Abstract
We prove continuity of the Riesz potential operator in optimal couples of rearrangement invariant function spaces defined in with the Lebesgue measure.
MSC:46E30, 46E35.
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1 Introduction
Let ℳ be the space of all locally integrable functions f on with the Lebesgue measure, finite almost everywhere, and let be the space of all non-negative locally integrable functions on with respect to the Lebesgue measure, finite almost everywhere. We shall also need the following two subclasses of . The subclass M consists of those elements g of for which there exists an such that is increasing. The subclass consists of those elements g of which are decreasing.
The Riesz potential operator , , is defined formally by
We shall consider rearrangement invariant quasi-Banach spaces E, continuously embedded in , such that the quasi-norm in E is generated by a quasi-norm , defined on with values in , in the sense that . In this way equivalent quasi-norms give the same space E. We suppose that E is nontrivial. Here is the decreasing rearrangement of f, given by
where is the distribution function of f, defined by
denoting the Lebesgue n-measure.
Note that , if .
There is an equivalent quasi-norm that satisfies the triangle inequality for some that depends only on the space E (see [1]).
We say that the norm is K-monotone (cf. [2], p.84, and also [3], p.305) if
Then is monotone, i.e., implies .
We use the notations or for non-negative functions or functionals to mean that the quotient is bounded; also, means that and . We say that is equivalent to if .
We say that the norm satisfies the Minkovski inequality if for the equivalent quasi-norm ,
For example, if E is a rearrangement invariant Banach function space as in [3], then by the Luxemburg representation theorem for some norm satisfying (1.2) and (1.3). More general example is given by the Riesz-Fischer monotone spaces as in [3], p.305.
Recall the definition of the lower and upper Boyd indices and . Let
be the dilation function generated by . Then
If is monotone, then the function is submultiplicative, increasing, , , hence . If is K-monotone, then by interpolation (analogously to [3], p.148), we see that . Hence in this case we have also .
Using the Minkovski inequality for the equivalent quasi-norm and monotonicity of , we see that
where . The main goal of this paper is to prove continuity of the Riesz potential operator in optimal couples of rearrangement invariant function spaces E and G, where . It is convenient to introduce the following classes of quasi-norms, where the optimality of is investigated. Let stand for all domain quasi-norms , which are monotone, rearrangement invariant, satisfying Minkowski’s inequality, and
Let consist of all target quasi-norms that are monotone, satisfy Minkowski’s inequality, , and
where is the characteristic function of the interval , . Note that technically it is more convenient not to require that the target quasi-norm is rearrangement invariant. Of course, the target space G is rearrangement invariant, since . Finally, let .
Definition 1.1 (Admissible couple)
We say that the couple is admissible for the Riesz potential if the following estimate is valid:
Moreover, is called domain quasi-norm (domain space), and (G) is called a target quasi-norm (target space).
For example, by Theorem 2.2 below (the sufficient part), the couple , , , is admissible if and v is related to w by the Muckenhoupt condition [4]:
Definition 1.2 (Optimal target quasi-norm)
Given the domain quasi-norm , the optimal target quasi-norm, denoted by , is the strongest target quasi-norm, i.e.,
for any target quasi-norm such that the couple , is admissible.
Definition 1.3 (Optimal domain quasi-norm)
Given the target quasi-norm , the optimal domain quasi-norm, denoted by , is the weakest domain quasi-norm, i.e.,
for any domain quasi-norm such that the couple , is admissible.
Definition 1.4 (Optimal couple)
The admissible couple , is said to be optimal if and .
The optimal quasi-norms are uniquely determined up to equivalence, while the corresponding optimal quasi-Banach spaces are unique.
2 Admissible couples
Here we give a characterization of all admissible couples . It is convenient to define the case as limiting and the case as sublimiting.
Theorem 2.1 (General case )
The couple is admissible if and only if
where
Proof First we prove
We are going to use real interpolation for quasi-Banach spaces. First we recall some basic definitions. Let be a couple of two quasi-Banach spaces (see [2, 5]) and let
be the K-functional of Peetre (see [2]). By definition, the K-interpolation space has a quasi-norm
where Φ is a quasi-normed function space with a monotone quasi-norm on with the Lebesgue measure and such that . Then (see [5])
where by we mean that X is continuously embedded in Y. If , , , we write instead of (see [2]).
Using the Hardy-Littlewood inequality , we get the well-known mapping property
and by the Minkovski inequality for the norm we get
Hence
therefore (see [2], Section 5.7)
implies
It is clear that (1.7) follows from (2.1) and (2.3).
Now we prove that (1.7) implies (2.1). To this end we choose the test function in the form , , so that for some positive constant c (cf. [6]). Then
whence
Note that , where
and
hence is decreasing, therefore
Thus, if (1.7) is given, then (2.4) implies (2.1). □
In the sublimiting case we can simplify the condition (2.1), replacing by . Here
Theorem 2.2 (Sublimiting case )
The couple is admissible if and only if
where we recall that
Proof Let , be an admissible couple, then
Since , it follows that , . Now we need to prove sufficiency of (2.6). We have
so
implies
□
In the subcritical case we have another simplification of (2.1).
