RESEARC H Open Access Boundedness of positive operators on weighted amalgams María Isabel Aguilar Cañestro and Pedro Ortega Salvador * * Correspondence: portega@uma. es Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain Abstract In this article, we characterize the pa irs (u, v) of positive measurable functions such that T maps the weighted amalgam ( L ¯ p ( v ) , ¯ q ) in (L p (u), ℓ q ) for all 1 < p , q , ¯ p , ¯ q < ∞ , where T belongs to a class of positive operators which includes Hardy operators, maximal operators, and fractional integrals. 2000 Mathematics Subject Classification 26D10, 26D15 (42B35) Keywords: Amalgams, Maximal operators, Weighted inequalities, Weights 1. Introduction Let u be a positive funct ion of one real variable and let p, q > 1. The amalgam (L p (u), ℓ q ) is the space of one variable real functions which are loc ally in L p (u) and globally in ℓ q . More precisely, (L p (u), q )={f : ||f || p ,u, q < ∞} , where ||f || p,u,q = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+1 n |f | p u ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q . These spaces were introduced by Wiener in [1]. The article [2] describes the role played by amalgams in Harmonic Analysis. Carton-Lebrun, Heinig, a nd Hoffmann studied in [3] the boundedness of the Hardy operator Pf (x)= x − ∞ |f | in weighted amalgam spaces. They characterized the pairs of weights (u, v) such that the inequality ||Pf || p,u,q ≤ C||f || ¯ p ,v, ¯ q (1:1) holds for al l f, with a constant C independent of f,whenever 1 < ¯ q ≤ q < ∞ .The characterization of the pairs (u, v) for (1.1) to hold in the case 1 < q < ¯ q < ∞ has been recently completed by Ortega and Ramírez ([4]), who have al so characterized the weak type inequality Pf p ,∞;u, q ≤ C f ¯ p ,v, ¯ q , Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 © 2011 Cañestro and Salvador; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://crea tivecommons.org/licens es/by/2.0), which permits unrestricted use, distribution, and re prod uction in any medium, provide d the origin al work is properly cited. where | |g|| p,∞;u,q = n∈Z ||gX (n,n+1) || q p,∞,u 1 q . There are s everal articles dealing wit h the boundedness in weighted amalgams of other operators different from Hardy’s one. Specifically, Carton-Lebrun, Heinig, and Hoffmann studied in [3] weighted inequalities in amalgams for the H ardy-Littlewood maxi mal operator as well as for some integral operators with kernel K(x, y) i ncreasing in the second variable and decreasing in the first one. On the other hand, Rakotondrat- simba ([5]) characterized some weighted inequalities in amalgams (corresponding to the cases 1 < ¯ p ≤ p < ∞ and 1 < ¯ q ≤ q < ∞ ) for the fractional maximal operators and the fractional i ntegrals. Finally, the authors characterized in [6] the weighted inequal- ities for some generalized Hardy operators, including the fractional integrals of order greater than one, in all cases 1 < p , ¯ p , q , ¯ q < ∞ , extending also results due to Heinig and Kufner [7]. Analyzing the results in the articles cited above, one can see some common features that lead to explo re the possibility of giving a general theorem characterizing the boundedness in weighted amalgams of a wide family of positive operators , and provid- ing, in such a way, a unified approach to the subject. This is the purpose of this article. 