Two weighted inequalities for convolution maximal operators

Publ. Mat. 46 (2002), 119–138 TWO WEIGHTED INEQUALITIES FOR CONVOLUTION MAXIMAL OPERATORS A. L. Bernardis∗and F. J. Mart´ ın-Reyes† Abstract Let ϕ:R→[0,∞) an integrable function such that ϕχ(−∞,0) =0 and ϕis decreasing in (0,∞). Let τhf(x)=f(x−h), with h∈R\{0}and fR(x)= 1 Rf(x R), with R>0. In this paper we characterize the pair of weights (u, v) such that the operators Mτhϕf(x) = supR>0|f|∗[τhϕ]R(x) are of weak type (p, p) with respect to (u, v), 1 <p<∞. 1. Introduction Let us consider the dilates ϕR(x)= 1 Rϕ(x R), R>0, of a nonnegative integrable function ϕdefined on the real line. It is well known that the study of the a.e. convergence of the convolutions f∗ϕRas R→0is related to the behavior of the maximal operator Mϕf(x) = sup R>0 |f|∗ϕR(x). If ϕbelongs to the set Fof the even functions ϕ:R→[0,∞), decreasing in [0,∞) with 0 <Rϕ=A<∞, then a classical result establishes that Mϕsatisfies the weighted weak type inequality {Mϕf>λ} u≤C λpR |f|pv,(1.1) 2000 Mathematics Subject Classification. 42B25. Key words. Weigthed inequalities, convolution maximal operators. ∗Supported by CONICET, PICT 98 (C´odigo 03-04186) and Prog. CAI+D - UNL. †Partially supported by D.G.E.S. grant (PB97-1097), Junta de Andaluc´ıa and UNL. 120 A. L. Bernardis, F. J. Mart´ ın-Reyes 1≤p<∞, if and only if (u, v) belongs to the Apclass of Muckenhoupt, i.e., if there exists C>0 such that for all a<b b a u1/p b a v1−p1/p ≤C(b−a),if 1 <p<∞,1 p+1 p=1, and Mu ≤Cv a.e.,if p=1, where Mf(x) = suph>01 2hx+h x−h|f(t)|dt is the (two -sided) Hardy-Littlewood maximal function. The above result is a consequence of the characterization for M[5] and the following pointwise inequalities: 2ϕ()Mf(x)≤Mϕf(x)≤AMf(x),(1.2) where is a positive real number such that ϕ()>0 (the existence of is guaranteed since we are assuming that ϕ≡ 0), together with the characterization of the weighted weak type inequalities for M(see [5] and [7]). The right inequality in (1.2) is a classical result (see [7]), the left one is an easy consequence of the inequalities 1 RR |f(y)|ϕx−y Rdy ≥1 Rx+R x−R ... dy ≥2ϕ()1 2R x+R x−R |f(y)|dy. Sharper estimates can be obtained if the function ϕis a member of the following set of functions: F+={ϕ:R→[0,∞):ϕχ(−∞,0) =0,ϕ decreasing in (0,∞) with 0 <Rϕ=A<∞} or F−={ϕ:ϕ(−x)∈ F+}. In fact, for almost all x∈Rwe have that, if ϕ∈F +and >0is such that ϕ()>0, then ϕ()M−f(x)≤Mϕf(x)≤AM−f(x),(1.3) and if ϕ∈F −and >0 is such that ϕ(−)>0, then ϕ(−)M+f(x)≤Mϕf(x)≤AM+f(x),(1.4) where M−f(x) = sup h>0 1 hx x−h |f(t)|dt and M+f(x) = sup h>0 1 hx+h x |f(t)|dt are the one-sided Hardy-Littlewood maximal functions. The right inequalities were proved by M. Lorente [2], the left ones can be obtained as in (1.2). By (1.3) and the characterization of the weighted weak type inequalities for M−(see [6] and [3]) we get that, if ϕ∈F +and Weighted Inequalities for Maximal Operators 121 1≤p<∞, then (1.1) holds if and only if (u, v) belongs to the Sawyer’s class A− p, i.e., if there exists C>0 such that for all a<b<c c b u1/p b