#### Abstract

The optimal couples of rearrangement invariant spaces for boundedness of a generalized maximal operator, associated with a quasiconcave function, have been characterized in terms of certain indices connected with rearrangement invariant spaces and quasiconcave functions.

#### 1. Introduction

Let be the space of all real-valued locally integrable functions on with the Lebesgue measure. For any positive function on , the generalized maximal operator is defined by where the supremum is taken with respect to all balls containing , and denotes the Lebesgue measure.

Note that for , is the classical Hardy-Littlewood maximal operator , and for , , we get the fractional maximal operator .

Let and be two classes of rearrangement invariant spaces (for definition, see Section 2). Take and , and assume that , where the symbol means that is bounded from into . We say that is an optimal domain space in the class if for any rearrangement invariant space , then it follows that , where by , we mean that is continuously embedded in . We say that is an optimal target space in the class if implies that for any rearrangement invariant space . Finally, the couple is called optimal in the class if is an optimal domain space in the class and is an optimal target space in the class .

The optimal couples for the fractional maximal operator have been characterized in [1]. The characterizations are based on certain conditions on the Boyd indices associated with rearrangement invariant spaces. Earlier, Jan Vybíral had considered the problem of optimal domain space for the operator in [2].

Our aim in this paper is to extend some of the results in [1] by means of replacing the function , , by an arbitrary quasiconcave function. By a quasiconcave function , we mean that is a positive function on such that is increasing and is decreasing. To our end, we will impose certain conditions on rearrangement invariant spaces as well as on by means of the Boyd indices associated with them.

The paper is organized as follows. In Section 2, we give the necessary background material. In particular, we define the rearrangement invariant spaces. It is worthy of mention that our rearrangement invariant spaces are more general than those in [3]. Particularly, we do not use Fatou property and duality arguments. For convenience, we will say that a couple is admissible if . The admissible and optimal couples are characterized in Sections 3 and 4, respectively.

#### 2. Preliminaries

##### 2.1. Rearrangement Invariant Spaces

Recall that the decreasing rearrangement of a measurable function on , denoted by , is defined as where is the distribution function of given as (see, e.g., [3]).

Let be the space of all nonnegative locally integrable functions on with the Lebesgue measure. As usual, a quasinorm satisfies the following conditions:(i) if and only if a.e.;(ii), ;(iii), .

For each quasinorm , there exists an equivalent quasinorm satisfying the triangle inequality , , for some (see [4]). We will say that the quasinorm satisfies Minkovski inequality if for the equivalent quasinorm , we have where the notation stands for the fact that is bounded above by a multiple of , the multiple being independent of any variables in and . Later we will use to indicate that both and hold.

A quasinorm is said to be monotone if for with , we have .

Let be a quasinorm, and let be the space consisting of all functions in for which the quasinorm is finite. The space is a rearrangement invariant in the sense that if and for , then . We will say that is the rearrangement invariant space generated by the quasinorm .

The rearrangement invariant Banach function spaces considered in [3] are a particular example of our rearrangement invariant spaces according to the Luxemburg representation theorem (see [3, page ]). More general examples are given by the Riesz-Fischer monotone spaces as in [3, page ].

##### 2.2. Boyd Indices

Let be a rearrangement invariant space generated by a quasinorm . The lower and upper Boyd indices of , denoted by and , are defined similarly to [5]. Let be the dilation function generated by . Suppose that is finite. Then, If is monotone, then the function is increasing. Hence, in view of and , we obtain . Furthermore, the submultiplicativity of will imply (see [6, page ]) that both indices are finite and given by the following limits:

Similarly, we can associate a pair of indices with a positive function on (see [7]). In this paper, we will need only one of them which we call as the lower Boyd index . It is defined by where is the dilation function generated by . If the function is quasiconcave, then (see [7, page ]).

We will make use of the assertions of the following proposition (cf. [3, Lemma , page ]) in the forthcoming sections.

Proposition 1. *Let be a rearrangement invariant space generated by a monotone quasinorm , and let be a quasiconcave function. Then, for any ,
*

*Proof. *We will derive only (10); the proof of (11) will be similar. Let be small enough such that . For this , there exists such that
and there also exists such that
Taking , we get
thus
We note that first integral on right hand side is convergent since . The second integral is also convergent since it is estimated from above by a convergent integral ; this completes the proof.

#### 3. Admissible Couples

In what follows, will be a fixed quasiconcave function. We will need to work with the following classes of rearrangement invariant spaces: consists of all rearrangement invariant spaces generated by a monotone quasinorm ; consists of all rearrangement invariant spaces generated by a quasinorm which is monotone and satisfies Minkovski inequality along with .