2. The results We consider an operator T acting on real measurable functions f of one real variable and define a sequence {T n } nÎℤ of local operators by T n f (x)=T(f X ( n−1,n+2 ) )(x) x ∈ (n − 1, n +2) . We assume that there exists a discrete operat or T d , i.e., which transforms sequences of real numbers in sequences of real numbers, verifying the following conditions: (i) There exists C > 0 such that for all non-negative functions f,alln Î ℤ and all x Î (n, n + 1), the inequality T f X (−∞,n−1) + f X (n+2,∞) (x) ≤ CT d ⎛ ⎝ ⎧ ⎨ ⎩ m m−1 f ⎫ ⎬ ⎭ ⎞ ⎠ (n ) (2:1) holds. (ii) There exists C > 0 such that for all sequences {a k } of non-negative real numbers and n Î ℤ, the inequality T d ( {a k } )( n ) ≤ CTf ( y ), (2:2) holds for all y Î (n, n + 1) and all non-negative f such that m m −1 f = a m for all m. We also assume that T verifies Tf = T |f|, T(lf)=|l|Tf, T(f + g)(x) ≤ Tf (x)+Tg (x) and Tf(x) ≤ Tg(x)if0≤ f (x) ≤ g(x). Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 2 of 12 We will say that an operator T verifying all the above conditions is admissible. There is a number of important admissible operators in Analysis. For instance: Hardy operators, Hardy-Littlewood maximal operators, Riemann-Liouvill e, and Weyl fractional integral operators, maximal fractional operators, etc. Our main result is the following one: Theorem 1. Let 1 < p , q , ¯ p , ¯ q < ∞ . Let u and v be positive locally integrable functions on ℝ and let T be an admissible operator. Then there exists a constant C >0such that the inequality | |Tf || p,u,q ≤ C||f || ¯ p ,v, ¯ q . (2:3) holds for all measurable functions f if and only if the following conditions hold: (i) T d is bounded from ¯ q ( {v n } ) to ℓ q ({u n }), where v n = n n−1 v 1− ¯ p − ¯ q ¯ p and u n = n+1 n u q p . (ii) (a) sup n ∈ Z || T n || (L ¯ p (v),L p (u)) < ∞ in the case 1 < ¯ q ≤ q < ∞ . (b) {||T n || ( L ¯ p ( v ) ,L p ( u )) }∈ s , with 1 s = 1 q − 1 ¯ q , in the case 1 < q < ¯ q < ∞ . The proof of Theorem 1 is contained in Sect. 3. Working as in Theorem 1, we can also prove the following weak type result: Theorem 2. Let 1 < p , q , ¯ p , ¯ q < ∞ . Let u and v be positive locally integrable functions on ℝ and let T be an admissible operator. Then there exists a constant C >0such that the inequality | |Tf || p,∞,u,q ≤ C||f || ¯ p ,v, ¯ q (2:4) holds for all measurable functions f if and only if the following conditions hold: (i) T d is bounded from ¯ q ( {v n } ) to ℓ q ({u n }),), with v n and un defined as in Theorem 1. (ii) (a) sup n ∈ Z || T n || (L ¯ p (v),L p,∞ (u)) < ∞ in the case 1 < ¯ q ≤ q < ∞ . (b) {||T n || ( L ¯ p ( v ) ,L p,∞ ( u )) }∈ s , with 1 s = 1 q − 1 ¯ q , in the case 1 < q < ¯ q < ∞ . If conditions on the weights u, v, and {u n }, {v n } characterizing the boundedness of the operators T n and T d , respectively, are available in the literature, we immediately obtain, by applying Theorems 1 and 2, conditions guaranteeing the boundedness of T between the weighted amalgams. In this sense, our result includes, as particular cases, most of the results cited above from the papers [3-7], as well as other corresponding to opera- tors whose behavior on weighted amalgams has not been studied yet. Thus, if M - is the one-sided Hardy-Littlewood maximal operator defined by M − f (x)=sup h>0 1 h x x − h |f | , Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 3 of 12 we have: (i) The discrete operator ( M - ) d , defined by (M − ) d ({a n })(j)= sup k≤j−1 1 j − k j−1 i = k | a i | , verifies conditions (2.1) and (2.2). (ii) The local operators M − n are defined by M − n f (x)= sup 0<h ≤ x−n+1 1 h x x − h |f |, x ∈ (n − 1, n +2) . (iii) If p = ¯ p and q = ¯ q , there are