Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces

Publ. Mat. 47 (2003), 71–102 BOUNDEDNESS OF THE WEYL FRACTIONAL INTEGRAL ON ONE-SIDED WEIGHTED LEBESGUE AND LIPSCHITZ SPACES S. Ombrosi and L. de Rosa Abstract In this paper we introduce the one-sided weighted spaces L− w(β), −1<β<1. The purpose of this definition is to obtain an extension of the Weyl fractional integral operator I+ αfrom Lp w into a suitable weighted space. Under certain condition on the weight w,wehave that L− w(0) coincides with the dual of the Hardy space H1 −(w). We prove for 0 <β<1, that L− w(β) consists of all functions satisfying a weighted Lipschitz condition. In order to give another characterization of L− w(β), 0 ≤β<1, we also prove a one-sided version of John-Nirenberg Inequality. Finally, we obtain necessary and sufficient conditions on the weight wfor the boundedness of an extension of I+ αfrom Lp w into L− w(β), −1<β<1, and its extension to a bounded operator from L− w(0) into L− w(α). 1. Notations, definitions and prerequisites Let E⊆Rbe a Lebesgue measurable set. We shall denote its Lebesgue measure by |E|and the characteristic function of Eby χE. As usual, a weight wis a measurable, non-negative and locally integrable function defined on R. Let wbe aweight. Given a Lebesgue measurable set E⊆R, its w-measure will be denote by w(E)=Ew(t)dt. 2000 Mathematics Subject Classification. Primary: 26A33; Secondary: 42B25. Key words. Weyl fractional integral, weigths, weighted Lebesgue and Lipschitz spaces, weighted BMO. This research has been partially supported by UBACYT 2000-2002 and CONICET. 72 S. Ombrosi, L. de Rosa Let 1 <p<∞. The weight wbelongs to the class A− pif there exists a constant Csuch that sup h>01 hpa+h a w(x)dx a a−h w(x)−1 p−1dxp−1≤C, for all real number a.Inasimilar way, wbelongs to A+ pif sup h>0 1 hpa a−h w(x)dx a+h a w(x)−1 p−1dxp−1 ≤C, for all real number a. The class A− 1is defined by the condition sup h>01 ha+h a w(x)dx≤Cw(a), for almost every real number a. The weight wbelongs to A+ 1if sup h>01 ha a−h w(x)dx≤Cw(a), for almost every a. These classes A− pand A+ pwere introduced by E. Sawyer in [12]. We recall three basic results on these weights. (i) For 1 <p<∞,aweight wbelongs to A− pif and only if w1−p belongs to A+ p, where 1 p+1 p=1. (ii) If 1 ≤p<q<∞, then A− p⊂A− q. (iii) If 1 <p<∞and wbelongs to A− p, then wbelongs to A− p−for some >0. The proof of (i) and (ii) are very simple and (iii) can be found in Proposition 3 in [3]. In the sequel, for each bounded interval I=[a, b]weshall denote I−=[a−|I|,a] and I+=[b, b +|I|]. Let 1 ≤q<∞.Aweight wsatisfies the condition RH−(q)ifthere exists a constant Csuch that for every bounded interval I. 1 |I|I w(x)qdx1/q ≤C1 |I|I− w(x)dx. We shall say that a weight wbelongs to D−if there exists a constant C such that for every