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.
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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