a v1−p1/p ≤C(c−a),if 1 <p<∞ and M+u≤Cv a.e.,if p=1. An analogous result holds with ϕ∈F −and (u, v)∈A+ pwhich is the same as A− pbut reversing the orientation of the real line. In this paper we are interested in the behavior of the convolution maximal operator associated to a translation of a function ϕ∈F(F+ or F−), i.e., if τhϕ(x)=ϕ(x−h) we wish to characterize (1.1) for the maximal operator Mτhϕf(x) = sup R>0 |f|∗[τhϕ]R(x). Clearly it is enough to work with functions ϕ∈F +since the results for ϕ∈F −are obtained similarly and the results for ϕ∈Ffollow from the corresponding ones for F+and F−. Examples of these operators are M− αf(x) = sup R>0 1 Rx−R x−2R |f(y)|x−R−y Rα dy, −1<α<0 and  M+ αf(x) = sup R>0 1 Rx+R x |f(y)|x+R−y Rα dy, −1<α<0. These operators were studied in [1] and [4] and are equal to Mτhϕwhere ϕ(t)=tαχ(0,1](t) with h= 1 and h=−1 respectively. Observe that in the above examples ϕ(0+) = limt→0+ϕ(t)=+∞.If ϕ∈F +and ϕ(0+) <+∞, the weighted weak type inequalities (1.1) are equivalent to conditions A− p,A+ por Apas it is shown in the following theorem which we shall prove in Section 2. Theorem 1.5. Let 1≤p<∞,ϕ∈F +and ϕ(0+) <+∞. Then (i) If h>0,(1.1) holds for Mτhϕif and only if (u, v)∈A− p. (ii) If h<0and supp(ϕ)⊂(0,|h|],(1.1) holds for Mτhϕif and only if (u, v)∈A+ p. (iii) If h<0and supp(ϕ)∩(|h|,∞)=∅,(1.1) holds for Mτhϕif and only if (u, v)∈Ap. 122 A. L. Bernardis, F. J. Mart´ ın-Reyes When ϕ(0+) = +∞, the situation is different. For example, the weighted weak type inequalities (1.1) for Mα( Mα) are equivalent to conditions which are strictly contained in A− p(A+ p). Therefore, there are weights in A− p(A+ p) which are not good weights for Mα( Mα). We shall dedicate Sections 3 and 4 to characterize the good weights for Mτhϕassuming only some restriction on the decreasingness of ϕ. More precisely we shall work in the rest of the paper with functions ϕ∈E + γ,δ, with γ>0, δ∈(0,1) and E+ γ,δ ={ϕ∈F +:ϕ(γ)>0 and tδϕ(t) is increasing in (0,γ]}. Observe that ϕ∈E + γ,δ implies that tϕ(t) is increasing in (0,γ]. Also notice that the functions ϕ(t)=tαχ(0,1](t), corresponding to the operators Mαand  Mα, belongs to E+ 1,−α. Other examples belonging to E+ γ,δ for some γand some δare the following: ϕ(t)=tαlog 1 tχ(0,1](t) with −1<α≤0 and ϕ(t) = (1 + log 1 t)χ(0,1](t)+tβχ(1,∞)(t), with β<−1. We shall prove the following characterizations of the weighted weak type (p, p) inequalities, 1 <p<∞, for Mτhϕ, under the assumption ϕ∈ E+ γ,δ. Notice that we always may assume that 0 <γ≤|h|. Theorem 1.6. Let 1<p<∞,h>0,0<γ≤h,δ∈(0,1) and ϕ∈E + γ,δ. The following statements are equivalent. (i) (1.1) holds for Mτhϕ. (ii) (u, v)∈A− p,ϕ,γ, i.e., there exists C>0such that c b u1/p b a v1−p(y)ϕpb−y c−aγdy1/p ≤Cc−a γ, for all a<b<c. Theorem 1.7. Let 1<p<∞,h<0,0<γ≤|h|,δ∈(0,1),ϕ∈ E+ γ,δ and assume that supp(ϕ)⊂(0,|h|]. The following statements are equivalent. (i) (1.1) holds for Mτhϕ. (ii) (u, v)∈ A+ p,ϕ,γ, i.e., there exists C>0such that b a u1/p c b v1−p(y)ϕpc−y c−aγdy1/p ≤Cc−a γ, for all a<b<c. Weighted Inequalities for Maximal Operators 123 