bounded interval I, w(I∪I+)≤Cw(I). Weyl Fractional Integral 73 It is well known that if w∈A− p,1≤p<∞, then w∈D−. Let wbe aweight, 1 ≤p<∞and fa measurable function. We shall say that fbelongs to Lp wif fp p,w =∞ −∞ |f(x)| w(x)p dx is finite. The function fbelongs to  Lp wif [f]p p,w = sup t>0 tpx∈R:|f(x)| w(x)>t  is finite. Let 0 <α<1. Given fa measurable function on R, its Weyl fractional integral is defined by I+ αf(x)=∞ x f(y) (y−x)1−αdy, whenever this integral is finite. In the sequel, the letter Cwill denote a positive finite constant not necessarily the same at each occurrence. If 1 ≤p≤∞then pwill be its conjugate exponent, that is, 1/p +1/p=1. Let wbe aweight and −1<β<1. Definition 1.1. We say that a locally integrable function fdefined on R belongs to Lw(β), if there exists a constant Csuch that 1 w(I)|I|βI |f(y)−fI|dy ≤C, for every bounded interval I, where fI=1 |I|If. The least constant C will be denoted fLw(β). The spaces Lw(β)were introduced by E. Harboure, O. Salinas and B. Viviani in [1]. They are a weighted version of the spaces Lλ,p, for p=1,defined by J. Peetre in [8]. If wbelongs to A− q,1≤q<2, then Lw(0) is the dual space of the one-sided weighted Hardy space H1 −(w), see [10] and [11]. 74 S. Ombrosi, L. de Rosa Definition 1.2. We say that a locally integrable function fdefined on R belongs to L− w(β), if there exists a constant Csuch that 1 w(I−)|I|βI |f(y)−fI|dy ≤C, for every bounded interval I. The least constant Csatisfying this inequality will be denoted fL− w(β). In the following definition, we consider a one-sided version of the classes H(α, p) defined in [1]. Definition 1.3. Let 0 <α<1 and 1 <p≤∞.Wesay that a weight w belongs to H−(α, p)ifthere exists a constant Csuch that for every bounded interval I=[a, b], the inequality |I|1 p−α+1 ∞ b w(y)p (y−a)(2−α)pdy1/p ≤Cw(I) |I|, holds. 2. Statement of the main results Lemma 4.1(iii) shows that if wbelongs to H−(α, p), 1 <p≤∞, then wbelongs to D−and therefore Lw(β)⊆L − w(β) for every β:−1< β<1. The next theorem states that wbelonging to D−is a sufficient condition for the equality of these spaces, whenever 0 ≤β<1. Theorem 2.1. Let 0≤β<1and let wbelong to D−. Then, the spaces Lw(β)and L− w(β)are equal, and their norms are equivalent. The next theorem gives us a characterization of the spaces Lw(β), 0≤β<1, whenever wbelongs to A− p.Inthe case β=0,weshall prove this result using Proposition 3.6, which states a one-sided weighted version of John-Nirenberg Inequality. Theorem 2.2. Let 0≤β<1and 1≤p<∞.Letwbeaweight such that wbelongs to A− p. Then, f∈L w(β)if and only if there exists a constant Csuch that I− |f(x)−fI+|qw(x)1−qdx ≤Cw(I−)|I|βq,(2.1) for all bounded interval Iand every q:1≤q≤p,q<∞. The following two theorems state a sufficient and necessary condition on the weight wto obtain extensions of I+ αdefined