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Boundedness of positive operators on weighted amalgams

  • María Isabel Aguilar Cañestro1 and
  • Pedro Ortega Salvador1Email author
Journal of Inequalities and Applications20112011:13

https://doi.org/10.1186/1029-242X-2011-13

Received: 8 October 2010

Accepted: 21 June 2011

Published: 21 June 2011

Abstract

In this article, we characterize the pairs (u, v) of positive measurable functions such that T maps the weighted amalgam in (L p (u), ℓ q ) for all , 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

AmalgamsMaximal operatorsWeighted inequalitiesWeights

1. Introduction

Let u be a positive function 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 locally in L p (u) and globally in ℓ q . More precisely,
where

These spaces were introduced by Wiener in [1]. The article [2] describes the role played by amalgams in Harmonic Analysis.

Carton-Lebrun, Heinig, and Hoffmann studied in [3] the boundedness of the Hardy operator in weighted amalgam spaces. They characterized the pairs of weights (u, v) such that the inequality
(1.1)
holds for all f, with a constant C independent of f, whenever . The characterization of the pairs (u, v) for (1.1) to hold in the case has been recently completed by Ortega and Ramírez ([4]), who have also characterized the weak type inequality

where

There are several articles dealing with 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 Hardy-Littlewood maximal operator as well as for some integral operators with kernel K(x, y) increasing in the second variable and decreasing in the first one. On the other hand, Rakotondratsimba ([5]) characterized some weighted inequalities in amalgams (corresponding to the cases and ) for the fractional maximal operators and the fractional integrals. Finally, the authors characterized in [6] the weighted inequalities for some generalized Hardy operators, including the fractional integrals of order greater than one, in all cases , 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 explore the possibility of giving a general theorem characterizing the boundedness in weighted amalgams of a wide family of positive operators, and providing, 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 }nof local operators by
We assume that there exists a discrete operator Td, i.e., which transforms sequences of real numbers in sequences of real numbers, verifying the following conditions:
  1. (i)
    There exists C > 0 such that for all non-negative functions f, all n and all x (n, n + 1), the inequality
    (2.1)
     
holds.
  1. (ii)
    There exists C > 0 such that for all sequences {a k } of non-negative real numbers and n , the inequality
    (2.2)
     

holds for all y (n, n + 1) and all non-negative f such that for all m.

We also assume that T verifies Tf = T |f|, Tf) = |λ| Tf, T(f + g)(x) ≤ Tf (x) + Tg (x) and Tf(x) ≤ Tg(x) if 0 ≤ f (x) ≤ g(x).

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-Liouville, and Weyl fractional integral operators, maximal fractional operators, etc.

Our main result is the following one:

Theorem 1. Let . Let u and v be positive locally integrable functions on and let T be an admissible operator. Then there exists a constant C > 0 such that the inequality
(2.3)
holds for all measurable functions f if and only if the following conditions hold:
  1. (i)

    T d is bounded from to q ({u n }), where and .

     
  2. (ii)

    (a) in the case .

     

     (b) , with , in the case .

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 . Let u and v be positive locally integrable functions on and let T be an admissible operator. Then there exists a constant C > 0 such that the inequality
(2.4)
holds for all measurable functions f if and only if the following conditions hold:
  1. (i)

    T d is bounded from to q ({u n }),), with v n and un defined as in Theorem 1.

     
  2. (ii)

    (a) in the case .

     

     (b) , with , in the case .

If conditions on the weights u, v, and {u n }, {v n } characterizing the boundedness of the operators T n and Td, 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 [37], as well as other corresponding to operators whose behavior on weighted amalgams has not been studied yet.

Thus, if M - is the one-sided Hardy-Littlewood maximal operator defined by
we have:
  1. (i)
    The discrete operator (M - )d, defined by
     
verifies conditions (2.1) and (2.2).
  1. (ii)
    The local operators are defined by
     
  2. (iii)

    If and , there are well-known conditions on the weights u, v, and {u n }, {v n } that characterize the boundedness of and (M - )d (see, for instance [810]).

     

Therefore, we obtain the following result:

Theorem 3. The following statements are equivalent:
  1. (i)

    M - is bounded from (L p (w), ℓ q ) to (L p (w), ℓ q ).

     
  2. (ii)

    M - is bounded from (L p (w), ℓ q ) to (L p,∞(w), ℓ q ).

     
  3. (iii)

    The next conditions hold simultaneously:

     
  4. (a)

    for all n, uniformly, and

     
  5. (b)
    the pair ({u n }, {v n }) verifies the discrete Sawyer's condition , i.e., there exists C > 0 such that
     

for all r, k with rk.

We can state a similar result for the one-sided maximal operator M+. In this case, the operator (M+)d defined by

verifies conditions (2.1) and (2.2). The theorem is the next one:

Theorem 4. The following statements are equivalent:
  1. (i)

    M + is bounded from (L p (w), ℓ q ) to (L p (w), ℓ q ).

     
  2. (ii)

    M + is bounded from (L p (w), ℓ q ) to (L p,∞(w), ℓ q ).

     
  3. (iii)

    The next conditions hold simultaneously:

     
  4. (a)

    for all n, uniformly, and

     
  5. (b)
    the pair ({u n }, {v n- 3}) verifies the discrete Sawyer's condition , i.e., there exists C > 0 such that
     

for all r, k with rk.

If M is the Hardy-Littlewood maximal operator, defined by

then M is admissible, with , and there are well-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:
  1. (i)

    M is bounded from (L p (w), ℓ q ) to (L p (w), ℓ q ).

     
  2. (ii)

    M is bounded from (L p (w), ℓ q ) to (L p,∞(w), ℓ q ).

