Optimal couples of rearrangement invariant spaces for the Riesz potential on the bounded domain
© Kang et al.; licensee Springer. 2014
Received: 15 October 2013
Accepted: 20 January 2014
Published: 10 February 2014
We prove continuity of the Riesz potential operator in optimal couples of rearrangement invariant function spaces defined in with the Lebesgue measure.
KeywordsRiesz potential operator rearrangement invariant function spaces real interpolation
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.
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 ).
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 .
For example, if E is a rearrangement invariant Banach function space as in , 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 , p.305.
If is monotone, then the function is submultiplicative, increasing, , , hence . If is K-monotone, then by interpolation (analogously to , p.148), we see that . Hence in this case we have also .
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)
Moreover, is called domain quasi-norm (domain space), and (G) is called a target quasi-norm (target space).
Definition 1.2 (Optimal target quasi-norm)
for any target quasi-norm such that the couple , is admissible.
Definition 1.3 (Optimal domain quasi-norm)
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 )
where by we mean that X is continuously embedded in Y. If , , , we write instead of (see ).
It is clear that (1.7) follows from (2.1) and (2.3).
Thus, if (1.7) is given, then (2.4) implies (2.1). □
Theorem 2.2 (Sublimiting case )
In the subcritical case we have another simplification of (2.1).
Theorem 2.3 (Case )
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)
hence . □
In the sublimiting case we can simplify the optimal target quasi-norm.
Now for the reverse, let , .
Taking the infimum, we get , hence . □
A simplification of the optimal target quasi-norm is possible also in the subcritical case .
We can construct an optimal domain quasi-norm by Theorem 2.1 as follows.
Definition 2.8 (Construction of an optimal domain quasi-norm)
Theorem 2.9 If is a given target quasi-norm, then the domain quasi-norm is optimal. Moreover, if , then the couple , is optimal.
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.
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 )
Moreover, the couple , is optimal.
For the reverse, we notice that , hence .
Now we give an example.
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)
Then the couple , is optimal.
Now it is easy to check that if .
then this couple is optimal. In particular, if , then and .
This study was supported by research funds from Dong-A University.
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