on certain spaces. Weyl Fractional Integral 75 Theorem 2.3. Let 0<α<1,1<p<∞and β=α−1/p. The following statements are equivalent. (i) The weight wbelongs to H−(α, p). (ii) The operator I+ αcanbeextended to a linear bounded operator  I+ α from  Lp winto L− w(β)by means of (2.2)  I+ α(f)(x)=−x x0 f(y)dy |y−x|1−α +∞ x01 |y−x|1−α−1−χ[x0,x0+1](y) (y−x0)1−αf(y)dy, for any x0∈R. (iii) The operator I+ αcanbeextended to a linear bounded operator  I+ α from Lp winto L− w(β), where  I+ αis defined as in (2.2). Theorem 2.4. Let wa weight and 0<α<1. The following statements are equivalent. (i) The weight wbelongs to H−(α, ∞). (ii) The operator I+ αcanbeextended to a linear bounded operator  I+ α:Lw(0) →L w(α)by means of  I+ α(f)(x)=∞ −∞ χ[x0,∞)(y) |y−x0|1−α−χ[x,∞)(y) |y−x|1−αf(y)dy, for an appropriate choice of x0∈R. Remark 2.5.Let 1 <p< 1 αand β=α−1/p<0. (i) It is easy to see that if wbelongs to RH−(1 1+β), then L−1/β w⊆ L− w(β). (ii) By Lemma 4.4 in [9], if wpbelongs to A− −βp+1 then wsatisfies the condition RH−(p), and taking into account that 1 1+β<p ,it follows that wbelongs to RH−(1 1+β). (iii) Theorem 6 in [4] states the fact that wpbelongs to A− −βp+1 is a necessary and sufficient condition for the boundedness of I+ α from Lp winto L−1/β w⊆L − w(β). (iv) If wpbelongs to A− −βp+1, since wp∈A− p+1,wehave that w belongs to H−(α, p). However, there exist weights wbelonging to H−(α, p) such that wpdoes not belong to A− p+1, for example, w(x)=|x|γfor −β≤γ<1−β, see Remark 4.3. 76 S. Ombrosi, L. de Rosa In consequence, if −1<β<0 and wpbelongs to A− −βp+1, the extension of I+ αin Theorem 2.3 can be obtained from Theorem 6 in [4]. But, (iv) shows that Theorem 2.3 can be applied to a larger class of weights. Remark 2.6.Let wbe aweight. We shall say that a locally integrable function fdefined on R,belongs to MW−(w)ifthere exists a constant C such that 1 |I| 1 ess infI−wI |f(y)−fI|dy ≤C, for every bounded interval I. (i) By Definition 1.2, it follows that MW−(w)⊆L − w(0). Moreover, if wbelongs to A− 1then Lw(0) ⊆MW−(w), and as a consequence of Theorem 2.1, L− w(0) = MW−(w). (ii) Following the same lines of Theorem 7 in [7], it can be seen that, in the case α=1/p, the weight wpbelongs to A− 1if and only if the operator I+ αis bounded from Lp winto MW−(w). Also see [2]. (iii) If wpbelongs to A− 1then, by Remark 4.3, wbelongs to H−(α, p). In consequence, the fact that wpbelongs to A− 1implies the boundedness of I+ αfrom Lp winto MW−(w), is contained in Theorem 2.3. 