q) = 0 for Q big enough. Hence, By Lemma 5.5 and Theorem 4.5, for a.e. x Q, A i f (x) c(n)i ( M f (x) +A f (x) ) (notice that here A i and A are taen with respect to different sparse families). From this and from (5.7), and using that the operatorais self-adoint, we have for any g 0, (A i f )gdx = R n R n f (A i g)dx 2 n +1 c n i α =1 α =1 R n f (A Dα,S α g)dx 2 n +1 = c n i (A Dα,S α f )gdx c ni sup A D,S f X g X. R n D,S Applying (5.4) yields (5.8), and the proof of Theorem 1.1 is complete.

19 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS Proof of Theorems 1.2 and 1.3 Proof of Theorem 1.2. Using the same argument as in the proof of (2.2), we have (A f )g = f Q R n g Q Q 2 M( f, g)dx,, E 2 M( f, g)dx 2 M Lp (v) L p (u 1 p ) L 1 f L p (v) g L p (u 1 p ). R n Taing the supremum over g with g L p (u 1 p ) = 1 gives A Lp (v) L p (u) 2 M L p (v) L p (u 1 p ) L 1. On the other hand, by Proposition 5.1, 2n (6.1) M( f, g)(x) 12 n M D α ( f, g)(x). Consider M d ( f, g) taen with respect to the standard dyadic grid. Suppose that f, g 0 and f, g L 1. We use exactly the same argument as in the Calderón- Zygmund decomposition. For c n (to be be specified below) and for Z, consider the sets = {x R n : M d ( f, g)(x) > c n}. Then = Q, where the cubes Q are pairwise disoint with fixed, and α =1 c n < f Q g Q 2 2n c n. From this and from Hölder s inequality, we have Q +1 = Qi +1 Qi +1 Q < c +1 2 n c +1 2 n Qi +1 Q ( Q ( f Q Q +1 i f Q +1 i g ) 1/2 ) 1/2 g 2 n c 1/2 n Q Hence, taing c n = 2 2(n+1), we see that the family {Q } is sparse, and M d ( f, g)(x) 2 2(n+1), f Q g Q χ Q (x). Therefore, R n M d ( f, g)dx 2 2(n+1), f Q g Q Q = 2 2(n+1) R n (A f )gdx.

20 160 ANDREI K. LERNER From this and (6.1), applying Hölder s inequality, we obtain 2 n M( f, g)(x)dx 4 48 n R n which completes the proof. α =1 R n (A Dα,S α f )gdx 2n 4 48 n A Dα,S α f Lp (u) g L p (u 1 p ) α = n sup A D,S Lp (v) L p (u) f Lp (v) g L p (u 1 p ), D,S Proof of Theorem 1.3. By Hölder s inequality (5.3), Hence f Q g Q fv 1/p Y,Q v 1/p Y,Q gu 1/p X,Q u 1/p X,Q. M( f, g)(x) c(u,v)m Y ( fv 1/p )(x)m X (gu 1/p )(x), where c(u,v) = sup Q u 1/p X,Q v 1/p Y,Q. Therefore, by the assumptions on X and Y and by the usual Hölder inequality, R n M( f, g)(x)dx c(u,v) M Y ( fv 1/p ) L p M X (gu 1/p ) L p c(u,v) f Lp (v) g L p (u 1 p ). Combining this estimate with Theorems 1.1 and 1.2 completes the proof. Added in proof. We have found [18] that the main result of this paper can be proved without the use of the Haar shift operators. This further simplifies the proof of the A 2 conecture. Almost simultaneously, a proof of the A 2 conecture based on a similar idea was obtained in [11]. REFERENCES [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New Yor, [2] D. Cruz-Uribe, J. M. Martell, and C. Pérez, Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conecture, Adv. Math. 216 (2007), [3] D. Cruz-Uribe, J. M. Martell, C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Birhäuser/Springer, Basel, AG, [4] D. Cruz-Uribe, J. M. Martell and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), [5] D. Cruz-Uribe, A. Rezniov and A. Volberg, Logarithmic bump conditions and the two-weight boundedness of Calderón-Zygmund operators, arxiv: [math. AP].

21 ESTIMATE OF CALDERÓN-ZYGMUND OPERATORS 161 [6] N. Fuii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), [7] T. P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), [8] T. P. Hytönen, Representation of singular integrals by dyadic operators, and the A 2 theorem, arxiv: [math. CA]. [9] T. P. Hytönen and M. Lacey, The A p A inequality for general Calderón Zygmund operators, Indiana Univ. Math. J., to appear; arxiv: [math. CA]. [10] T. P. Hytönen, M. T. Lacey, H. Martiainen, T. Orponen, M. C. Reguera, E. T. Sawyer, and I. Uriarte-Tuero, Wea and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces, J. Anal. Math. 118 (2012), [11] T. P. Hytönen, M. T. Lacey, and C. Pérez, Non-probabilistic proof of the A 2 theorem, and sharp weighted bounds for the q-variation of singular integrals, arxiv: [math. CA]. [12] T. P. Hytönen and C. Pérez, Sharp weighted bounds involving A, Anal. PDE 6 (2013), [13] T. P. Hytönen, C. Pérez, S. Treil, and A. Volberg, Sharp weighted estimates for dyadic shifts and the A 2 conecture, J. Reine Angew. Math., to appear; arxiv: [math. CA]. [14] M. T. Lacey, An A p -A inequality for the Hilbert transform, Houston Math. J. 38 (2012), [15] M. T. Lacey, On the A 2 inequality for Calderón-Zygmund operators, arxiv: [math. CA]. [16] M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, Two weight inequalities for discrete positive operators, arxiv: [math. CA]. [17] A. K. Lerner, A pointwise estimate for the local sharp maximal function with applications to singular integrals, Bull. London Math. Soc. 42 (2010), [18] A. K. Lerner, A simple proof of the A 2 conecture, Int. Math. Res. Not. IMRN doi: /imrn/rns145. [19] A. K. Lerner, Mixed A p -A r inequalities for classical singular integrals and Littlewood-Paley operators, J. Geom. Anal. 23 (2013), [20] A. K. Lerner, S. Ombrosi, and C. Pérez, A 1 bounds for Calderón-Zygmund operators related to a problem of Mucenhoupt and Wheeden, Math. Res. Lett. 16 (2009), [21] F. Nazarov, A. Rezniov, S. Treil, and A. Volberg, A Bellman function proof of the L 2 conecture, J. Anal. Math. 121 (2013), [22] F. Nazarov, A. Rezniov, V. Vasuynin, and A. Volberg, A 1 conecture: wea norm estimates of weighted singular operators and Bellman functions, preprint. [23] C. Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), [24] C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted L p -spaces with different weights, Proc. London Math. Soc.(3) 71 (1995), [25] E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), [26] S. Treil, Sharp A 2 estimates of Haar shifts via Bellman function, arxiv: [math. CA]. Andrei K. Lerner DEPARTMENT OF MATHEMATICS BAR-ILAN UNIVERSITY RAMAT GAN, ISRAEL alerner@netvision.net.il (Received February 17, 2012)