Optimal Couples of Rearrangement Invariant Spaces for Generalized Maximal Operators

where the supremum is taken with respect to all balls B containing x, and | ⋅ | denotes the Lebesgue measure. Note that for φ(t) = t, M φ is the classical HardyLittlewood maximal operator M, and for φ(t) = t, 0 < s < n, we get the fractional maximal operatorM. Let S d and S t be two classes of rearrangement invariant spaces (for definition, see Section 2). Take E ∈ S d and G ∈ S t , and assume thatM φ : E → G, where the symbol T : A → B means that T is bounded from A into B. We say that E is an optimal domain space in the class S d ifM φ : E 󸀠 → G for any rearrangement invariant space E󸀠 ∈ S d , then it follows that


Introduction
Let  loc be the space of all real-valued locally integrable functions on R  with the Lebesgue measure.For any positive function  on (0, ∞), 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 () =  1−/ , 0 <  < , 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   →  1−/ , 0 <  < , by an arbitrary quasiconcave function.By a quasiconcave function , we mean that  is a positive function on (0, ∞) 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.

Rearrangement Invariant Spaces.
Recall that the decreasing rearrangement of a measurable function  on R  , 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 (0, ∞) with the Lebesgue measure.As usual, a quasinorm  :  → [0, ∞] satisfies the following conditions: For each quasinorm , there exists an equivalent quasinorm   satisfying the triangle inequality , for some  ∈ (0, 1] (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.
Let   :  → [0, ∞] be a quasinorm, and let  be the space consisting of all functions  in  loc for which the quasinorm ‖‖  :=   ( * ) is finite.The space  is a rearrangement invariant in the sense that if  ∈  and  * =  * for  ∈  loc , 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 62]).More general examples are given by the Riesz-Fischer monotone spaces as in [3, page 305].

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 ℎ  is finite.Then, If   is monotone, then the function ℎ  is increasing.Hence, in view of ℎ  (1) = 1 and ℎ  ()ℎ  (1/) ≥ 1, we obtain 0 ≤   ≤   .Furthermore, the submultiplicativity of ℎ  will imply (see [6, page 244]) that both indices are finite and given by the following limits: Similarly, we can associate a pair of indices with a positive function  on (0, ∞) (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 ℎ  generated by .If the function  is quasiconcave, then 0 ≤   ≤ 1 (see [7, page 54]).
We will make use of the assertions of the following proposition (cf.[3, Lemma 5.9, page 147]) 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  > 0, Proof.We will derive only (10); the proof of (11) will be similar.Let  be small enough such that   +   − 2 > 1.
For this , there exists 0 <  1 < 1 such that and there also exists 0 <  2 < 1 such that Taking  = min( 1 ,  2 ), we get thus We note that first integral on right hand side is convergent since   +   − 2 > 1.The second integral is also convergent since it is estimated from above by a convergent integral ∫ 1  V − (V/V); this completes the proof.

Admissible Couples
In what follows,  will be a fixed quasiconcave function.We will need to work with the following classes of rearrangement invariant spaces: 0 consists of all rearrangement invariant spaces  generated by a monotone quasinorm   ;  ,1 consists of all rearrangement invariant spaces  generated by a quasinorm   which is monotone and satisfies Minkovski inequality along with   +   > 1.
In the formulation of our results, we will need also a subclass  0 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 3.13].
Theorem 2. Let ,  ∈  0 ; 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  ∈  ,1 .Then, where Proof.We start off with a simple change of variables to have Since the function As   is monotone and satisfies Minkowski's inequality, we get )) V V . (24) Finally, using the definitions of ℎ  and ℎ  along with the quasiconcavity of , we obtain and the required estimate follows thanks to (10).
Theorem 4. Let  ∈  ,1 and  ∈  0 .Then, the couple (, ) is admissible if and only if where Remark 5. Note that if  is an identity function, then the condition   +   > 1, which turns into   > 0, is not needed since  =  if  ∈  0 .
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  ∈  0 .Denoting we note that  ≈  +   as  is increasing.Then, by Theorem 2, it will suffice to establish that   (  ) ≲   ().
Remark 6.By Theorem 1.3 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.

Optimal Couples
We introduce two more classes of rearrangement invariant spaces as follows: ,2 consists of all rearrangement invariant spaces  generated by a quasinorm   which is monotone and satisfies Minkovski inequality along with   +   > 1 and   < 1;  ,1 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.Then, the couple (, ) is optimal in the class ( 0 ,  ,1 ).
To see that  is the optimal domain space in the class  0 , let   ∈  0 be another rearrangement invariant space such that the couple (  , ) is admissible.Take  ∈   and apply Theorem 2 to obtain where we get   →  as desired.
Remark 11.The statements of Theorems 9 and 10 generalize those of Theorems 2.10 and 3.1 in [1], respectively.