3. The spaces L w (β) and L − w (β) The next lemma will be used in the proof of Theorem 2.1. Lemma 3.1. Let −1<β<1,falocally integrable function defined on R, and w∈D−. The following statements are equivalent. (i) f∈L − w(β). (ii) There exists a constant Csuch that for every a∈Rand h>0, 1 w([a−h/2,a])hβa+h a |f(y)−f[a+h/2,a+h]|dy ≤C. (iii) There exists a constant Csuch that for every a∈Rand h>0, 1 w([a−h/2,a])hβa+h a |f(y)−f[a+h,a+3h]|dy ≤C. The constants Cin (ii) and (iii) are equivalent to fL− w(β). Weyl Fractional Integral 77 Proof: (i) ⇒(ii). Using (i) and taking into account that w∈D−,we have a+h/2 a |f(y)−f[a+h/2,a+h]|dy ≤a+h/2 a |f(y)−f[a+h/4,a+h/2]|dy+2a+h a+h/4 |f(y)−f[a+h/2,a+h]|dy ≤3a+h/2 a |f(y)−f[a,a+h/2]|dy +5a+h a+h/4 |f(y)−f[a+h/4,a+h]|dy ≤CfL− w(β)w([a−h/2,a])hβ+CfL− w(β)w([a−h/2,a+h/4])hβ ≤CfL− w(β)w([a−h/2,a])hβ. From these inequalities and using (i) again, we have the estimate a+h a |f(y)−f[a+h/2,a+h]|dy =a+h/2 a |f(y)−f[a+h/2,a+h]|dy +a+h a+h/2 |f(y)−f[a+h/2,a+h]|dy ≤CfL− w(β)w([a−h/2,a])hβ+CfL− w(β)w([a, a +h/2])hβ ≤CfL− w(β)w([a−h/2,a])hβ, which shows that (ii) holds. In a similar way it can be proved that (ii) ⇒(iii) and (iii) ⇒(i). As we have already mencioned if wbelongs to D−then, for every −1<β<1wehave the inclusion Lw(β)⊆L − w(β). In order to prove Theorem 2.1, it will be sufficient to show that L− w(β)⊆L w(β). 78 S. Ombrosi, L. de Rosa Proof of Theorem 2.1: We suppose that f∈L − w(β). Let a∈Rand h>0. For each j≥0wedefine aj=a+h/2j. Then, a+h/2 a |f(y)−f[a+h/2,a+h]|dy = ∞  j=1 aj aj+1 |f(y)−f[a+h/2,a+h]|dy ≤ ∞  j=1 aj aj+1 |f(y)−f[aj,aj−1]|dy+ ∞  j=2 h 2j+1 |f[aj,aj−1]−f[a1,a0]| =I+II. (3.1) Taking into account that for each j≥2, |f[aj,aj−1]−f[a1,a0]|≤2j haj−1 aj |f−f[a+h/2,a+h]| it follows that, II ≤ ∞  j=2 1 2aj−1 aj |f−f[a+h/2,a+h]|=1 2a+h/2 a |f(y)−f[a+h/2,a+h]|dy. Then, by (3.1) a+h/2 a |f(y)−f[a+h/2,a+h]|dy ≤2I.(3.2) Now, using (iii) of Lemma 3.1 and keeping in mind that β≥0wehave that, I≤C ∞  j=1 h 2jβ w([aj+2,a j+1]) ≤Chβw([a, a +h/4]).(3.3) Weyl Fractional Integral 79 From (3.2) and (3.3), and taking into account that f∈L − w(β), we get a+h a |f(y)−f[a+h/2,a+h]|dy =a+h/2 a |f(y)−f[a+h/2,a+h]|dy +a+h a+h/2 |f(y)−f[a+h/2,a+h]|dy ≤Chβw([a, a +h/4]) + Chβw([a, a +h/2]) ≤Chβw([a, a +h]). Therefore, a+h a |f(y)−f[a,a+h]|dy ≤3a+h a |f(y)−f[a+h/2,a+h]|dy ≤Chβw([a, a +h]), which shows that f∈L w(β). Remark 3.2.Let −1<β<0 and w(t)=e−t. The weight wbelongs to A− 1however, we only have the strict inclusion Lw(β)⊂L − w(β). For example, given a>1weconsider the function f(t)=e−at,t≥0 1,t<0. We observe, using Remark 2.5(i), that f∈L − w(β). On the other hand, 1 hβw([0,h])h 0 |f−f[h,2h]|=1 hβ(1 −e−h)1−e−ah a−e−ah a(1−e−ah) =(1 −e−ah)2 hβ(1 −e−h)a, which tends to infinite whenever htends to infinite. This implies that f/∈L w(β). The next proposition will be used in the proof of Theorem 2.2. 