Theorem 1.8. Let 1<p<∞,h<0,0<γ≤|h|,δ∈(0,1),ϕ∈E + γ,δ and assume that supp(ϕ)∩(|h|,∞)=∅. The following statements are equivalent. (i) (1.1) holds for Mτhϕ. (ii) (u, v)∈ A+ p,ϕ,γ ∩Ap. Taking into account the results for Mαand  Mαwe see that the class of good weights for Mτhϕwill depend on the behavior of ϕclose to zero. This is our starting point to analize the operator Mτhϕ. In fact, given ϕ∈F +,h∈R,h= 0 and γ>0 small enough, let us say γ≤|h|,we write ϕ=ϕχ(0,γ]+ϕχ(γ,∞). Then if we denote Mϕ,h,γ := Mτh(ϕχ(0,γ])and Mϕ,h,∞:= Mτh(ϕχ(γ,∞)) we get the following pointwise inequalities: max {Mϕ,h,γ,M ϕ,h,∞}≤Mτhϕ≤Mϕ,h,γ +Mϕ,h,∞.(1.9) Therefore, Mτhϕsatisfies (1.1) if and only if (1.1) holds for Mϕ,h,γ and Mϕ,h,∞. The study of Mϕ,h,∞is completely similar to the study of Mτhϕwith ϕ(0+) <∞. The difficult part is concentrated in the local operator Mϕ,h,γ. The operators Mϕ,h,γ have the following explicit expressions: Mϕ,h,γf(x) = sup R>0 1 Rx−|h|R x−(|h|+γ)R |f(y)|ϕx−|h|R−y Rdy if h>0 and Mϕ,h,γf(x) = sup R>0 1 Rx+|h|R x+(|h|−γ)R |f(y)|ϕx+|h|R−y Rdy if h<0. We may observe that the operators Mϕ,h,γ are of different geometric nature depending on the sign of h.Ifh>0, the integrals are taken over intervals I⊂(−∞,x) and ϕis evaluated in a point which depends on the distance of yto the end point of Inearer to x, while if h<0 the integrals are computed over intervals I⊂(x, ∞) and ϕis evaluated in a point which depends on the distance of yto the end point of Ifarer from x. The paper is organized as follows: Section 2 and 3 are devoted to the proof of Theorems 1.5 and 1.6 respectively, while we give the proofs of Theorems 1.7 and 1.8 in Section 4. Throughout the paper h,γand δare real numbers, h=0,γ>0 with γ≤|h|,0<δ<1 and the classes E+ γ,δ are the ones defined above. The 124 A. L. Bernardis, F. J. Mart´ ın-Reyes functions uand vwill be weights, i.e., positive measurable functions. Finally, pstands for the conjugate exponent of p,1<p<∞, and the letter Cmeans a positive constant that may change from one line to another. 2. Proof of Theorem 1.5 Let ϕ∈F +and ϕ(0+) <+∞. Without loss generality we can assume that ϕ(0) = ϕ(0+). The proof of Theorem 1.5 is based on the following lemma. Lemma 2.1. Let >0be such that ϕ()>0. There exist positive constants C1and C2such that (i) If h>0, C1ϕ()hM−f(x)≤Mτhϕf(x)≤ϕ(0)h+∞ 0 ϕM−f(x). (ii) If h<0and supp(ϕ)⊂(0,|h|], C2ϕ()|h|M+f(x)≤Mτhϕf(x)≤ϕ(0)|h|M+f(x). (iii) If h<0,supp(ϕ)∩(|h|,∞)=∅and >|h|, 2ϕ() min{|h|,+h}Mf(x)≤Mτhϕf(x)≤2ϕ(0)|h|+∞ |h| ϕMf(x). Before proving the above lemma we define the following maximal operators: N− µf(x) = sup T>0 1 Tx−T x−µT |f(y)|dy for µ>1 and N+ ηf(x) = sup T>0 1 Tx+T x+ηT |f(y)|dy for 0 <η<1. The above operators are pointwise equivalent to M−and M+respectively. In fact, we have the following proposition. Proposition 2.2. There exist positive constants C1and C2such that (i) C1M−f(x)≤N− µf(x)≤µM−f(x)and (ii) C2M+f(x)≤N+ ηf(x)≤M+f(x). Weighted Inequalities for Maximal Operators 125 Proof: The right inequalities in (i) and (ii) are obvious. In order to prove the left inequality in (i), we