In the formulation of our results, we will need also a subclass of , formed of all decreasing functions in .

We note here that the admissibility of a couple (i.e., ) is equivalent to the following estimate:

Our starting point is the following characterization of all admissible couples which is essentially a reformulation of the sharp rearrangement inequality which was proved in [2, Theorem ].

Theorem 2. *Let ; then the couple is admissible if and only if
**
where
*

The one-dimensional operator is rather complicated; however, we can replace it by a simpler one in the condition (17) by exploiting the next estimate.

Lemma 3. *Assume that . Then,
**
where
*

*Proof. *We start off with a simple change of variables to have
Since the function is increasing in , therefore
As is monotone and satisfies Minkowski’s inequality, we get
Now we observe that the function is increasing in ; thus,
Finally, using the definitions of and along with the quasiconcavity of , we obtain
and the required estimate follows thanks to (10).

Theorem 4. *Let and . Then, the couple is admissible if and only if
**
where
*

*Remark 5. *Note that if is an identity function, then the condition , which turns into , is not needed since if .

*Proof. *Assume first that the couple is admissible. Then, in view of the condition (17), , and the monotonicity of , the condition (26) follows immediately.

Conversely, assume that the condition (26) holds, and fix . Denoting
we note that as is increasing. Then, by Theorem 2, it will suffice to establish that . Since both the functions and are decreasing in , we have
where . Observe that . Therefore, using the monotonicity of and applying the condition (26), we obtain . The proof will thus be complete if we show that . To this end, we set . Then, by definition of , we have
whenever . As is increasing, we further have
whenever , whence we get
by taking supremum over all . Finally, multiplying both sides by , using monotonicity of , and applying Lemma 3, we arrive at
as desired. The proof is complete.

*Remark 6. *By Theorem in [8], the conditions (17) and (26) are also equivalent if is a rearrangement invariant space as in [3]. The key ingredient of the proof of the above mentioned theorem is the Hardy-Littlewood-Póyla principle (see [3, Theorem 4.6, page 61]) which is not guaranteed in our rearrangement invariant spaces.

#### 4. Optimal Couples

We introduce two more classes of rearrangement invariant spaces as follows: consists of all rearrangement invariant spaces generated by a quasinorm which is monotone and satisfies Minkovski inequality along with and ; consists of all rearrangement invariant spaces generated by a quasinorm which is monotone and satisfies Minkovski inequality along with .

First we construct optimal couples with the aid of Theorem 2. We will need the next estimate which can be proved by using Minkovski inequality and (11) as in Lemma 3.

Lemma 7. *Let ; then,
**
where
*

*Remark 8. *In fact, the above estimate is two sided because the reverse estimate holds trivially since if .

Theorem 9. *Let be a given rearrangement invariant space. Let be the rearrangement invariant space generated by the quasinorm as follows:
**
Then, the couple is optimal in the class .*

*Proof. *Clearly, the couple is admissible by Theorem 2. To see that is the optimal domain space in the class , let be another rearrangement invariant space such that the couple is admissible. Take and apply Theorem 2 to obtain
where we get as desired.

It now remains to establish that is the optimal target space in the class . Let be another rearrangement space such that the couple is admissible, and take . Set
to observe that because . Consequently,
again by Theorem 2. It follows, by definition of , that . In view of , we can write the following:
Since , an application of Lemma 7 yields
Hence, the required embedding will follow if we show that . For this, let us note that
Since , thus . Therefore, by [7, Corollary 3, page 57], we get . This gives as desired. The proof is complete.

Now we apply Theorem 4 to construct optimal couples.

Theorem 10. *Let be a given rearrangement invariant space. Let be the rearrangement invariant space generated by the quasinorm as follows:
**
Then, the couple is optimal in the class .*

*Proof. *Let . Then, by the definition of ,
Since , we can apply Lemma 7 to to obtain . Hence, the couple is admissible by Theorem 4.

To see that is the optimal target space in the class , let be another rearrangement space such that the couple is admissible. Given that , consider again the function . Then, . Thus,
by Theorem 4. Therefore,
whence we get the desired embedding .

Our remaining task is to show that is the optimal domain space in the class . To do so, let be another rearrangement invariant space such that the couple is admissible. Take and note that . Thus,
which gives by Theorem 4. It follows that as required. The proof is complete.

*Remark 11. *The statements of Theorems 9 and 10 generalize those of Theorems and in [1], respectively.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.