86 S. Ombrosi, L. de Rosa By (3.15) and (3.14), we have II ≤Cs γp i,k w(H− i,k)=Cs γp i w(Hi)≤Cµsp−1 γpw(I−). Then, (3.16) and (3.17) imply that A(λ, I)≤CµA(λ−γ) s+sp−1 γpw(I−). From this inequality, (ii) follows as in Theorem 3 of [6]. Proposition 3.7. Let 0<β<1and 1<p<∞.Letwbeaweight such that w1+ β 1−βpbelongs to A− p. Then, f∈L w(β)if and only if there exists a constant Csuch that (2.1) holds for all bounded interval Iand every q:1≤q≤p/(1 −β). Proof: Suppose that (2.1) holds for every q:1≤q≤p/(1 −β). Taking q=1it is easy to show that f∈L w(β). Conversely, let fbelong to Lw(β). We observe that it will be sufficient to consider q=p/(1−β), because from this case and applying H¨older’s inequality we obtain (2.1) for every 1 ≤q<p /(1 −β). Given a bounded interval Iand using Proposition 3.3, we have that I− |f(x)−fI+|qw(x)1−qdx ≤I−1 |I+|I+ |f(x)−f(y)|dyq w(x)1−qdx ≤CI− w(x)1−q1 |I+|I+x+|y−x| 2 x w(z) (z−x)1−βdz +y+|y−x| 2 y w(z) (z−y)1−βdzdyq dx ≤CI− w(x)1−qx+3|I| 2 x w(z) (z−x)1−βdzq dx +C |I+|qI− w(x)1−qI+y+3|I| 2 y w(z) (z−y)1−βdz dyq dx =A+B. (3.19) Weyl Fractional Integral 87 If we denote J=I−∪I∪I+then we have the estimate A≤CI− w(x)1−qI+ β(wχJ)(x)qdx. Our hypothesis w1+ β 1−βp∈A− pis equivalent to w1−p 1−β∈A+ p,(3.20) where p=1+ q sand 1 s=1 q+β. Then, by Theorem 6 in [4]itfollows that A≤C∞ −∞ w(x)−s q|wχJ(x)|sdxq/s =CJ w(x)s/q dxq/s . Since q/s =qβ +1 >1, applying H¨older’s inequality and taking into account that w∈D−we obtain A≤CJ w(x)dx |J|q s−1≤Cw(I−)|I|βq.(3.21) Let us estimate B.Ifweset J=I+∪I++ ∪I+++, then B≤C |I+|qI− w(x)1−qI+ I+ β(wχJ)(y)dyq dx. Applying H¨older’s inequality, B≤C |I+|qI− w(x)1−qdxI+ w(y)dyq/q I+ w(y)1−qI+ β(wχJ)(y)qdx. From (3.20), it follows that w1−q∈A+ qthen, we have that B≤CI+ w(y)1−qI+ β(wχJ)(y)qdx. Proceeding as in the estimation of Aand taking into account that w∈ D−we obtain B≤Cw(I−)|I|βq.(3.22) As consequence of (3.19), (3.21) and (3.22) we get (2.1) and the proof of this proposition is complete. Proof of Theorem 2.2: We shall prove that fbelonging to Lw(β)isa sufficient condition for (2.1) holds. The fact that (2.1) is a necessary condition follows as in the previous proposition. For that, we shall consider different cases. First of all, we assume that β=0and f∈L w(0). If w∈A− 1we have that (2.1) is an immediate consequence of Proposition 3.6(i). If 88 S. Ombrosi, L. de Rosa w∈A− p,1<p<∞,wehave that w∈A− p−for some >0. Then, by Proposition 3.6(ii), and proceeding as in Theorem 4 of [6], we obtain that fsatisfies (2.1). Let 0 <β<1 and 1 <p<∞. Since the weight wbelongs to A− p there exists 0 <α<βsuch that w1+ α 1−αpbelongs to A− p. Proceeding as in (3.19), we have that I− |f(x)−fI+|qw(x)1−qdx ≤CI− w(x)1−q1 |I+|I+x+|y−x| 2 x w(z) (z−x)1−βdz +y+|y−x| 2 y w(z) (z−y)1−βdzdyq dx ≤C|I|(β−α)qI− w(x)1−qx+3|I| 2 x w(z) (z−x)1−αdzq dx +C |I|(β−α−1)qI− w(x)1−qI+y+3|I| 2 y w(z) (z−y)1−αdz dyq dx =|I|(β−α)q(A+B). Substituting in the proof of the previous proposition αfor βin the estimation of Aand Bwe obtain this case. Finally, we suppose that 0 <β<1 and p=1. Since the weight w belongs to A− 1it follows that wbelongs to A− sfor every 1 <s< ∞. Then, by the previous case we obtain that (2.1) holds for every 1≤q<∞. 