may assume that M−f(x)<∞. Let sbe such that 1/µ<s<1. Then, there exists T>0 such that sM−f(x)≤1 µT x x−µT |f(y)|dy =1 µT x−T x−µT |f(y)|dy +1 µT x x−T |f(y)|dy ≤1 µN− µf(x)+ 1 µM−f(x). Then, since s>1/µ we obtain (i) with C1=µs −1. The left inequality in (ii) is proved similarly. In fact, assume that M+f(x)<∞and let s be such that η<s<1. Then, there exists T>0 such that sM+f(x)≤1 Tx+T x |f(y)|dy =1 Tx+ηT x |f(y)|dy +1 Tx+T x+ηT |f(y)|dy ≤ηM+f(x)+N+ ηf(x). Then, since η<swe obtain (ii) with C2=s−η. Proof of Lemma 2.1: (i) First, notice that τh(ϕ) is dominated by ϕ(0)χ(0,h]+τh(ϕ)∈F +. Therefore, by (1.3) we get the right inequality of (i). On the other hand, we fix µ=h+ h>1 and since ϕis decreasing we have that 1 RR |f(y)|ϕx−y−hR Rdy ≥1 Rx−hR x−(+h)R ... dy ≥hϕ()1 hR x−hR x−µhR |f(y)|dy. 126 A. L. Bernardis, F. J. Mart´ ın-Reyes Taking supremum over R>0 we have that Mτhϕf(x)≥hϕ()N− µf(x) and using Proposition 2.2(i) we obtain statement (i). (ii) By the hypothesis on hand on the support of ϕwe can easily see that τh(ϕ) is dominated by ϕ(0)χ[h,0] ∈F −and by (1.4) we get that Mτhϕf(x)≤ϕ(0)|h|M+f(x). Let us fix η=|h|− |h|. Then (ii) follows by the inequalities 1 RR |f(y)|ϕx−y−hR Rdy ≥1 Rx+|h|R x+(|h|−)R ... dy ≥|h|ϕ()1 |h|Rx+|h|R x+η|h|R |f(y)|dy, taking supremum over R>0 and applying Proposition 2.2(ii). (iii) The function τh(ϕ) is dominated by a sum of two functions: φ1= ϕ(0)χ[h,0] ∈F −and φ2=τh(ϕ)χ(0,∞)∈F +. Therefore, using (1.3) and (1.4) we get that Mτhϕf(x)≤ϕ(0)|h|M+f(x)+∞ |h|ϕM−f(x)≤ 2ϕ(0)|h|+∞ |h|ϕMf(x). On the other hand, if ν= min{|h|,+h}, then 1 RR |f(y)|ϕx−y−hR Rdy ≥1 Rx+|h|R x−(+h)R ... dy ≥2νϕ()1 2νR x+νR x−νR |f(y)|dy. Therefore, taking supremum over R>0 we complete the proof of the lemma. Now, Theorem 1.5 follows from Lemma 2.1 together with the characterizations of the weighted weak type (p, p) inequalities for M−,M+ and M. 3. Proof of Theorem 1.6 We shall start studying the local part Mϕ,h,γ . More precisely, we shall prove the following theorem. Weighted Inequalities for Maximal Operators 127 Theorem 3.1. Let 1<p<∞,h>0,0<γ≤h,δ∈(0,1) and ϕ∈E + γ,δ. The following statements are equivalent. (i) (1.1) holds for Mϕ,h,γ. (ii) (u, v)∈A− p,ϕ,γ. First, we notice that if ϕ,hand γare as in Theorem 3.1 and β= h+γ h>1 then we have Mϕ,h,γf(x) = sup R>0 1 Rx−hR x−βhR |f(y)|ϕx−hR −y Rdy. In order to prove Theorem 3.1, we define the following noncentered version of this operator Nϕ,h,γf(x) = sup (a,b)∈Ax γ b−ab a |f(y)|ϕb−y b−aγdy, where Ax={(a, b):b<xand b−a≥γ h(x−b)}. The operators Mϕ,h,γ and Nϕ,h,γ are pointwise equivalent for ϕ∈E + γ,δ. Proposition 3.2. If h>0,0<γ≤h,β=h+γ h,δ∈(0,1) and ϕ∈E + γ,δ, then Mϕ,h,γf(x)≤Nϕ,h,γf(x)≤β γϕ(γ)γ 0 ϕ(y)dy +2 Mϕ,h,γf(x). Proof: The first inequality is obvious. To prove the second one let us consider x∈Rand (a, b)∈A x. Let Rbe the positive number such that a=x−βhR. Observe that x−b≤hR. Let mbe the nonnegative integer number such that x−hR βm≤b<x−hR βm+1 . Then b a |f(y)|ϕb−y b−aγdy =m−1  k=0 x−hR βk x−hR βk−1 +x−hR βm x−hR βm−1 +b x−hR βm(... dy) =I+II +III, 134 A. L. Bernardis, F. J. Mart´ ın-Reyes Now, by the definition of φwe obtain 1 Rx+|h|R x |f(y)|φx+|h|R−y Rdy =ϕ(γ) Rx+(|h|−γ)R x |f(y)|dy +1 Rx+|h|R x+(|h|−γ)R |f(y)|ϕx+|h|R−y Rdy. Taking supremum over R>0 and using (4.4) we have Mφ,h,|h|f(x)≤ϕ(γ)(|h|−γ)M+f(x)+Mϕ,h,γf(x) ≤ϕ(γ)|h|M+f(x)+Mϕ,h,γf(x) ≤CM ϕ,h,γf(x), as we wished to prove. Once Lemma 4.3 has been proved we are able to show that Theorem 4.1 for |h|>γfollows from Theorem 4.1 from h=−γ. In fact, let us assume that Theorem 4.1 is proved for h=−γ. By Lemma 4.3, we can easily see that (i) is equivalent to (u, v)∈ A+ p,φ,|h|, i.e., there exists C>0 such that b a u1/p c b v1−p(y)φpc−y c−a|h|dy1/p ≤Cc−a |h|, for all a<b<c. It only remains to prove that  A+ p,φ,|h|and  A+ p,ϕ,γ are equivalent. The implication (u, v)∈ A+ p,φ,|h|⇒(u, v)∈ A+ p,ϕ,γ is a consequence of the increasingness of tφ(t)in(0,|h|] while the converse follows from the fact that φis decreasing. Proof of Theorem 4.1 for h=−γ:Notice that in this case Mϕ,h,γf(x) = sup R>0 1 Rx+|h|R x |f(y)|ϕx+|h|R−y Rdy = sup c>x γ c−xc x |f(y)|ϕc−y c−xγdy. (i) ⇒(ii). Let a<b<c. Let vnand ϕnbe as in the proof of Theorem 3.1 and let us consider f(y)=v1−p n(y)ϕp−1 nc−y c−aγχ(b,c)(y). Weighted Inequalities for Maximal Operators 135 Using that tϕ(t) is increasing in (0,γ] and ϕ≥ϕn, we have for all x∈(a, b), Mϕ,h,γf(x)≥γ c−xc b v1−p n(y)ϕp−1 nc−y c−aγϕc−y c−xγdy ≥γ c−ac b v1−p n(y)ϕp nc−y c−aγdy ≡λ. Then (ii) follows applying (i) and letting ntend to ∞. The implication (ii) ⇒(i) follows, as in the proof of Theorem 3.1, from the following proposition. Proposition 4.5. Let 1<p<∞,γ>0,δ∈(0,1),ϕ∈E + γ,δ,h= −γand (u, v)∈ A+ p,ϕ,γ. Then, there exists C>0such that for every measurable function f Mϕ,h,γf(x)≤CM+ u|f|pvu−1(x)1/p . Proof: Let x∈R. Let {xi}be the decreasing sequence in [x, c] defined by x0=cand xi+1 x u=xi xi+1 u=1 2xi x u. Then, c x |f(y)|ϕc−y c−xγdy = ∞  i=0 xi xi+1 |f(y)|ϕc−y c−xγdy. The rest of the proof follows in a similar way as in the proof of Proposition 3.3. In fact, by taking qi=Uxi−c U−1with U=c−x xi−xi+1 >1wecan prove that c−y c−x≥xi−y xi−xi+1 if and only if y∈[qi,x i]. Then xi xi+1 |f(y)|ϕc−y c−xγdy =qi xi+1 ... dy+xi qi ... dy=I+II. 136 A. L. Bernardis, F. J. Mart´ ın-Reyes Since ϕis decreasing, the H¨older inequality, the hypothesis (u, v)∈  A+ p,ϕ,γ and the definition of the sequence {xi}give II ≤xi qi |f(y)|ϕxi−y xi−xi+2 γdy ≤xi xi+1 ... dy ≤xi xi+1 |f|pv1/p xi xi+1 v1−p(y)ϕpxi−y xi−xi+2 γdy1/p ≤Cxi x |f|pv1/p xi+1 xi+2 u−1/p xi−xi+2 γ ≤Cxi−xi+2 γM+ u|f|pvu−1(x)1/p . To estimate Iwe shall use that c−y c−x<xi−y xi−xi+1 if and only if y<q iand the fact that tδϕ(t) is increasing in (0,γ]. Then, I=qi xi+1 |f(y)|ϕc−y c−xγdy ≤qi xi+1 |f(y)|ϕxi−y xi−xi+2 γg(y)dy, where g(y)=c−y c−x−δxi−y xi−xi+2 δ. Since gis decreasing in (xi+2,q i), we have I≤c−xi+2 c−x−δxi xi+1 |f(y)|ϕxi−y xi−xi+2 γdy. With the same argument as in the boundedness of IV in the proof of Theorem 3.1, using that (c−y)−δis increasing, we get that I≤Cc−xi+2 c−x−δxi−xi+2 γM+ u|f|pvu−1(x)1/p ≤C γxi xi+2 c−y c−x−δ dyM+ u|f|pvu−1(x)1/p . Now, adding up in i, we obtain