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# Boundedness of positive operators on weighted amalgams

- María Isabel Aguilar Cañestro
^{1}and - Pedro Ortega Salvador
^{1}Email author

**2011**:13

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

© Cañestro and Salvador; licensee Springer. 2011

**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

- Amalgams
- Maximal operators
- Weighted inequalities
- Weights

## 1. Introduction

*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,

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

*u*,

*v*) such that the inequality

*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

*T*acting on real measurable functions

*f*of one real variable and define a sequence {

*T*

_{ n }}

_{n∈ℤ}of local operators by

*T*

^{d}, i.e., which transforms sequences of real numbers in sequences of real numbers, verifying the following conditions:

- (i)

- (ii)

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|*, *T*(λ*f*) = *|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*ℝ

*and let T be an admissible operator. Then there exists a constant C*> 0

*such that the inequality*

*holds for all measurable functions f if and only if the following conditions hold:*

- (i)
*T*^{d}*is bounded from**to*ℓ^{ q }({*u*_{ n }}),*where**and*. - (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*ℝ

*and let T be an admissible operator. Then there exists a constant C*> 0

*such that the inequality*

*holds for all measurable functions f if and only if the following conditions hold:*

- (i)
*T*^{d}*is bounded from**to*ℓ^{ q }({*u*_{ n }}),),*with v*_{ n }*and un defined as in Theorem 1*. - (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 *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 operators whose behavior on weighted amalgams has not been studied yet.

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)
*for all n*,*uniformly*,*and* - (b)

*for all r*, *k* ∈ ℤ *with r* ≤ *k*.

*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:*

- (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)
*for all n*,*uniformly*,*and* - (b)

*for all r*, *k* ∈ ℤ *with r* ≤ *k*.

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:*

- (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)

*for all r*, *k* ∈ ℤ *with r* ≤ *k*.

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.

*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 < *p* ≤ *q* < ∞ 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
.

- (i)
- (ii)
- (iii)
- (iv)

The result reads as follows.

**Theorem 6**.

*M*

_{ α }

*is bounded from*

*to*(

*L*

^{ p }(

*u*), ℓ

^{ q })

*if and only if*

- (i)
*in the case**and*, sup_{n∈ℤ}*J*_{n}< ∞*and J*< ∞; - (ii)
*in the case**and*,*and J*< ∞; - (iii)
*in the case**and*, {*J*_{ n }}_{ n }∈ ℓ^{ s },*where*,*and*; - (iv)
*in the case**and*,*and*.

## 3. Proof of Theorem 1

*n*∈ ℤ and let

*f*be a non-negative function supported in (

*n -*1,

*n*+ 2). Then, on one 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 .

*a*

_{ n }} be a sequence of non-negative real numbers and . For each

*n*∈ ℤ,

*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,

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:*

- (i)

*holds for all sequences*{

*a*

_{ n }}

*of real numbers*.

- (ii)
*The sequence**belongs to the space l*^{ s }.

*a*

_{ m }} is a a sequence of non-negative real numbers and

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

where .

*I*

_{2}. 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

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

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