Differential transforms of Cesàro averages in weighted spaces

Publ. Mat. 52 (2008), 101–127 DIFFERENTIAL TRANSFORMS OF CES` ARO AVERAGES IN WEIGHTED SPACES A. L. Bernardis and F. J. Mart´ ın-Reyes Abstract In this paper we obtain convergence results for the series of differences of Ces`aro averages along lacunary sequences in the setting of weighted Lp-spaces. These results give some information about how the Ces`aro averages converge. The paper extends results of an earlier work by R. L. Jones and J. Rosenblatt. The operators considered are essentially convolution operators given by kernels more singular than the ones in the article by Jones and Rosenblatt. 1. Introduction The Lebesgue’s Differentiation theorem in the real line establishes that if fis locally integrable then lim ε→0+ 1 εZx+ε x f(t)dt =f(x) for almost every x. We can interpret the above limit as Ces`aro (C, 1) right-continuity of fat x(see [7]). In general, for α > −1, we say that fis (C, 1 + α) right-continuous at xif lim ε→0+Dεf(x) = f(x), where the Ces`aro-αaverages Dεfare defined by Dεf(x) = 1 + α ε1+αZx+ε x (x+ε−t)αf(t)dt. 2000 Mathematics Subject Classification. 42B25. Key words. Differential transforms, Ces`aro averages, weight. This research has been partially supported by Spanish goverment Grant MTM20058350-C03-02. The first author was supported in part by CAI+D-UNL and CONICET. The second author was supported by Junta de Andaluc´ıa Grant FQM 354. 102 A. L. Bernardis, F. J. Mart´ ın-Reyes If α= 0 then the Ces`aro-αaverages are the usual averages. It is easy to see that the (C, 1 + α) right-continuity of fat ximplies the (C, 1 + β) right-continuity of fat xfor all β > α > −1. We are interested in obtaining information about the convergence of Dεf. In this paper we are going to study the convergence through lacunary sequences. Let us consider the Ces`aro averages associated to a lacunary sequence, that is, (1.1) Dkf(x) = Dεkf(x) = 1 + α ε1+α kZx+εk x (x+εk−t)αf(t)dt, where {εk}k∈Zis a ρ-lacunary sequence of positive numbers, that is, εk+1/εk≥ρ > 1 for all k∈Z. In order to study the limit of the averages Dk, it is natural to consider the maximal operator M+ α,kf(x) = sup k∈Z Dk|f|(x). From known results about approximations of the identity we have that the operator M+ α,k is of weak type (1,1) and of strong type (p, p) for p > 1. As a consequence, we have that if p > 1 and f∈Lp(dx) then limk→−∞ Dkf=fand limk→+∞Dkf= 0 almost everywhere and in the Lp(dx)-norm. For the limit case p=1 it is proved that limk→−∞ Dkf=f and limk→+∞Dkf= 0 almost everywhere and in measure for all f∈ L1(dx). However, it follows from the results in [7] that the maximal operator M+ αf(x) = sup ε>0 Dε|f|(x) is of strong type (p, p) for p > 1/(1 + α), it is of restricted weak type (1 1+α,1 1+α) and it is not of weak type ( 1 1+α,1 1+α) for α < 0. (Notice that if α≥0, the operator M+ αis pointwise equivalent to the onesided Hardy-Littlewood maximal operator M+ 0which we simply denote by M+. For that reason we are considering only the case α≤0.) The study of the boundedness of M+ αon weighted spaces can be found in [11] (see also [2]). In the setting of Ergodic Theory the convergence of Ces`aro-αaverages appears, for instance, in [3] and [5]. As we said above, we wish to obtain some information about how the convergence of Dkfoccurs. In [6] Jones and Rosenblatt discuss the problem for the usual averages (case α= 0) in connection with the convergence of ergodic averages. To obtain information about how the convergence of the sequence {Dk}occurs they study the behavior of the Differential Transforms in Weighted Spaces 103 series of the differences (1.2) ∞ X k=−∞ vk[Dkf(x)−Dk−1f(x)], where {vk}is a bounded sequence of real or complex numbers (we shall say that {vk}is a multiplying sequence and we shall write kvkk∞= supk|vk|). In [6], Jones and Rosenblatt work in Lebesgue spaces Lp(dx) while in [1] the results were extended to weighted Lebesgue spaces. In order to make precise statements about the convergence properties of (1.2), we begin by pointing out that Dkf(x) = ϕk∗f(x) where ϕk(x) = 1 εkϕ(x/εk) and ϕ(x) = (1 + α)(1 + x)αχ(−1,0)(x). For each N∈Z2, N= (N1, N2) with N1< N2we define the sum (1.3) Tα Nf(x) = N2 X k=N1 vk[Dkf(x)−Dk−1f(x)] = Kα N∗f(x), where Kα N(x) = N2 X k=N1 vk[ϕk(x)−ϕk−1(x)]. Our goal is to prove convergence results of Tα Nf(x) as N= (N1, N2) tends to (−∞,+∞) which means that N1→ −∞ and N2→+∞. As usual, to prove the a.e. convergence, we shall study the boundedness of the associated maximal operator T∗f(x) = sup N∈Z2 |Tα Nf(x)| in the setting of the weighted spaces Lp(w), that is, the space of all measurable functions fsuch that kfkLp(w)=RR|f|pw1/p <∞. Since the operators Tα Nare convolution operators with kernels Kα N supported in (−∞,0), the weights that appear are the one sided A+ pweights defined by E. Sawyer [12]. We say that a weight wbelongs to the class A+ 1[12], [9], if there exists Csuch that M−w(x)≤Cw(x) a.e., where M−is the left-sided Hardy-Littlewood maximal function defined as M−f(x) = sup ε>0 1 εZε 0 |f(x−t)|dt. 104 A. L. Bernardis, F. J. Mart´ ın-Reyes For 1 < p < ∞, we say that wbelongs to A+ p[12] if there exists Csuch that for any three points a < b < c Zb a w!1 pZc b w1−p′1 p′ ≤C(c−a), where p+p′=pp′. The next theorem contains a relation between the maximal operator M+ αand A+ pclasses (see [11] and [2]). We shall use this theorem in our proofs. Theorem 1.4. Let −1< α ≤0. If 1/(1+α)< p < ∞and w∈A+ p(1+α) then there exists a constant Csuch that (1.5) Z|M+ αf(x)|pw(x)dx ≤CZ|f(x)|pw(x)dx for all f∈Lp(w). In what follows, we state the main results of this paper. Theorem 1.6. Let {εk}be a