Boundedness of positive operators on weighted amalgams

RESEARCH Open Access

Boundedness of positive operators on weightedamalgamsMaría Isabel Aguilar Cañestro and Pedro Ortega Salvador*

* Correspondence: [email protected]álisis Matemático, Facultad deCiencias, Universidad de Málaga,29071 Málaga, Spain

Abstract

In this article, we characterize the pairs (u, v) of positive measurable functions suchthat T maps the weighted amalgam (Lp̄(v), �q̄) in (Lp (u), ℓq) for all 1 < p, q, p̄, q̄ < ∞,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. IntroductionLet u be a positive function of one real variable and let p, q > 1. The amalgam (Lp(u),

ℓq) is the space of one variable real functions which are locally in Lp(u) and globally in

ℓq. More precisely,

(Lp(u), �q) = {f : ||f ||p,u,q < ∞},

where

||f ||p,u,q =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∑n∈Z

⎛⎝ n+1∫

n

|f |pu⎞⎠qp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

.

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 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 ||p̄,v,q̄ (1:1)

holds for all f, with a constant C independent of f, whenever 1 < q̄ ≤ q < ∞. The

characterization of the pairs (u, v) for (1.1) to hold in the case 1 < q < q̄ < ∞ has

been recently completed by Ortega and Ramírez ([4]), who have also characterized the

weak type inequality∥∥Pf∥∥p,∞;u,q ≤ C∥∥f∥∥p̄,v,q̄,

Aguilar Cañestro and Ortega Salvador Journal of Inequalities and Applications 2011, 2011:13http://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 CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

mailto:[email protected]

mailto:[email protected]

http://creativecommons.org/licenses/by/2.0

where ||g||p,∞;u,q =

{∑n∈Z

||gX(n,n+1)||qp,∞,u

}1q.

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, Rakotondrat-

simba ([5]) characterized some weighted inequalities in amalgams (corresponding to

the cases 1 < p̄ ≤ p < ∞ and 1 < q̄ ≤ q < ∞) for the fractional maximal operators and

the fractional integrals. 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, p̄, q, q̄ < ∞, 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 provid-

ing, in such a way, a unified approach to the subject. This is the purpose of this article.

2. The resultsWe consider an operator T acting on real measurable functions f of one real variable

and define a sequence {Tn}nÎℤ of local operators by

Tnf (x) = T(fX(n−1,n+2))(x) x ∈ (n − 1,n + 2).

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:

(i) There exists C > 0 such that for all non-negative functions f, all n Î ℤ and all x

Î (n, n + 1), the inequality

T(fX(−∞,n−1) + fX(n+2,∞)

)(x) ≤ CTd

⎛⎝

⎧⎨⎩

m∫m−1

f

⎫⎬⎭

⎞⎠ (n) (2:1)

holds.

(ii) There exists C > 0 such that for all sequences {ak} of non-negative real numbers

and n Î ℤ, the inequality

Td({ak})(n) ≤ CTf (y), (2:2)

holds for all y Î (n, n + 1) and all non-negative f such that∫ mm−1 f = am 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) if 0 ≤ f (x) ≤ g(x).


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-Liouville, and Weyl

fractional integral operators, maximal fractional operators, etc.

Our main result is the following one:

Theorem 1. Let 1 < p, q, p̄, q̄ < ∞. 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

||Tf ||p,u,q ≤ C||f ||p̄,v,q̄. (2:3)

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

(i) Td is bounded from �q̄({vn})to ℓq({un}), where

vn =(∫ n

n−1 v1−p̄′ )−

q̄p̄′ and

un =(∫ n+1

n u)qp.

(ii) (a) supn∈Z

||Tn||(Lp̄(v),Lp(u)) < ∞in the case 1 < q̄ ≤ q < ∞.

(b) {||Tn||(Lp̄(v),Lp(u))} ∈ �s, with1s=1q

− 1q̄, in the case 1 < q < q̄ < ∞.

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, p̄, q̄ < ∞. 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

||Tf ||p,∞,u,q ≤ C||f ||p̄,v,q̄ (2:4)

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

(i) Td is bounded from �q̄({vn})to ℓq ({un}),), with vn and un defined as in Theorem 1.

(ii) (a) supn∈Z

||Tn||(Lp̄(v),Lp,∞(u)) < ∞in the case 1 < q̄ ≤ q < ∞.

(b) {||Tn||(Lp̄(v),Lp,∞(u))} ∈ �s, with1s=1q

− 1q̄, in the case 1 < q < q̄ < ∞.

If conditions on the weights u, v, and {un}, {vn} characterizing the boundedness of the

operators Tn 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 [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) = suph>0

1h

x∫x−h

|f |,


Page 3 of 12

we have:

(i) The discrete operator (M -)d, defined by

(M−)d({an})(j) = supk≤j−1

1j − k

j−1∑i=k

| ai|,

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

1h

x∫x−h

|f |, x ∈ (n − 1, n + 2).