well-known conditions on the weights u, v, and {u n }, {v n } that characterize the boundedness of M − n and (M - ) d (see, for instance [8-10]). Therefore, we obtain the following result: Theorem 3. The following statements are equivalent: (i) M - is bounded from (L p (w), ℓ q ) to (L p (w), ℓ q ). (ii) M - is bounded from (L p (w), ℓ q ) to (L p,∞ (w), ℓ q ). (iii) The next conditions hold simultaneously: (a) w ∈ A − p, ( n−1,n+2 ) for all n, uniformly, and (b) the pair ({u n }, {v n }) verifies the discrete Sawyer’s condition S − q , i.e., there exists C >0such that k j =r ((M − ) d ({v 1−q n })) q (j)u j ≤ C k j =r v 1−q j , for all r, k Î ℤ with r ≤ k. We can state a similar result for the one-sided maximal operator M + . In this case, the operator (M + ) d defined by (M + ) d ({a n })(j)= sup k≥ j +3 1 k − j − 2 k i= j +3 | a i | , verifies conditions (2.1) and (2.2). The theorem is the next one: Theorem 4. The following statements are equivalent: (i) M + is bounded from (L p (w), ℓ q ) to (L p (w), ℓ q ). (ii) M + is bounded from (L p (w), ℓ q ) to (L p,∞ (w), ℓ q ). (iii) The next conditions hold simultaneously: (a) w ∈ A + p, ( n−1,n+2 ) for all n, uniformly, and (b) the pair ({u n }, {v n-3 }) verifies the discrete Sawyer’ s condition S + q , i.e., there exists C >0such that k j =r ((M + ) d ({v 1−q n })) q (j)u j ≤ C k j =r v 1−q j , Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 4 of 12 for all r, k Î ℤ with r ≤ k. If M is the Hardy-Littlewood maximal operator, defined by Mf (x)=sup x∈I 1 |I| I |f | , then M is admissible, with M d ({a n })(j)= sup r≤ j ≤k 1 k − r +1 k i=r | a i | ,andtherearewell- known results, due to Muckenhoupt ([11]) and Sawyer ([12]), which characterize the boundedness of M in weighted Lebesgue spaces. Applying Theorems 1 and 2, we get the following result: Theorem 5. The following statements are equivalent: (i) M is bounded from (L p (w), ℓ q ) to (L p (w), ℓ q ). (ii) M is bounded from (L p (w), ℓ q ) to (L p,∞ (w), ℓ q ). (iii) The next conditions hold simultaneously: (a) w Î A p,(n-1,n+2) for all n, uniformly, and (b) the pair ({u n }, {v n }) verifies t he discrete two-sided Sawyer’sconditionS q , i.e., there exists C >0such that k j =r (M d ({v 1−q n }) q (j)u j ≤ C k j =r v 1−q j for all r, k Î ℤ with r ≤ k. This result improves the one obtained by Carton-Lebrun, Heinig an d Hofmann in [3], in the sense that the conditions we give are necessary and sufficient for the bound- edness of the maximal operator in the amalgam (L p (w), ℓ q ), while in [3] only sufficient conditons were given. We also prove the equivalence between the strong type inequal- ityandtheweaktypeinequality.Theequivalence(i)⇔ (iii) in Theorem 5 is included in Rakotondratsimba’s paper [5], where the proof of the admissibility of M can also be found. Finally, we will apply our results to the fractional maximal operator M a ,0<a <1, defined by M α f (x)= sup c<x<d 1 ( d − c ) 1−α d c | f | . The proof of the admissibility of M a , with the obvious M d α , is implied in Rakoton- dratsimba’s paper ([5]). Verb itsky ([13]) in the case 1 <q < p < ∞ and Sawyer ([12]) in the case 1 <p ≤ q < ∞ characterized the bo undednes s of M a from L p to L q (w). These results allow us to give necessary and sufficient conditions on the weight u for M a to be bounded from ( L ¯ p , ¯ q ) to ( L p ( u ) , q ) . Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 5 of 12 Before stating the theorem, we introduce the notation: (i) If 1 < ¯ q < ∞ , we define H : ℤ ® ℝ by H(i)= sup r≤i≤k 1 ( k − r +1 ) 1−α ¯ q k j=r u j . (ii) If 1 < ¯ q ≤ q , we define J =sup r≤k ||X [r,k] M d α (X [r,k] )|| q ({u j }) ( k − r +1 ) 1 ¯ q . (iii) If 1 < ¯ p < ∞ and n Î ℤ, we define for x Î (n-1, n +2) H n (x)= sup x∈I⊂ ( n−1,n+2 ) 1 |I| 1−α ¯ p I u . (iv) If 1 < ¯ p < p and n Î ℤ, we define J n =sup I⊂(n−1,n+2) ||X I M α (X I )|| L p (u) | I | 1 ¯ p . The result reads as follows. Theorem 6. M a is bounded from ( L ¯ p , ¯ q ) to (L p (u), ℓ q ) if and only if (i) in the case 1 < ¯ p ≤ p < ∞ and 1 < ¯ q ≤ q < ∞ , sup nÎℤ J n < ∞ and J < ∞; (ii) in the case 1 < p < ¯ p < ∞ and 1 < ¯ q ≤ q < ∞ , sup n∈Z || H n || L p ¯ p −p ( u ) < ∞ and J < ∞; (iii) in the case 1 < ¯ p ≤ p < ∞ and 1 < q < ¯ q < ∞ ,{J n } n Î ℓ s , where 1 s = 1 q − 1 ¯ q , and H ∈ q ¯ q−q ({u j } ) ; (iv) in the case 1 < p < ¯ p < ∞ and 1 < q < ¯ q < ∞ , | |H n || L p ¯ p −p ( u ) ∈ s and H ∈ q ¯ q−q ({u j } ) . 3. Proof of Theorem 1 Let us suppose that the inequality ( 2.3) holds. Let n Î ℤ and let f be a non-negative function supported in (n-1, n + 2). Then, on one hand, | |f || ¯ p,v, ¯ q = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝ n n−1 f ¯ p v ⎞ ⎠ ¯ q ¯ p + ⎛ ⎝ n+1 n f ¯ p v ⎞ ⎠ ¯ q ¯ p + ⎛ ⎝ n+2 n+1 f ¯ p v ⎞ ⎠ ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q ≤ C ¯ p, ¯ q ⎛ ⎝ n+2 n−1 f ¯ p v ⎞ ⎠ 1 ¯ p , Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 6 of 12 and, on the other hand, | |Tf || p,u,q ≥ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝ n n−1 (Tf ) p u ⎞ ⎠ q p + ⎛ ⎝ n+1 n (Tf ) p u ⎞ ⎠ q p + ⎛ ⎝ n+2 n+1 (Tf ) p u ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q ≥ C p,q ⎛ ⎝ n+2 n−1 (Tf ) p u ⎞ ⎠ 1 p ≥ C p,q ⎛ ⎝ n+2 n−1 (T n f ) p u ⎞ ⎠ 1 p = C p , q ||T n f || p ,u . Therefore, by (2.3), T n is bounded and ||T n || ( L ¯ p ( v ) ,L p ( u )) ≤ C , where C is a positive con- stant in dependent of n. Then (ii)a holds independently of the relationship between q and ¯ q . Let us prove that if 1 < q < ¯ q < ∞ , then (ii)b also holds. It is well known that ||T n || (L ¯ p (v),L p (u)) =sup {f :||f || L ¯ p ( v ) =1} ||T n f || L p (u ) .Therefore,foreachn there exists a non-negative measurable function f n , with support in (n-1, n +2)and with ||f n || ( L ¯ p ( v ) , ( n−1,n+2 )) = 1 , such that ||T n || (L ¯ p (v),L p (u)) < ||T n f n || L p (u) + 1 2 |n| . Since 1 2 |n| ∈ s ,toprovethat {||T n || ( L ¯ p ( v ) ,L p ( u )) }∈ s it suffices to see that {||T n f n || L p ( u ) }∈ s . Let {a n } be a sequence of non-negative real numbers and f = n a n f n . For each n Î ℤ, f(x) ≥ a n f n (x) and then Tf (x) ≥ a n T n f n (x) for all x Î (n-1, n + 2). Thus, | |Tf || p,u,q ≥ C ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n∈Z n+2 n−1 a p n (T n f n ) p u q p ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 1 q = C n∈Z a q n ||T n f n || q L p (u) 1 q . Then, from (2.3) we deduce n∈Z a q n ||T n f n || q L p (u) 1 q ≤ C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+2 n−1 f ¯ p v ⎞ ⎠ ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q ≤ C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z a ¯ q n ⎛ ⎝ n+2 n−1 f ¯ p n v ⎞ ⎠ ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q = C n∈Z a ¯ q n . Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 7 of 12 This means that the identity operator is bounded from ¯ q to q ||T n f n || q L p (u) . Then {||T n f n || L p ( u ) }∈ s , by applying the following lemma (see [4]). Lemma 1. Let 1 < q < ¯ q < ∞ and 1 s = 1 q − 1 ¯ q . Suppose that {u n } and {v n } are sequences of positive real numbers. The following statements are equivalent: (i) There exists C >0 such that the inequality n∈Z (|a n |u n ) q 1 q ≤ C n∈Z (|a n |v n ) ¯ q 1 ¯ q holds for all sequences {a n } of real numbers. (ii) The sequence {u n v −1 n } belongs to the space l s . On the other hand, let us prove that (i) holds. If {a m } is a a sequence of non-negative real numbers and f = m∈Z a m χ (m−1,m) ⎛ ⎝ m m−1 ν 1− ¯ p ⎞ ⎠ − 1 ν 1− ¯ p , then m m−1 f = a m , m m−1 f ¯ p v = a ¯ p m m m−1 v 1− ¯ p 1− ¯ p and by the properties of the operator T we have ||Tf || p,u,q = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+1 n (Tf) p (x)u(x)dx ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q ≥ C ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+1 n T d ⎛ ⎝ ⎧ ⎨ ⎩ m m−1 f ⎫ ⎬ ⎭ ⎞ ⎠ p (n)u(x)dx ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q = C ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z T d ({a m }) q (n) ⎛ ⎝ n+1 n u(x)dx ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q = ||T d {a m }|| q {u n } ) . Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 8 of 12 Applying (2.3) we obtain ||T d {a m }|| q ({u n }) ≤ C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+1 n f ¯ p v ⎞ ⎠ ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q = C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z a ¯ q n ⎛ ⎝ n n−1 v 1− ¯ p ⎞ ⎠ − ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q = ||a n || ¯ q ( {v n } ) , which means that the discrete operator T d is bounded from ¯ q ( {v n } ) to ℓ q ({u n }), as we wished to prove. Conversely, let us suppose that (i) and (ii) hold. Then, we have ||Tf || p,u,q ≤ C ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+1 n (T(f χ (−∞,n−1) + f χ (n+2,∞) )) p u ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q + C ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+1 n (Tf χ (n−1,n+2) ) p u ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q ≤ C ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z (T d ({a m })(n)) q ⎛ ⎝ n+1 n u ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q + C ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+1 n (T n f ) p u ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 1 q = C ( I 1 + I 2 ) , where a m = m m −1 f . Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 9 of 12 Applying that T d is bounded from ¯ q ( {v n } ) to ℓ q ({u n }) and Hölder inequality, we obtain I 1 ≤ C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z a ¯ q n ⎛ ⎝ n n−1 v 1− ¯ p ⎞ ⎠ − ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q = C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n n−1 f ⎞ ⎠ ¯ q ⎛ ⎝ n n−1 v 1− ¯ p ⎞ ⎠ − ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q ≤ C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n n−1 f ¯ p v ⎞ ⎠ ¯ q ¯ p ⎛ ⎝ n n−1 v 1− ¯ p ⎞ ⎠ ¯ q ¯ p ⎛ ⎝ n n−1 v 1− ¯ p ⎞ ⎠ − ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q = C ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n n−1 f ¯ p v ⎞ ⎠ ¯ q ¯ p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ 1 ¯ q = C||f || ¯ p ,v, ¯ q . Now we estimate I 2 .If 1 < ¯ q ≤ q < ∞ , since (ii)a holds, we know that the operators T n are uniformly bounded from L p (u,(n -1,n + 2)) to L ¯ p ( v, ( n − 1, n +2 )) and then I 2 ≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+2 n−1 (T n f ) p u ⎞ ⎠ q p ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 1 q ≤ C ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+2 n−1 f ¯ p v ⎞ ⎠ q ¯ p ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 1 q ≤ C ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n∈Z ⎛ ⎝ n+2 n−1 f ¯ p v ⎞ ⎠ ¯ q ¯ p ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 1 ¯ q ≤ C||f || ¯ p ,v, ¯ q . Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 10 of 12 [...]... http://www.journalofinequalitiesandapplications.com/content/2011/1/13 doi:10.1186/1029-242X-2011-13 Cite this article as: Aguilar Cañestro and Ortega Salvador: Boundedness of positive operators on weighted amalgams Journal of Inequalities and Applications 2011 2011:13 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance... 2011 References 1 Wiener N: On the representation of functions by trigonometric integrals Math Z 1926, 24:575-616 2 Fournier JJF, Stewart J: Amalgams of Lp and ℓq Bull Am Math Soc 1985, 13(1):1-21 3 Carton-Lebrun C, Heinig HP, Hofmann SC: Integral operators on weighted amalgams Stud Math 1994, 109(2):133-157 4 Ortega Salvador P, Ramírez Torreblanca C: Hardy operators on weighted amalgams Proc Roy Soc... Rakotondratsimba Y: Fractional maximal and integral operators on weighted amalgam spaces J Korean Math Soc 1999, 36(5):855-890 6 Aguilar Cañestro MI, Ortega Salvador P: Boundedness of generalized Hardy operators on weighted amalgam spaces Math Inequal Appl 2010, 13(2):305-318 7 Heinig HP, Kufner A: Weighted Friedrichs inequalities in amalgams Czechoslovak Math J 1993, 43(2):285-308 8 Andersen K: Weighted. .. maximal function Trans Am Math Soc 1972, 165:207-226 12 Sawyer ET: A characterization of a two-weight norm inequality for maximal operators Stud Math 1982, 75:1-11 13 Verbitsky IE: Weighted norm inequalities for maximal operators and Pisier’s Theorem on factorization through Lp, ∞ Integr Equ Oper Theory 1992, 15:124-153 Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011,... q−q ⎪ ⎪ ¯ ⎬ q ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 1 s n∈Z ≤ C||f ||p,v,¯ ¯ q This finishes the proof of the theorem Acknowledgements This research has been supported in part by MEC, grant MTM 2008-06621-C02-02, and Junta de Andalucía, Grants FQM354 and P06-FQM-01509 Authors’ contributions Both authors participated similarly in the conception and proofs of the results Both authors read and approved the final manuscript Competing... inequalities for maximal functions associated with general measures Trans Am Math Soc 1991, 326:907-920 9 Martín-Reyes FJ, Ortega Salvador P, de la Torre A: Weighted inequalities for one-sided maximal functions Trans Am Math Soc 1990, 319(2):517-534 10 Sawyer ET: Weighted inequalities for the one-sided Hardy-Littlewood maximal functions Trans Am Math Soc 1986, 297:53-61 11 Muckenhoupt B: Weighted norm inequalities...Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13 http://www.journalofinequalitiesandapplications.com/content/2011/1/13 Page 11 of 12 ¯ Let us suppose, finally, that 1 < q < q < ∞ Then (ii)b holds and, therefore, I2 ≤ C ≤C ⎧ ⎪ ⎪ ⎨ ⎛ n+2 ⎝ ⎪ ⎪ n∈Z ⎩ n−1 ⎧ ⎪ ⎪ ⎨ ⎫1 ⎞q... Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Page 12 of 12 . Salvador: Boundedness of positive operators on weighted amalgams. Journal of Inequalities and Applications 2011 2011:13. Submit your manuscript to a journal and benefi t from: 7 Convenient online. above, one can see some common features that lead to explo re the possibility of giving a general theorem characterizing the boundedness in weighted amalgams of a wide family of positive operators. inequalities, Weights 1. Introduction Let u be a positive funct ion of one real variable and let p, q > 1. The amalgam (L p (u), ℓ q ) is the space of one variable real functions which are loc ally in