ρ-lacunary sequence, {vk}a multiplying sequence and −1< α ≤0. If 1/(1 + α)< p < ∞and w∈A+ p(1+α)then there exists a constant Csuch that kT∗fkLp(w)≤CkfkLp(w), for all functions f∈Lp(w). Theorem 1.7. Let {εk}be a ρ-lacunary sequence, {vk}a multiplying sequence and −1< α ≤0. If wr∈A+ 1for some r > 1/(1+α)then there exists a constant Csuch that w({x∈R:|T∗f(x)|> λ})≤C λkfkL1(w), for all λ > 0and all functions f∈L1(w). Notice that, under the assumptions of the last theorem, that is, wr∈ A+ 1for some r > 1/(1 + α), we have that T∗is of strong type (p, p) with respect to w(x)dx for all p > 1. This statement is easily seen applying the last two theorems and interpolation arguments. Remark 1.8.It follows from Theorem 1.4 and the arguments in the proof of Theorem 1.7 that the conclusions of Theorems 1.6 and 1.7 hold for the maximal operator M+ α,k. Using the theorems and proving the a.e. convergence in the Schwartz class we obtain the following result. Differential Transforms in Weighted Spaces 105 Theorem 1.9. Let {εk}be a ρ-lacunary sequence and let {vk}be a multiplying sequence. Assume that wis a weight. (i) If −1< α ≤0,1/(1 + α)< p < ∞and w∈A+ p(1+α)then Tα Nfconverge a.e. and in Lp(w)-norm for all f∈Lp(w)as N= (N1, N2)tends to (−∞,+∞). (ii) If wr∈A+ 1for some r > 1/(1 + α)then Tα Nfconverge a.e. and in measure for all f∈L1(w)as N= (N1, N2)tends to (−∞,+∞). We remark that applying Theorems 1.6 and 1.7 with w≡1 we have that T∗is of weak type (1,1) and of strong type (p, p), 1 < p < ∞, with respect to the Lebesgue measure. The corresponding results of convergence also holds. The organization of the paper is as follows. Section 2 is devoted to state properties of the lacunary sequences. In Section 3 we prove some properties of the kernels Kα N. Section 4 is devoted to prove uniform boundedness of the operators Tα N, while Theorems 1.6, 1.7, and 1.9 are proved in Sections 5, 6 and 7, respectively. Throughout the paper, we shall use the notations introduced in this section and the letter Cwill mean a positive constant not necessarily the same at each occurrence. 2. Lacunary sequences We establish in this section some properties of the ρ-lacunary sequence {εk}. The next proposition shows that, without loss of generality, we may assume that (2.1) 1 < ρ ≤εk+1 εk ≤ρ2. Proposition 2.2. Given the ρ-lacunary sequence {εk}and the multiplying sequence {vk}, we can define a ρ-lacunary sequence {ηk}and a multiplying sequence {wk}verifying the following properties: (i) 1 < ρ ≤ηk+1 ηk≤ρ2and ||vk||∞=||wk||∞. (ii) For all N= (N1, N2)there exists M= (M1, M2)with Tα N= ˜ TM, where ˜ TMis the operator defined in (1.3) for the new sequences {ηk}and {wk}. The proof is exactly as in the case α= 0 (see [1]). It follows from this proposition that it is enough to prove all the results of this paper in the case of a ρ-lacunary