(iii) If p = p̄ and q = q̄, there are well-known conditions on the weights u, v, and {un},

{vn} 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 (Lp(w), ℓq) to (Lp(w), ℓq).

(ii) M- is bounded from (Lp(w), ℓq) to (Lp,∞(w), ℓq).

(iii) The next conditions hold simultaneously:

(a) w ∈ A−p,(n−1,n+2)for all n, uniformly, and

(b) the pair ({un}, {vn}) verifies the discrete Sawyer’s condition S−q , i.e., there exists

C > 0 such that

k∑j=r((M−)d({v1−q′

n }))q(j)uj ≤ Ck∑j=r

v1−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({an})(j) = supk≥j+3

1k − j − 2

k∑i=j+3

| ai|,

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


(i) M + is bounded from (Lp(w), ℓq) to (Lp(w), ℓq).

(ii) M + is bounded from (Lp(w), ℓq) to (Lp,∞ (w), ℓq).


(a) w ∈ A+p,(n−1,n+2)for all n, uniformly, and

(b) the pair ({un}, {vn-3}) verifies the discrete Sawyer’s condition S+q, i.e., there

exists C > 0 such that

k∑j=r((M+)d({v1−q′

n }))q(j)uj ≤ Ck∑j=r

v1−q′j ,


Page 4 of 12


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

Mf (x) = supx∈I

1|I|

∫I

|f |,

then M is admissible, with Md({an})(j) = supr≤j≤k

1k − r + 1

k∑i=r

| ai|, 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:


(i) M is bounded from (Lp(w), ℓq) to (Lp(w), ℓq).

(ii) M is bounded from (Lp(w), ℓq) to (Lp,∞(w), ℓq).


(a) w Î Ap,(n-1,n+2) for all n, uniformly, and

(b) the pair ({un}, {vn}) verifies the discrete two-sided Sawyer’s condition Sq, i.e.,

there exists C > 0 such that

k∑j=r(Md({v1−q′

n })q(j)uj ≤ Ck∑j=r

v1−q′j


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 bound-

edness of the maximal operator in the amalgam (Lp(w), ℓq), while in [3] only sufficient

conditons were given. We also prove the equivalence between the strong type inequal-

ity 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 Ma, 0 <a < 1,

defined by

Mαf (x) = supc<x<d

1

(d − c)1−α

d∫c

| f |.

The proof of the admissibility of Ma, with the obvious Mdα, is implied in Rakoton-

dratsimba’s paper ([5]).

Verbitsky ([13]) in the case 1 <q <p < ∞ and Sawyer ([12]) in the case 1 <p ≤ q < ∞

characterized the boundedness of Ma from Lp to Lq(w). These results allow us to give

necessary and sufficient conditions on the weight u for Ma to be bounded from

(Lp̄, �q̄) to (Lp(u), �q).


Page 5 of 12

Before stating the theorem, we introduce the notation:

(i) If 1 < q̄ < ∞, we define H : ℤ ® ℝ by

H(i) = supr≤i≤k

1

(k − r + 1)1−αq̄

k∑j=r

uj.

(ii) If 1 < q̄ ≤ q, we define

J = supr≤k

||X[r,k]Mdα(X[r,k])||�q({uj})

(k − r + 1)

1q̄

.

(iii) If 1 < p̄ < ∞ and n Î ℤ, we define for x Î (n - 1, n + 2)

Hn(x) = supx∈I⊂(n−1,n+2)

1

|I|1−αp̄

∫Iu.

(iv) If 1 < p̄ < p and n Î ℤ, we define

Jn = supI⊂(n−1,n+2)

||XIMα(XI)||Lp(u)

|I|1p̄

.

The result reads as follows.

Theorem 6. Ma is bounded from (Lp̄, �q̄)to (Lp(u), ℓq) if and only if

(i) in the case 1 < p̄ ≤ p < ∞and 1 < q̄ ≤ q < ∞, supnÎℤ Jn < ∞ and J < ∞;

(ii) in the case1 < p < p̄ < ∞and1 < q̄ ≤ q < ∞, supn∈Z||Hn||L

pp̄−p (u)

< ∞and J < ∞;

(iii) in the case 1 < p̄ ≤ p < ∞and 1 < q < q̄ < ∞, {Jn}n Î ℓs, where

1s=1q

− 1q̄,

and H ∈ �q

q̄−q({uj});

(iv) in the case 1 < p < p̄ < ∞and 1 < q < q̄ < ∞, ||Hn||L

pp̄−p (u)

∈ �sand

H ∈ �q

q̄−q({uj}).