I+II ≤Cc−x γ2−δ 1−δM+ u|f|pvu−1(x)1/p, and we are done. Weighted Inequalities for Maximal Operators 137 As in the case h>0, we obtain the characterizations for Mϕ,h,∞from Theorem 1.5. Theorem 4.6. Let 1≤p<∞,ϕ∈F +,h<0and 0<γ≤|h|. Then (i) If supp(ϕ)⊂(0,|h|]and γ=|h|, then Mϕ,h,∞≡0. (ii) If supp(ϕ)∩(|h|,∞)=∅and γ=|h|,(1.1) holds for Mϕ,h,∞if and only if (u, v)∈A− p. (iii) If supp(ϕ)⊂(0,|h|]and γ<|h|,(1.1) holds for Mϕ,h,∞if and only if (u, v)∈A+ p. (iv) If supp(ϕ)∩(|h|,∞)=∅and γ<|h|,(1.1) holds for Mϕ,h,∞if and only if (u, v)∈Ap. Proof: (i) is obvious. As in the proof of Theorem 3.4, taking ψ= τ−γ(ϕχ(γ,∞))∈F +the operator Mϕ,h,∞is equal to Mτh+γψ. In the case (ii), Mτh+γψ=Mψand therefore, (ii) follows from one of the results cited in the introduction. In the cases (iii) and (iv) we have that h+γ<0 and applying Theorem 1.5(ii) and (iii) we are done. Now we shall prove Theorems 1.7 and 1.8. Proof of Theorem 1.7: The proof follows as the proof of Theorem 1.6 using Theorem 4.1, Theorem 4.6(i) and (iii), the inequalities (1.9) and the fact that  A+ p,ϕ,γ ⊂A+ pwhich is a consequence of the decreasingness of ϕ. Proof of Theorem 1.8: It follows from Theorem 4.1, Theorem 4.6(ii) and (iv) and inequalities (1.9). Remark 4.7.We have not studied in this paper the case p= 1. The study of the weighted weak type inequality (1,1) for Mτhϕwill appear in a forthcoming paper on weighted restricted weak type inequalities for this operator and 1 ≤p<∞(notice that the restricted weak type (1,1) inequality for Mτhϕis equivalent to the weak type (1,1) inequality [8]). References [1] A. L. Bernardis and F. J. Mart´ ın-Reyes, Two weighted inequalities for maximal functions related to Ces`aro convergence, J. Austral. Math. Soc. Ser. A (to appear). [2] M. Lorente, The convergence in L1of singular integrals in Harmonic Analysis and Ergodic Theory, J. Fourier Anal. Appl. 5(6) (1999), 617–638. 138 A. L. Bernardis, F. J. Mart´ ın-Reyes [3] F. J. Mart´ ın-Reyes, New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Proc. Amer. Math. Soc. 117(3) (1993), 691–698. [4] F. J. Mart´ ın-Reyes and A. de la Torre, Some weighted inequalities for general one-sided maximal operators, Studia Math. 122(1) (1997), 1–14. [5] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. [6] E. Sawyer, Weighted inequalities for the one-sided HardyLittlewood maximal functions, Trans. Amer. Math. Soc. 297(1) (1986), 53–61. [7] E. M. Stein,“Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals”, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, N.J., 1993. [8] E. M. Stein and G. Weiss,“Introduction to Fourier analysis on Euclidean spaces”, Princeton Mathematical Series 32, Princeton University Press, Princeton, N.J., 1971. A. L. Bernardis: IMAL - CONICET G¨uemes 3450 (3000) Santa Fe Argentina E-mail address:[email protected] F. J. Mart´ın-Reyes: Departamento de An´alisis Matem´atico Facultad de Ciencias Universidad de M´alaga 29071 M´alaga Spain E-mail address:[email protected] Primera versi´o rebuda el 8 de mar¸c de 2001, darrera versi´o rebuda el 13 de novembre de 2001.