sequence satisfying (2.1). For 106 A. L. Bernardis, F. J. Mart´ ın-Reyes this reason, in the rest of the paper we assume that {εk}satisfies (2.1) without saying it explicitly. Observe that, under this assumption, the following properties hold: (2.3) 1 ρ2(m−n) ≤εn εm ≤1 ρm−n ,for all m > n. In fact, these inequalities follow from (2.1) and the equality εn εm =εn εn+1 εn+1 εn+2 ···εm−1 εm . If we denote by βthe smallest positive integer such that 1/ρ + (1/ρ)β≤1, we get from (2.3) that (2.4) εi+εm≤εm+1 for all m≥i+β−1. 3. Properties of the kernels Kα N The next lemma will allow us to prove the uniform boundedness on L2(dx) of the operators Tα N. Lemma 3.1. Let −1< α ≤0and ϕ(x) = (1 + α)(1 + x)αχ(−1,0)(x). The Fourier transform of ϕ, defined by ˆϕ(ξ) = Z∞ −∞ ϕ(x) exp(−ixξ)dx, has the following properties: (i) There exists a constant Cdepending on αsuch that |ˆϕ(ξ)| ≤ C |ξ|1+α for all ξsuch that |ξ|>1. (ii) |dˆϕ dξ (ξ)| ≤ 1for all ξ. Proof: Since (ii) is obvious, we shall only prove (i). Let ξbe such that |ξ|>1. Then |ˆϕ(ξ)| 1 + α≤Z−1+ 1 |ξ| −1 (1 + x)αexp(−ixξ)dx+Z0 −1+ 1 |ξ| (1 + x)αexp(−ixξ)dx =I+II. Clearly I≤Z−1+ 1 |ξ| −1 (1 + x)αdx =1 (1 + α)|ξ|1+α. Differential Transforms in Weighted Spaces 107 Integrating by parts II ≤−1 iξ +1 iξ|ξ|αexp −iξ −1 + 1 |ξ| +α iξ Z0 −1+ 1 |ξ| (1 + x)α−1exp(−ixξ)dx ≤1 |ξ|+1 |ξ|α+1 +1 |ξ|(1 + 1 |ξ|α)≤4 |ξ|α+1 . Lemma 3.2. Let −1< α ≤0. There exists a constant Cdepending only on α,ρand kvkk∞such that sup N |d Kα N(ξ)|= sup N N2 X k=N1 vkˆϕ(εkξ)−ˆϕ(εk−1ξ)≤C, for all ξ∈R. Proof: Clearly |d Kα N(0)|= 0. For fixed ξ∈R,ξ6= 0, let k0be such that εk0−1<1/|ξ| ≤ εk0. Then |d Kα N(ξ)| ≤ ∞ X k=−∞ |vk|| ˆϕ(εkξ)−ˆϕ(εk−1ξ)| ≤ kvkk∞ k0 X k=−∞ ···+kvkk∞ ∞ X k=k0+1 ···=I+II. To estimate Inotice that, by using the mean value theorem, the second property in Lemma 3.1 and (2.1), we get that |ˆϕ(εkξ)−ˆϕ(εk−1ξ)| ≤ C(εk−εk−1)|ξ| ≤ C(ρ2−1)εk−1|ξ|. Then, using (2.3) and by the election of k0we get I≤Ckvkk∞|ξ|(ρ2−1) k0 X k=−∞ εk−1≤Ckvkk∞|ξ|(ρ2−1)εk0−1 +∞ X i=0 1 ρi ≤C. 108 A. L. Bernardis, F. J. Mart´ ın-Reyes To estimate II we observe that if k≥k0+ 1 then |ξεk| ≥ |ξεk−1| ≥ |ξεk0| ≥ 1. Therefore, by Lemma 3.1 and the election of k0, II ≤ kvkk∞ ∞ X k=k0+1|ˆϕ(εkξ)|+|ˆϕ(εk−1ξ)| ≤Ckvkk∞ |ξ|1+α ∞ X k=k0+1 1 ε1+α k−1 ≤Ckvkk∞ |ξ|1+αε1+α k0 +∞ X i=0 1 ρ1+αi ≤C. Lemma 3.3. Let −1< α ≤0. There exists a constant Cdepending only on α,ρand kvkk∞such that sup N |Kα N(x)| ≤ +∞ X k=−∞ |vk(ϕk(x)−ϕk−1(x))| ≤C |x|+C +∞ X k=−∞ (εk+1 +x)α ε1+α k+1 χ(−εk+1,−εk)(x), (3.4) for all x6= 0. Consequently Kα(x) = +∞ X k=−∞ vk(ϕk(x)−ϕk−1(x)) is defined for all x∈Rand Z{x:|x|<R} |x||Kα(x)|dx ≤CR for all R > 0; the same inequality holds for Kα Nwith a constant independent of N. Proof: It is clear that Kα N(x) = 0 for x≥0. For negative x, there exists k0such that −εk0+1 ≤x < −εk0, and since the sequence εkis