3. Proof of Theorem 1Let 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 ||p̄,v,q̄ =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎛⎝ n∫n−1

f p̄v

⎞⎠q̄p̄+

⎛⎝ n+1∫

n

f p̄v

⎞⎠q̄p̄+

⎛⎝ n+2∫n+1

f p̄v

⎞⎠q̄p̄

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

≤ Cp̄,q̄

⎛⎝ n+2∫

n−1

f p̄v

⎞⎠1p̄,


Page 6 of 12

and, on the other hand,

||Tf ||p,u,q ≥

⎧⎪⎪⎪⎨⎪⎪⎪⎩

⎛⎝ n∫n−1

(Tf )pu

⎞⎠qp+

⎛⎝ n+1∫

n

(Tf )pu

⎞⎠qp+

⎛⎝ n+2∫n+1

(Tf )pu

⎞⎠qp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

≥ Cp,q

⎛⎝ n+2∫n−1

(Tf )pu

⎞⎠1p

≥ Cp,q

⎛⎝ n+2∫n−1

(Tnf )pu

⎞⎠1p

= Cp,q||Tnf ||p,u.

Therefore, by (2.3), Tn is bounded and ||Tn||(Lp̄(v),Lp(u)) ≤ C, where C is a positive con-

stant independent of n. Then (ii)a holds independently of the relationship between q

and q̄. Let us prove that if 1 < q < q̄ < ∞, then (ii)b also holds.

It is well known that ||Tn||(Lp̄(v),Lp(u)) = sup{f :||f ||Lp̄(v)=1}

||Tnf ||Lp(u). Therefore, for each n

there exists a non-negative measurable function fn, with support in (n - 1, n + 2) and

with ||fn||(Lp̄(v),(n−1,n+2)) = 1, such that ||Tn||(Lp̄(v),Lp(u)) < ||Tnfn||Lp(u) + 12|n|.

Since

{12|n|

}∈ �s, to prove that {||Tn||(Lp̄(v),Lp(u))} ∈ �s it suffices to see that

{||Tnfn||Lp(u)} ∈ �s.

Let {an} be a sequence of non-negative real numbers and f =∑nanfn. For each n Î ℤ,

f(x) ≥ anfn (x) and then Tf (x) ≥ anTnfn (x) for all x Î (n - 1, n + 2). Thus,

||Tf ||p,u,q ≥ C

⎧⎪⎪⎨⎪⎪⎩

∑n∈Z

(n+2∫n−1

apn(Tnfn)pu

)qp

⎫⎪⎪⎬⎪⎪⎭

1q

= C{∑n∈Z

aqn||Tnfn||qLp(u)}1q .

Then, from (2.3) we deduce

{∑n∈Z

aqn||Tnfn||qLp(u)}1q ≤ C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

⎛⎝ n+2∫n−1

f p̄v

⎞⎠q̄p̄

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

≤ C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

aq̄n

⎛⎝ n+2∫n−1

f p̄n v

⎞⎠q̄p̄

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

= C

{∑n∈Z

aq̄n

}.


Page 7 of 12

This means that the identity operator is bounded from �q̄ to �q({

||Tnfn||qLp(u)}). Then

{||Tnfn||Lp(u)} ∈ �s, by applying the following lemma (see [4]).

Lemma 1. Let 1 < q < q̄ < ∞and1s=1q

− 1q̄. Suppose that {un} and {vn} are

sequences of positive real numbers. The following statements are equivalent:

(i) There exists C >0 such that the inequality

{∑n∈Z

(|an|un)q}1q ≤ C

{∑n∈Z

(|an|vn)q̄}1q̄

holds for all sequences {an} of real numbers.

(ii) The sequence {unv−1n }belongs to the space ls.

On the other hand, let us prove that (i) holds. If {am} is a a sequence of non-negative

real numbers and

f =∑m∈Z

amχ(m−1,m)

⎛⎝ m∫m−1

ν1−p̄′

⎞⎠

−1

ν1−p̄′,

then∫ mm−1 f = am,

∫ mm−1 f

p̄v = ap̄m(∫ m

m−1 v1−p̄′)1−p̄ 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

⎞⎠qp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

≥ C

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∑n∈Z

⎛⎝ n+1∫

n

Td

⎛⎝

⎧⎨⎩

m∫m−1

f

⎫⎬⎭

⎞⎠

p

(n)u(x) dx

⎞⎠q

p

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

= C

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∑n∈Z

Td({am})q(n)⎛⎝ n+1∫

n

u(x) dx

⎞⎠qp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

= ||Td{am}||�q{un}).