Theorem 2.3 (Case )
The couple is admissible if and only if
where
Proof Let be admissible, then
As
we have
For the reverse, it is enough to check that (2.7) implies (2.1) for , or
As
so
Here we use
□
2.1 Optimal quasi-norms
Here we give a characterization of the optimal domain and optimal target quasi-norms. We can define an optimal target quasi-norm by using Theorem 2.1.
Definition 2.4 (Construction of the optimal target quasi-norm)
For a given domain quasi-norm we set
Then
Theorem 2.5 Let be a given domain quasi-norm. Then , the couple , is admissible and the target quasi-norm is optimal. By definition,
Proof To see that is a quasi-norm, we first prove (1.6), for that we first prove
Take and consider an arbitrary such that, for , . By the Hardy inequality . Then,
Hence
Taking the infimum over all h such that , we get (2.10). Hence , also . And these two together give (1.6). is indeed a quasi-norm on . Since , which gives . Also
Hence , is admissible couple. Now we are going to prove that is optimal. For this purpose, suppose that the couple is admissible. Then by Theorem 2.1,
Therefore if , , then
so taking the infimum on the right-hand side, we get
hence . □
In the sublimiting case we can simplify the optimal target quasi-norm.
Theorem 2.6 If be a given domain quasi-norm. Then for ,
i.e.,
Proof If , , then , therefore
and taking the infimum, we get
Now for the reverse, let , .
Then
so
which gives, since ,
and this implies
which gives
Taking the infimum, we get , hence . □
A simplification of the optimal target quasi-norm is possible also in the subcritical case .
Theorem 2.7 Let be a given domain quasi-norm. Then for ,
i.e.,
Proof If , , then
Therefore
and taking the infimum, we get
For the reverse, let . Then . As
we get
whence
where we use
Therefore, taking the infimum we arrive at
□
We can construct an optimal domain quasi-norm by Theorem 2.1 as follows.
Definition 2.8 (Construction of an optimal domain quasi-norm)
For a given target quasi-norm , we construct an optimal domain quasi-norm by
Theorem 2.9 If is a given target quasi-norm, then the domain quasi-norm is optimal. Moreover, if , then the couple , is optimal.
Proof Since , so
it follows that is a quasi-norm. To prove the property (1.5), we notice that
The couple , is admissible since . Moreover, is optimal, since for any admissible couple we have , . Therefore,
To check that if , the couple , is optimal, we need only to prove that is an optimal target quasi-norm, i.e., , where is defined by (2.11), since . We have , where , , therefore,
implies
since
so
Now we define
For , since , we have , therefore . Also and . Therefore,
For , since we have , therefore using , Minkowski’s inequality, and monotonicity of , we have
Thus
hence . □
Example 2.10 If consists of all bounded continuous functions such that and , then and , i.e., and the couple E, G is optimal.
Example 2.11 Let with and let
Then, the couple E, G is optimal and . In particular, this is true if v is slowly varying since then and .
2.2 Subcritical case
Here we suppose that .
Theorem 2.12 (Sublimiting case )
For a given domain quasi-norm with , we have
i.e.,
Moreover, the couple , is optimal.
Proof If , , then for , , whence
Taking the infimum, we get
For the reverse, we notice that , hence .
It remains to prove that the domain quasi-norm is also optimal. Let , be an admissible couple in . Then
□
Now we give an example.
Example 2.13 Let
where and are slowly varying. Then we have , . Now by applying the previous theorem, we get
and the couple is optimal.
In the limiting case , the formula for the optimal target quasi-norm is more complicated.
Theorem 2.14 (Limiting case)
Let
where is a monotone quasi-norm with , , and let
Define
Then the couple , is optimal.
Proof Note that
Indeed, . Hence the above embedding follows. Consequently, . On the other hand,
Hence . This together with gives . Then from the conditions on it follows that . Also, and . On the other hand, if , then
For every , we can find a , such that for all . Then for ,
Now it is easy to check that if .
To prove that we need to check that the couple , is admissible. We write for ,
Then
To prove that the target space is optimal, notice first that
If , then by [2]
If then obviously and
whence
On the other hand,
implies , which gives , which implies , and therefore
proving optimality of G. To check optimality of E, we notice that
Hence
□
Example 2.15 Let , consisting of all such that as , w is slowly varying. Then . If , where and
then this couple is optimal. In particular, if , then and .
References
Köthe G: Topologisch Lineare Räume. Springer, Berlin; 1966.
Bergh J, Löfström J: Interpolation Spaces, an Introduction. Springer, Berlin; 1976.
Bennett C, Sharpley R: Interpolation of Operators. Academic Press, Boston; 1988.
Muckenhoupt B: Hardy’s inequality with weights. Stud. Math. 1972, 44: 31–38.
Brudnyı̌ YA, Krugliak NY: Interpolation Spaces and Interpolation Functors. North-Holland, Amsterdam; 1991.
Cianchi A: Symmetrization and second-order Sobolev inequalities. Ann. Mat. Pura Appl. 2004, 183: 45–77. 10.1007/s10231-003-0080-6
Acknowledgements
This study was supported by research funds from Dong-A University.
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Kang, S.M., Rafiq, A., Nazir, W. et al. Optimal couples of rearrangement invariant spaces for the Riesz potential on the bounded domain. J Inequal Appl 2014, 60 (2014). https://doi.org/10.1186/1029-242X-2014-60
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DOI: https://doi.org/10.1186/1029-242X-2014-60