increasing, we get that χ(−εk,0)(x) = 0 for every k≤k0, and χ(−εk,0)(x) = 1 for Differential Transforms in Weighted Spaces 109 every k > k0. Thus we obtain |Kα N(x)| ≤ +∞ X k=−∞ |vk(ϕk(x)−ϕk−1(x))| ≤C ∞ X k=k0+1 (εk+x)α ε1+α k ≤C ∞ X k=k0+2 (εk−εk0+1)α ε1+α k +(εk0+1 +x)α ε1+α k0+1 ! ≤C ∞ X k=k0+2 1 εk +(εk0+1 +x)α ε1+α k0+1 ! ≤C εk0+1 +C(εk0+1 +x)α ε1+α k0+1 ≤C |x|+C(εk0+1 +x)α ε1+α k0+1 . To prove the integral inequality we take k0such that εk0≤R≤εk0+1. Then using the estimate (3.4) we get Z{x:|x|<R} |x||Kα(x)|dx ≤CR +C k0 X k=−∞ Z−εk −εk+1 |x|(εk+1 +x)α ε1+α k+1 dx ≤CR +C k0 X k=−∞ Z−εk −εk+1 (εk+1 +x)α εα k+1 dx ≤CR +C k0 X k=−∞ εk≤CR +Cεk0≤CR. In what follows we shall prove that the kernels Kα Nverify a one-sided smoothness condition uniformly. The proof is similar to the one in [13]. To prove this result we shall need the following lemma which follows easily taking into account the supports of the characteristic functions and using the property (2.4) of the sequence {εk}. We omit the proof. 116 A. L. Bernardis, F. J. Mart´ ın-Reyes Tα m,M (f3)(0) = 0. Then |Tα m,M f(0)| ≤ |Tα m,M f1(0)|+|Tα m,M f2(0)| =I+II. It is clear that I≤C M X k=m−1 1 ε1+α kZεm−1 0 (εk−y)α|f(y)|dy ≤C M X k=m−11 ρ(1+α)(k−m+1) 1 ε1+α m−1Zεm−1 0 |f(y)|(εm−1−y)αdy! ≤CM+ αf(0). On the other hand II =1 εm−1−βZεm−1−β 0 |Tα m,M f2(0)|dx ≤1 εm−1−βZεm−1−β 0 |Tα −M,M f(x)|dx +1 εm−1−βZεm−1−β 0 |Tα −M,M f1(x)|dx +1 εm−1−βZεm−1−β 0 |Tα m,M f2(0) −Tα m,M f2(x)|dx +1 εm−1−βZεm−1−β 0 |Tα −M,m−1f2(x)|dx =A1+A2+A3+A4. (If m=−Mwe understand that A4= 0.) It is obvious that A1≤M+(|Tα −M,M f|)(0). Differential Transforms in Weighted Spaces 117 For the second term, we use the uniform boundedness on Lsof the operators T−M,M given in Theorem 4.1 (ii). Thus A2≤1 εm−1−βZεm−1−β 0 |Tα −M,M f1(x)|sdx1/s ≤C1 εm−1−βZεm−1 0 |f(x)|sdx1/s ≤C M+ sf(0), where in the last inequality we have used condition (2.3). Now we estimate A3. Let x∈(0, εm−1−β), s > 1/(1 + α) and 1/s + 1/s′= 1, by H¨older inequality and Lemma 3.6 we have that |Tα m,M f2(0) −Tα m,M f2(x)| =Zy>εm−1 [Kα N(−y)−Kα N(x−y)]f2(y)dy = ∞ X j=m−1 Zεj+1 εj |Kα N(x−y)−Kα N(−y)|s′dy!1/s′ Zεj+1 0 |f(y)|sdy1/s ≤C M+ sf(0). Finally, we estimate A4. First, it is clear that A4≤C εm−1−βZεm−1−β 0 m−1 X k=−M 1 ε1+α kZ∞ εm−1 (εk+x−y)αχ(x,x+εk)(y)|f(y)|dy dx. By using (2.4) we get that for x∈(0, εm−1−β) and k≤m−2, x+εk≤ εm−1−β+εm−2≤εm−1. Therefore, the sum in the above inequality reduces to the term k=m−1. Thus, we have A4≤C εm−1−βZεm−1−β 0 1 ε1+α m−1Zx+εm−1 εm−1 (εm−1+x−y)α|f(y)|dy dx ≤C εm−1−βZεm−1−β 0 1 (x+εm−1)1+αZx+εm−1 0 (εm−1+x−y)α|f(y)|dy dx ≤CM+ αf(0). Putting all these inequalities together we are done. 118 A. L. Bernardis, F. J. Mart´ ın-Reyes Proof of Theorem 1.6: As in Theorem 4.4, since w∈A+ p(1+α)there is an s > 1/(1 + α) such that p/s > 1 and w∈A+ p/s. By Theorem 5.1 we have T∗ Mf(x)≤CM+(|Tα −M,M f|)(x) + M+ sf(x) + M+ αf(x). The operator M+ sis bounded in Lp(w) because w∈A+ p/s. Theorems 1.4 and 4.4 and the relation A+ p(1+α)⊂A+ pgive the uniform boundedness of the others two operators. Consequently, letting Mincrease to infinity, we see that the same holds for the operator T∗and we are done. 6. Proof of Theorem 1.7 We begin studying the behavior of T∗on the functions of compact support and average zero. For this, we shall need the following remark. Remark 6.1.It is clear that w∈A+ 1implies the following condition: there exists Csuch that for any M > 1 and every interval I= (a, a +h) Za a−Mh w≤CMh ess inf{w(x) : x∈I}. It follows from this property that if wsatisfies A+ 1then the following onesided doubling property holds: there exists Csuch that for any M > 1 Za+h a−Mh w≤(CM + 1) Za+h a w. Lemma 6.2. Let abe supported on I= (0, εi)and such that RIa= 0. Let −1< α ≤0. Assume that wis a weight such that wr∈A+ 1for some r > 1/(1 + α). Then there exists C > 0, independent of a, such that Zz<−εi+β T∗a(z)w(z)dz ≤CZI |a(z)|w(z)dz. Proof: Let us write Zz<−εi+β T∗a(z)w(z)dz = ∞ X m=i+βZ−εm −εm+1 T∗a(z)w(z)dz. Let z∈(−εm+1,−εm). Then, for fixed N∈Z2we get that |Tα Na(z)| ≤ C ∞ X k=−∞ ZI [ϕk(z−u)−ϕk−1(z−u)]a(u)du. Differential Transforms in Weighted Spaces 119 Observe that if z∈(−εm+1,−εm) and u∈Iwe have that z−u∈ (−εm+2,−εm). Then z−u∈(−εk,0) for all k≥m+ 2 and z−u6∈ (−εk,0) for all k≤m. Therefore, for all z∈(−εm+1,−εm), |Tα Na(z)| ≤ CZI (εm+1 +z−u)α ε1+α m+1 χ(−εm+1,0)(z−u)a(u)du +CZI (εm+2 +z−u)α ε1+α m+2 a(u)du +C ∞ X k=m+3 ZI (εk+z−u)α ε1+α k a(u)du =Am(z) + Bm(z) + Cm(z). Notice that Z−εm −εm+1 Am(z)w(z)dz =Z−εm+1+2εi −εm+1 ···+Z−εm −εm+1+2εi ··· ≤CZ−εm+1+2εi −εm+1 w(z)ZI |εm+1 +z−u|α ε1+α m+1 |a(u)|du dz +CZ0 −εm+1+2εi w(z)ZI (z+εm+1−u)α ε1+α m+1 a(u)dudz. Now, doing the same with Bm(z) we get Z−εm −εm+1 Bm(z)w(z)dz ≤Z−εm+2+2εi −εm+2 ···+Z−εm −εm+2+2εi ··· ≤CZ−εm+2 +2εi −εm+2 w(z)ZI |εm+2 +z−u|α ε1+α m+2 |a(u)|du dz +CZ0 −εm+2+2εi w(z)ZI (z+εm+2−u)α ε1+α m+2 a(u)dudz. 120 A. L. Bernardis, F. J. Mart´ ın-Reyes Consequently, Zz<−εi+β T∗a(z)w(z)dz ≤ ∞ X m=i+βZεm −εm+1 (Am(z) + Bm(z) + Cm(z)) w(z)dz ≤C ∞ X m=i+βZ−εm+1+2εi −εm+1 w(z)Zεi 0 |εm+1 +z−u|α ε1+α m+1 |a(u)|du dz +C ∞ X m=i+βZ0 −εm+1+2εi w(z)Zεi 0 (z+εm+1 −u)α ε1+α m+1 a(u)dudz +C ∞ X m=i+βZ−εm −εm+1 w(z) ∞ X k=m+3Zεi 0 (εk+z−u)α ε1+α k a(u)dudz =I+II +III. Now we shall prove that each sum is dominated by CRεi 0|a(u)|w(u)du. By Fubini’s theorem and H¨older inequality with exponents r > 1/(1 + α) and r′, 1/r + 1/r′= 1, we obtain for the first sum the following inequalities: I≤C ∞ X m=i+βZεi 0 |a(u)|Z−εm+1+2εi −εm+1 w(z)|εm+1 +z−u|α ε1+α m+1 dz du ≤C ∞ X m=i+βZεi 0 |a(u)|1 ε1+α m+1 Z−εm+1+2εi −εm+1 wr(z)dz!1/r (2εi−u)α+1/r′du ≤C ∞ X m=i+βZεi 0 |a(u)|1 ε1+α m+1 Z0 −εm+1 wr(z)dz!1/r εα+1/r′ idu. Let us fix r > 