Page 8 of 12

Applying (2.3) we obtain

||Td{am}||�q({un}) ≤ C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

⎛⎝ n+1∫

n

f p̄v

⎞⎠q̄p̄

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

= C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

aq̄n

⎛⎝ n∫n−1

v1−p̄′

⎞⎠

−q̄p̄′

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

= ||an||�q̄({vn}),

which means that the discrete operator Td is bounded from �q̄({vn}) to ℓq ({un}), 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(fχ(−∞,n−1) + fχ(n+2,∞)))pu

⎞⎠qp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

+ C

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∑n∈Z

⎛⎝ n+1∫

n

(Tfχ(n−1,n+2))pu

⎞⎠q

p

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

≤ C

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∑n∈Z

(Td({am})(n))q⎛⎝ n+1∫

n

u

⎞⎠qp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

+ C

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∑n∈Z

⎛⎝ n+1∫

n

(Tnf )pu

⎞⎠qp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

1q

= C(I1 + I2),

where am =∫ mm−1 f .


Page 9 of 12

Applying that Td is bounded from �q̄({vn}) to ℓq ({un}) and Hölder inequality, we

obtain

I1 ≤ C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

aq̄n

⎛⎝ n∫n−1

v1−p̄′

⎞⎠

−q̄p̄′

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

= C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

⎛⎝ n∫n−1

f

⎞⎠

q̄⎛⎝ n∫n−1

v1−p̄′

⎞⎠

−q̄p̄′

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

≤ C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

⎛⎝ n∫n−1

f p̄v

⎞⎠q̄p̄⎛⎝ n∫n−1

v1−p̄′

⎞⎠

q̄p̄′

⎛⎝ n∫n−1

v1−p̄′

⎞⎠

−q̄p̄′

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

= C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑n∈Z

⎛⎝ n∫n−1

f p̄v

⎞⎠q̄p̄

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q̄

= C||f ||p̄,v,q̄.

Now we estimate I2. If 1 < q̄ ≤ q < ∞, since (ii)a holds, we know that the operators

Tn are uniformly bounded from Lp(u, (n - 1, n + 2)) to Lp̄(v, (n − 1, n + 2)) and then

I2 ≤

⎧⎪⎪⎨⎪⎪⎩

∑n∈Z

⎛⎝ n+2∫n−1

(Tnf )pu

⎞⎠

qp

⎫⎪⎪⎬⎪⎪⎭

1q

≤ C

⎧⎪⎪⎨⎪⎪⎩

∑n∈Z

⎛⎝ n+2∫n−1

f p̄v

⎞⎠

qp̄

⎫⎪⎪⎬⎪⎪⎭

1q

≤ C

⎧⎪⎪⎨⎪⎪⎩

∑n∈Z

⎛⎝ n+2∫n−1

f p̄v

⎞⎠

q̄p̄

⎫⎪⎪⎬⎪⎪⎭

1q̄

≤ C||f ||p̄,v,q̄.


Page 10 of 12

Let us suppose, finally, that 1 < q < q̄ < ∞. Then (ii)b holds and, therefore,

I2 ≤ C

⎧⎪⎪⎨⎪⎪⎩

∑n∈Z

⎛⎝ n+2∫n−1

Tnf pu

⎞⎠

qp

⎫⎪⎪⎬⎪⎪⎭

1q

≤ C

⎧⎪⎪⎨⎪⎪⎩

∑n∈Z

(||Tn||(Lp̄(v),Lp(u)))q⎛⎝ n+2∫n−1

f p̄v

⎞⎠

qp̄

⎫⎪⎪⎬⎪⎪⎭

1q

≤ C

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎛⎜⎜⎝∑

n∈Z

⎛⎝ n+2∫n−1

f p̄v

⎞⎠

q̄p̄

⎞⎟⎟⎠

qq̄ (∑

n∈Z

(||Tn||(Lp̄(v),Lp(u))) qq̄q̄−q

) q̄−qq̄

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1q

= C

⎧⎪⎪⎨⎪⎪⎩

∑n∈Z

⎛⎝ n+2∫n−1

f p̄v

⎞⎠

q̄p̄

⎫⎪⎪⎬⎪⎪⎭

1q̄ (∑

n∈Z

(||Tn||(Lp̄(v),Lp(u)))s) 1

s

≤ C||f ||p̄,v,q̄.

This finishes the proof of the theorem.

AcknowledgementsThis research has been supported in part by MEC, grant MTM 2008-06621-C02-02, and Junta de Andalucía, GrantsFQM354 and P06-FQM-01509.

Authors’ contributionsBoth authors participated similarly in the conception and proofs of the results. Both authors read and approved thefinal manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 8 October 2010 Accepted: 21 June 2011 Published: 21 June 2011

References1. 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 ℓ

q. 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 Edinburgh 2010,

140A:175-188.5. 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 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 for the Hardy 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.


Page 11 of 12

doi:10.1186/1029-242X-2011-13Cite this article as: Aguilar Cañestro and Ortega Salvador: Boundedness of positive operators on weightedamalgams. Journal of Inequalities and Applications 2011 2011:13.

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