4. The classes H − (α, p) The next lemma states necessary conditions for that a weight wbelongs to H−(α, p). Lemma 4.1. Let 1<p≤∞.Ifw∈H−(α, p)then, (i) wpbelongs to ∈D−, (ii) wbelongs to ∈RH−(p), (iii) wbelongs to ∈D−. Weyl Fractional Integral 89 Proof: The proof of (i) and (ii) are similar to ones of Lemma 3.7 and Lemma 3.8, in [1], respectively. Applying H¨older’s inequality and (ii), we obtain (iii). Lemma 4.2. Let wbe a weight. The following conditions are equivalent. (a) w∈H−(α, p). (b) w∈RH−(p)and there exist positive constants Cand such that, wp([a, a +θt]) ≤Cθ(2−α)p−wp([a, a +t]), for every a∈R,t>0and θ≥1. (c) There exist positive constants Cand such that, wp([a, a +θt]) θt 1/p ≤Cθ1 p+1−α− pw([a−t, a]) t, for every a∈R,t>0and θ≥1. Proof: (a) ⇒(b). By Lemma 4.1(ii) we have that w∈RH−(p). Let I=[a, a +t]. Applying H¨older’s inequality and keeping in mind that w∈H−(α, p), wp(I) |I|≥w(I) |I|p ≥C|I|(1 p−α+1)p∞ a+t w(y)p (y−a)(2−α)pdy ≥C|I|(1 p−α+1)p k≥0 1 (2k+1t)(2−α)pa+2k+1t a+2kt w(y)pdy. (4.1) Since i≥k1 2(2−α)pi=C1 2(2−α)pk,by(4.1) and applying Fubini’s Theorem, wp(I) |I|≥C|I|(1 p−α+1)p1 t(2−α)p k≥0a+2k+1t a+2kt w(y)pdy  i≥k1 2(2−α)pi =C|I|(1 p−α+1)p i≥0 1 (2it)(2−α)p i  k=0 a+2k+1t a+2kt w(y)pdy =C|I|(1 p−α+1)p i≥0 1 (2it)(2−α)pa+2i+1t a+t w(y)pdy. 90 S. Ombrosi, L. de Rosa Therefore, wp(I) |I|≥C|I|(1 p−α+1)p i≥0 1 (2it)(2−α)pa+2i+1t a w(y)pdy ≥C|I|(1 p−α+1)p i≥02i+1t 2it wp([a, a +s]) s(2−α)p ds s =C|I|(1 p−α+1)p∞ t wp([a, a +s]) s(2−α)p ds s. In consequence, ∞ t wp([a, a +s]) s(2−α)p ds s≤Cwp([a, a +t]) t(2−α)p. Now, using Lemma 3.3 in [1] with ϕ(s)=wp([a, a+s]) and r=(2−α)p, there exist Cand such that ϕ(θt)≤Cθr−ϕ(t), for every t>0 and θ≥1. That is, wp([a, a +θt]) ≤Cθ(2−α)p−wp([a, a +t]), for every t>0 and θ≥1, This completes the proof of (a) ⇒(b). (b) ⇒(a). Let I=[a, a +t]. If (b) holds, we have that ∞ a+t w(y)p (y−a)(2−α)pdy1/p =∞  k=0 a+2k+1t a+2kt w(y)p (y−a)(2−α)pdy1/p ≤∞  k=0 1 (2kt)(2−α)pwp([a+t, a +t+2 k+1t])1/p ≤C∞  k=0 (2k+1)(2−α)p− (2kt)(2−α)pwp([a+t, a +2t])1/p ≤C1 ta+2t a+t w(y)pdy1/p t1 p−2+α. (4.2) Weyl Fractional Integral 91 Using the hypothesis w∈RH−(p)weobtain that (4.2) is bounded by C1 ta+t a w(y)dy t 1 p−2+α=Cw([a, a +t]) t1 p+2−α, which shows that w∈H−(α, p). The proof of (b) ⇒(c) is very simple and we shall omit it. (c) ⇒(b). Taking θ=1in (c) we have that w∈RH−(p). Using (c) and H¨older’s inequality, wp([a−t, a +θt]) θt 1/p =wp([a−t, a]) θt +wp([a, a +θt]) θt 1/p ≤wp([a−t, a]) θt 1/p +Cθ1 p+1−α− pwp([a−t, a]) t1/p . We can suppose that 1 p+1−α− p>0, then taking into account that θ≥1 wp([a−t, a −t+θt]) θt 1/p ≤wp([a−t, a +θt]) θt 1/p ≤Cθ1 p+1−α− pwp([a−t, a]) t1/p . From these inequalities with a=b+twe obtain that wp([b, b +θt]) ≤Cθ(2−α)p−wp([b, b +t]), which completes the