1/(1 + α) such that wr∈A+ 1. By (6.1) we get that I≤CZεi 0 |a(u)|w(u)du ∞ X m=i+βεi εm+1 α+1/r′ ≤CZεi 0 |a(u)|w(u)du. Differential Transforms in Weighted Spaces 121 We notice that we have used twice that αr′+ 1 >0. We shall estimate now the second sum. First we write II as C ∞ X m=i+β ∞ X ℓ=1Z−εm+1+2ℓ+1εi −εm+1+2ℓεiZεi 0 a(u)(z+εm+1−u)α ε1+α m+1 duw(z)χ(−∞,0)(z)dz. Using that Rεi 0a= 0, the mean value theorem and the fact that z∈ (−εm+1 + 2ℓεi,−εm+1 + 2ℓ+1εi) we obtain that Zεi 0 a(u)(z+εm+1 −u)αdu≤CZεi 0 |a(u)|(z+εm+1 −u)α−1u du ≤CZεi 0 |a(u)|(2ℓεi)α−1εidu. Therefore, II is bounded by C ∞ X m=i+β (εi)α (εm+1)1+α Zεi 0 |a(u)|du ∞ X ℓ=1 2ℓ(α−1) Z−εm+1+2ℓ+1εi −εm+1+2ℓεi w(z)χ(−∞,0)(z)dz. Let us fix again r > 1/(1 + α) such that wr∈A+ 1. By H¨older inequality and (6.1) we get Z−εm+1+2ℓ+1εi −εm+1+2ℓεi w(z)χ(−∞,0)(z)dz ≤ Z0 −εm+1 wr!1/r (2ℓεi)1/r′ ≤Cε1/r m+1 ess infx∈(0,εi)w(x)(2ℓεi)1/r′ and consequently II ≤C ∞ X m=i+βεi εm+1 α+1/r′Zεi 0 |a(u)|w(u)du∞ X ℓ=1 2ℓ(α−1+1/r′) ≤CZεi 0 |a(u)|w(u)du. In order to estimate the third sum, we use again that Rεi 0a= 0 and the mean value theorem. Then we have III ≤C ∞ X m=i+βZ−εm −εm+1 w(z) ∞ X k=m+3 Zεi 0 (εk+z−u)α−1 ε1+α k u|a(u)|du dz ≤C ∞ X m=i+β ∞ X k=m+3 Zεi 0 εi|a(u)|Z−εm −εm+1 (εk+z−u)α−1 ε1+α k w(z)dz du. 122 A. L. Bernardis, F. J. Mart´ ın-Reyes Obvious inequalities, (2.3), (2.4) and w∈A+ 1give for almost every u∈ (0, εi) Z−εm −εm+1 (εk+z−u)α−1w(z)dz ≤(εk−εm+1 −εi)α−1Z−εm −εm+1 w(z)dz ≤C(εm+1 +εi)(εm+3 −εm+1 −εi)α−1w(u) ≤Cεα m+3w(u). Consequently, III ≤CZεi 0 |a(u)|w(u)duεi ∞ X m=i+β εα m+3 ∞ X k=m+3 1 ε1+α k ≤CZεi 0 |a(u)|w(u)duεi ∞ X m=i+β 1 εm+3 ≤CZεi 0 |a(u)|w(u)du. Corollary 6.3. Let −1< α ≤0and let wbe a weight such that wr∈A+ 1 for some r > 1/(1 + α). Let abe supported on I= (x∗, x∗+h)and such that RIa= 0. If A=ρ2(β+1) there exists Cindependent of x∗,hand a, such that Zz<x∗−Ah T∗a(z)w(z)dz ≤CZI |a(z)|w(z)dz. Proof: First notice that it is sufficient to prove the corollary for x∗= 0. Choose isuch that εi−1≤h < εi. Then ais supported on (0, εi) and has integral zero. Furthermore, by (2.3), −Ah < −εi+βand by the lemma Zz<−Ah T∗a(z)w(z)dz ≤Zz<−εi+β T∗a(z)w(z)dz ≤CZI |a(z)|w(z)dz. Once we have Corollary 6.3 and Theorem 1.6 the proof of Theorem 1.7 is straightforward (see for instance [1]). We include it for the sake of completeness. Proof of Theorem 1.7: Let Oλ={x:M+f(x)> λ}. It is well known that if {Ii}are the connected components of Oλ, then λ=1 |Ii|RIif=fIi. We decompose fas f=fχR\Oλ+XfIiχIi+X(f−fIi)χIi. Differential Transforms in Weighted Spaces 123 As usual fχR\Oλ+PfIiχIiwill be denoted by gand P(f−fIi)χIi= Pbiby b. Observe that each bihas support on Iiand average zero. Now, ZR |g(y)|w(y)dy ≤ZR\Oλ |f(y)|w(y)dy +Xw(Ii)fIi =ZR\Oλ |f(y)|w(y)dy +λXw(Ii) =ZR\Oλ |f(y)|w(y)dy +λw(Oλ) ≤CZR |f(y)|w(y)dy, (6.4) because the operator M+fis of weak