     
  3. (iii)

    The next conditions hold simultaneously:

     
  4. (a)

    w A p,(n-1,n+2) for all n, uniformly, and

     
  5. (b)
    the pair ({u n }, {v n }) verifies the discrete two-sided Sawyer's condition S q , i.e., there exists C > 0 such that
     

for all r, k with rk.

This result improves the one obtained by Carton-Lebrun, Heinig and Hofmann in [3], in the sense that the conditions we give are necessary and sufficient for the boundedness 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 inequality and the weak type inequality. The equivalence (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 α , 0 < α < 1, defined by

The proof of the admissibility of M α , with the obvious , is implied in Rakotondratsimba's paper ([5]).

Verbitsky ([13]) in the case 1 < q < p < ∞ and Sawyer ([12]) in the case 1 < pq < ∞ characterized the boundedness of M α from L p to L q (w). These results allow us to give necessary and sufficient conditions on the weight u for M α to be bounded from to .

Before stating the theorem, we introduce the notation:
  1. (i)
    If , we define H : by
     
  2. (ii)
    If , we define
     
  3. (iii)
    If and n , we define for x (n - 1, n + 2)
     
  4. (iv)
    If and n , we define
     

The result reads as follows.

Theorem 6. M α is bounded from to (L p (u), ℓ q ) if and only if
  1. (i)

    in the case and , supn J n < ∞ and J < ∞;

     
  2. (ii)

    in the case and , and J < ∞;

     
  3. (iii)

    in the case and , {J n } n s , where , and ;

     
  4. (iv)

    in the case and , and .

     

3. Proof of Theorem 1

Let us suppose that the inequality (2.3) holds. Let n and let f be a non-negative function supported in (n - 1, n + 2). Then, on one hand,
and, on the other hand,

Therefore, by (2.3), T n is bounded and , where C is a positive constant independent of n. Then (ii)a holds independently of the relationship between q and . Let us prove that if , then (ii)b also holds.

It is well known that . Therefore, for each n there exists a non-negative measurable function f n , with support in (n - 1, n + 2) and with , such that .

Since , to prove that it suffices to see that .

Let {a n } be a sequence of non-negative real numbers and . For each n , f(x) ≥ a n f n (x) and then Tf (x) ≥ a n T n f n (x) for all x (n - 1, n + 2). Thus,
Then, from (2.3) we deduce

This means that the identity operator is bounded from to . Then , by applying the following lemma (see [4]).

Lemma 1. Let and . Suppose that {u n } and {v n } are sequences of positive real numbers. The following statements are equivalent:
  1. (i)
    There exists C > 0 such that the inequality
     
holds for all sequences {a n } of real numbers.
  1. (ii)

    The sequence 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
then and by the properties of the operator T we have
Applying (2.3) we obtain

which means that the discrete operator Td is bounded from to ℓ q ({u n }), as we wished to prove.

Conversely, let us suppose that (i) and (ii) hold. Then, we have

where .

Applying that Td is bounded from to ℓ q ({u n }) and Hölder inequality, we obtain
Now we estimate I2. If , since (ii)a holds, we know that the operators T n are uniformly bounded from L p (u, (n - 1, n + 2)) to and then
Let us suppose, finally, that . Then (ii)b holds and, therefore,

This finishes the proof of the theorem.

Declarations

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’ Affiliations

(1)
Análisis Matemático, Facultad de Ciencias, Málaga, Spain

References

  1. Wiener N: On the representation of functions by trigonometric integrals. Math Z 1926, 24: 575–616. 10.1007/BF01216799MathSciNetView ArticleGoogle Scholar
  2. Fournier JJF, Stewart J: Amalgams of L p and ℓ q . Bull Am Math Soc 1985,13(1):1–21. 10.1090/S0273-0979-1985-15350-9MATHMathSciNetView ArticleGoogle Scholar
  3. Carton-Lebrun C, Heinig HP, Hofmann SC: Integral operators on weighted amalgams. Stud Math 1994,109(2):133–157.MATHMathSciNetGoogle Scholar
  4. Ortega Salvador P, Ramírez Torreblanca C: Hardy operators on weighted amalgams. Proc Roy Soc Edinburgh 2010, 140A: 175–188.View ArticleGoogle Scholar
  5. Rakotondratsimba Y: Fractional maximal and integral operators on weighted amalgam spaces. J Korean Math Soc 1999,36(5):855–890.MATHMathSciNetGoogle Scholar
  6. Aguilar Cañestro MI, Ortega Salvador P: Boundedness of generalized Hardy operators on weighted amalgam spaces. Math Inequal Appl 2010,13(2):305–318.MATHMathSciNetGoogle Scholar
  7. Heinig HP, Kufner A: Weighted Friedrichs inequalities in amalgams. Czechoslovak Math J 1993,43(2):285–308.MATHMathSciNetGoogle Scholar
  8. Andersen K: Weighted inequalities for maximal functions associated with general measures. Trans Am Math Soc 1991, 326: 907–920. 10.2307/2001788MATHView ArticleGoogle Scholar
  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. 10.2307/2001252MATHView ArticleGoogle Scholar
  10. Sawyer ET: Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Trans Am Math Soc 1986, 297: 53–61. 10.1090/S0002-9947-1986-0849466-0MATHMathSciNetView ArticleGoogle Scholar
  11. Muckenhoupt B: Weighted norm inequalities for the Hardy maximal function. Trans Am Math Soc 1972, 165: 207–226.MATHMathSciNetView ArticleGoogle Scholar
  12. Sawyer ET: A characterization of a two-weight norm inequality for maximal operators. Stud Math 1982, 75: 1–11.MATHMathSciNetGoogle Scholar
  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. 10.1007/BF01193770MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Cañestro and Salvador; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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