proof. Remark 4.3.It is easy to see that if wpbelongs to A− 1then, w∈ H−(α, p). On the other hand, applying Lemma 4.2 (b) ⇒(a), it follows that if w(x)=|x|γwith 0 <γ<1/p −α+1,then wbelongs to H−(α, p), but wdoes not belong to A− 1.For0<α<1/p,asanimmediate consequence of Lemma 4.2 (c) ⇒(a) it follows that if wpbelongs to A− p+1 then, wbelongs to H−(α, p). The next two lemmas show that if wbelongs to H−(α, p), 1 <p< ∞, then there exists η>0 such that wbelongs to H−(α, q) for every q:p−η<q<p+η. 92 S. Ombrosi, L. de Rosa Lemma 4.4. Let 1<p<∞and w∈H−(α, p). Then, there exists δ0∈(0,1) such that w∈H−(α, (pδ))for any δ:δ0<δ≤1. Proof: It is a simple variant of Lemma 3.13 in [1]. Lemma 4.5. Let 1<p<∞and w∈H−(α, p). Then, there exists τ0>1such that w∈H−(α, (pτ))for any 1≤τ≤τ0. Proof: Since w∈RH−(p) applying Theorem 5.3 in [9], there exists τ0>1 such that for every τ:1≤τ≤τ0there exists a constant Csuch that 1 c−bc b w(y)pτdy1 pτ ≤C1 b−ab a w(y)dy ≤C1 b−ab a w(y)pdy1 p (4.3) for every a<b<cwith c−b=2(b−a). Let I=[a, b]. Using (4.3) we have that, ∞ b w(y)pτ (y−a)(2−α)pτdy = k≥02k|I|≤y−a≤2k+1|I| w(y)pτ (y−a)(2−α)pτdy ≤ k≥0 1 (2k|I|)(2−α)pτ2k|I|≤y−a≤2k+1|I| w(y)pτdy ≤C k≥0 1 (2k|I|)(2−α)pτ−11 2k|I|2k−1|I|≤y−a≤2k|I| w(y)pdyτ . (4.4) Taking into account that τ>1, (4.4) is bounded by C k≥0 2k|I|1 2k|I|2k−1|I|≤y−a≤2k|I| w(y)p (y−a)(2−α)pdyτ ≤C|I|1−τ|I| 2≤y−a w(y)p (y−a)(2−α)pdyτ . Weyl Fractional Integral 93 Keeping in mind that w∈H−(α, p)wehave, ∞ b w(y)pτ (y−a)(2−α)pτdy ≤C|I|1−τw([a, a +|I|/2]) |I||I|−1/p+α−1pτ =Cw(I) |I| 1 |I|1 (pτ)−α+1 pτ , which implies that w∈H−(α, (pτ)). Lemma 4.6. Let 1<p 1<p 2<∞. Suppose that w∈H−(α, pi)for i=1,2. Then w∈H−(α, p)for every p:p1<p<p 2. Proof: This is an one-sided version of Lemma 3.15 in [1]. Lemma 4.7. Let 1<p<∞and w∈RH−(p). There exists a constant Csuch that for every f∈ Lp wand every bounded interval I=[a, b], if we denote  I−=[a−|I| 2,a]then, I |f(x)|dx ≤Cw( I−) |I|1/p [f]p,w. Proof: Since w∈RH−(p)byTheorem 5.3 in [9], there exists s>p  such that w∈RH−(s), that is, there exists a constant Csuch that for every bounded interval I, 1 |I|I w(x)sdx1/s ≤Cw( I−) |I|. From this fact, the proof follows as in Lemma 4.1 of [1]. Lemma 4.8. Let 1<p<∞and w∈H−(α, p). Then there exists a constant Csuch that for every f∈ Lp wand every bounded interval I= [a, b],∞ b |f(y)| (y−a)2−αdy ≤Cw(I) |I|2+ 1 p−α[f]p,w. Proof: Taking into account Lemma 4.4 and Lemma 4.5, the proof of this lemma is similar to one in Lemma 4.4 of [1]. 94 S. Ombrosi, L. de Rosa Lemma 4.9. Let α>0and δ≥0such that 0<α+δ<1.Letw∈D−. For a<b,wedenote c=a+b 2and I=[c, b]. Then, for every f∈L w(δ), there exists a constant Csuch that, ∞ b |f(y)−fI| (y−a)2−αdy ≤CfLw(δ)∞ c w(y) (y−a)2−α−δdy.(i) b a |f(y)−fI| (y−a)1−αdy ≤CfLw(δ)c a w(y) (y−a)1−α−δdy.