type (1,1) with respect to w. For each interval I= (a, a +h), let us denote by I∗the interval (a−Ah, a +h), where A=ρ2(β+1). We also denote by f Oλthe union of all the intervals I∗ i. Observe that w({x:T∗f(x)> λ})≤w({x:T∗g(x)> λ/2}) + w(f Oλ) +w({x /∈f Oλ:T∗b(x)> λ/2}) = I+II +III. The one-sided doubling property of the weight and the weak type (1,1) inequality for M+give II =w(∪iI∗ i)≤Cw(Oλ)≤C λZ|f(y)|w(y)dy. On the other hand, since A+ 1implies condition A+ pfor any p > 1/(1+α), Theorem 1.6 implies that T∗is a bounded operator in Lp(w). Then we have w({x:T∗g(x)> λ/2})≤C λpZ(T∗g(y))pw(y)dy ≤C λpZ|g(y)|pw(y)dy ≤C λZ|g(y)|w(y)dy ≤C λZ|f(y)|w(y)dy. 124 A. L. Bernardis, F. J. Mart´ ın-Reyes Observe that in the last two inequalities we have used |g| ≤ λand (6.4). Finally, by using Corollary 6.3 and the one-sided nature of the operator T∗,we have III ≤C λZR\ f Oλ T∗b(x)w(x)dx ≤C λX iZR\I∗ i T∗bi(x)w(x)dx ≤C λX iZIi |bi(x)|w(x)dx. Since the Ii’s are disjoint and b(x) = bi(x) on each Iithe last term is bounded by C λZ|b(x)|w(x)dx=C λZ|f(x)−g(x)|w(x)dx≤C λZ|f(x)|w(x)dx. 7. Proof of Theorem 1.9 By Theorems 1.6 and 1.7 it is clear that it suffices to prove the a.e. convergence in the Schwartz’s class S. Theorem 7.1. The functions Tα Nψ(x)converge for all ψ∈ S and for every x∈Ras N= (N1, N2)tends to (−∞,+∞). Further, if ψhas compact support and Kα(x) = P+∞ k=−∞ vk(ϕk(x)−ϕk−1(x)) then lim N→(−∞,+∞)Tα Nψ(x) = Kα∗ψ(x) for all xoutside the support of ψ∈ S. Proof: It will suffice to show that Tα −M,0ψ(x) and Tα 0,M ψ(x) converge as M→+∞. In fact, we shall prove that |Tα −M,0ψ(x)−Tα −N,0ψ(x)|+|Tα 0,M ψ(x)−Tα 0,N ψ(x)| =|Tα −M,−N−1ψ(x)|+|Tα N+1,M ψ(x)|=I+II is small for N < M and Nbig enough. First, let us observe that from (2.3) we have that for all s > 0 there exists a positive constant C such that for each m≤n (7.2) n X k=−∞ εs k≤C εs nand ∞ X k=m 1 εs k ≤C1 εs m . Differential Transforms in Weighted Spaces 125 Since RKα N= 0 for every N∈Z2, by using the mean value theorem and (7.2) we get that I=ZKα −M,−N−1(x−y)[ψ(y)−ψ(x)] dy ≤2||ψ′||L∞||vk||∞Z−N−1 X k=−M−1 ϕk(u)|u|du ≤C −N−1 X k=−M−1 1 εα kZ0 −εk (εk+u)αdu ≤C −N−1 X k=−M−1 εk≤Cε−N−1, which is small when Nis big enough. In order to estimate II, we choose a number s > 1/(1 + α). By H¨older’s inequality and using (7.2) again we have II ≤2||vk||∞ZM X k=N ϕk(x−y)|ψ(y)|dy ≤C M X k=N 1 ε1+α kZx+εk x |ψ(y)|(εk+x−y)αdy ≤C||ψ||Ls M X k=N 1 ε1+α kZx+εk x (x+εk−y)αs′dy1/s′ ≤C||ψ||Ls M X k=N 1 ε1/s k ≤C ε1/s N ||ψ||Ls, which can be done small taking Nbig enough. Let us denote the support of ψby supp(ψ). Let x6∈ supp(ψ). Then there exists δ > 0 such that |x−y| ≥ δfor all y∈supp(ψ). Let k0∈Z such that εk0< δ ≤εk0+1. Therefore, from Lemma 3.3 and taking into account the support of Kα Nwe get that |Kα N(x−y)| ≤ C δ+CX k≥k0 (εk+1 +x−y)α ε1+α k+1 χ(−εk+1,−εk)(x−y).