(ii) Proof: The proof of (i) and (ii) are similar, then we only prove (i). For every j≥0, let Ij=[a+2 j|I|,a+2 j+1|I|]. We observe that I0=[a+|I|,a+2|I|]=[c, b]=I. Since f∈L w(δ)wehave that, ∞ b |f(y)−fI| (y−a)2−αdy= ∞  j=1 a+2j+1|I| a+2j|I| |f(y)−fI| (y−a)2−αdy ≤ ∞  j=1 1 (2j|I|)2−αa+2j+1|I| a+2j|I| |f(y)−fI0|dy ≤ ∞  j=1 1 (2j|I|)2−αa+2j+1|I| a+2j|I| |f(y)−fIj|dy +2j|I| j  k=1 |fIk−fIk−1| ≤ ∞  j=1 1 (2j|I|)1−αCfLw(δ)w(Ij)(2j|I|)δ−1 + j  k=1 1 |Ik−1|Ik−1 |f(y)−fIk|dy. (4.5) Using that f∈L w(δ) and w∈D−we obtain the estimate, 1 |Ik−1|Ik−1 |f(y)−fIk|dy ≤CfLw(δ)w(Ik−1)(2k−1|I|)δ−1. Weyl Fractional Integral 95 Then applying Fubini’s Theorem, (4.5) is bounded by CfLw(δ) ∞  j=1 1 (2j|I|)1−α j  k=0 w(Ik)(2k|I|)δ−1 =CfLw(δ) ∞  k=0 w(Ik)(2k|I|)δ−1 ∞  j=k 1 (2j|I|)1−α =CfLw(δ) ∞  k=0 1 (2k|I|)2−α−δa+2k+1|I| a+2k|I| w(y)dy ≤CfLw(δ)∞ c w(y) (y−a)2−α−δdy, as we wanted to prove. 5. Proof of Theorems 2.3 and 2.4 Proof of Theorem 2.3: (i) ⇒(ii). Let w∈H−(α, p) and x0∈R. Given f∈ Lp wlet  I+ α(f) define as in (2.2). Choose a bounded interval I= [a, a +h]. We consider I0=[a+2h, x0]ifa+2h≤x0and I0=∅if x0<a+2h, and we also define I1=[x0,a+2h]ifx0<a+2hand I1=∅in the other case. We set aI=I0 f(y) (y−a)1−αdy +∞ x01−χI1(y) (y−a)1−α−1−χ[x0,x0+1](y) (y−x0)1−αf(y)dy. We shall show that aIis a finite constant. Suppose that x0<a+2h. Let nbe apositive integer such that a+2 nh>x 0+1and |a−x0|≤2n−1h. Then, aI=a+2nh x0 +∞ a+2nh1−χ[x0,a+2h](y) (y−a)1−α−1−χ[x0,x0+1](y) (y−x0)1−αf(y)dy =J1+J2. For each y≥a+2nh,byMean Value Theorem, there exists θ:0<θ<1 such that,  1 (y−a)1−α−1 (y−x0)1−α≤C|x0−a| |y−θa −(1 −θ)x0|2−α≤C|x0−a| |y−a|2−α. 102 S. Ombrosi, L. de Rosa [6] B. Muckenhoupt and R. L. Wheeden,Weighted bounded mean oscillation and the Hilbert transform, Studia Math. 54(3) (1975/76), 221–237. [7] B. Muckenhoupt and R. L. Wheeden,Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. [8] J. Peetre,Onthe theory of Lp,λ spaces, J. Functional Analysis 4 (1969), 71–87. [9] M. S. Riveros and A. de la Torre,Onthe best ranges for A+ p and RH+ r,Czechoslovak Math. J. 51(126), no. 2 (2001), 285–301. [10] L. de Rosa and C. Segovia,Weighted Hpspaces for one sided maximal functions, in: “Harmonic analysis and operator theory” (Caracas, 1994), Contemp. Math. 189, Amer. Math. Soc., Providence, RI, 1995, pp. 161–183. [11] L. de Rosa and C. Segovia, Dual spaces for one-sided weighted Hardy spaces, Rev. Un. Mat. Argentina 40(3–4) (1997), 49–71. [12] E. Sawyer,Weighted inequalities for the one-sided HardyLittlewood maximal functions, Trans. Amer. Math. Soc. 297(1) (1986), 53–61. Departamento de Matem´atica Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad Universitaria, Pabell´on I 1428 Ciudad de Buenos Aires Argentina E-mail address:[email protected] E-mail address:[email protected] Primera versi´o rebuda el 12 de desembre de 2001, darrera versi´o rebuda el 8 de juliol de 2002.