Abstract
Given and sequences of integers and such that , the generalized mixed norm space is defined as those sequences such that where . The necessary and sufficient conditions for a sequence to belong to the space of multipliers , for different sequences and of intervals in , are determined.
1. Introduction
Let be the space of complex valued sequences with the locally convex vector topology given by means of the seminorms where . Given two Banach spaces continuously contained in , we write for the space of multipliers from into . More precisely, We will use the notation and for the sequence where and .
Of course for the classical spaces, one easily sees that where . We use the notation to mean whenever and whenever .
The above result can be extended (see [1]) to the class of mixed norm sequence spaces, denoted , which are defined by the condition
Theorem 1. Let . Then
In particular, the Köthe dual of , defined by , becomes for and .
Also multipliers between sequence spaces given by Taylor coefficients of holomorphic functions in the disk have been deeply studied in the literature. Since the time of Hardy and Littlewood, mixed norm and related spaces have been used to study function spaces on the unit disk and later to study multipliers between such spaces. Special emphasis has been put on the case where the spaces and correspond to the sequence space of Taylor coefficient of analytic functions such as Hardy spaces, Bergman spaces, mixed norm spaces of analytic functions, and so forth. The theory of Hardy spaces and mixed norm spaces of analytic functions was originated in the work of Hardy and Littlewood (see [2, 3]) who implicitly considered the spaces of functions such that Their work on these spaces was continued by Flett and Sledd (see [4–8]) and later on by Pavlović (see [9, 10]). Multipliers on Hardy spaces were in fashion for a long time and much work was done on them and related spaces. However nowadays complete descriptions of multipliers between Hardy spaces for certain values of and remain still open. The reader is referred to the surveys (see [11, 12]) for lots of results and references. Also many results on multipliers between mixed norm spaces of analytic functions have been established in the last decades (see [13–15] and references thereby). For such a purpose, the use of solid spaces (sequence spaces whose norm depends only on the size of the coefficients), and in particular spaces, is a rather important tool. It is worth mentioning that the smallest solid space contained or which contains one of classical Hardy, Bergman, and is actually for some values , , and (see [14, 15]) and this last space can be identified with certain weighted , due to Plancherel’s theorem.
Another appearance of mixed norm spaces comes with the use of lacunary sequences, that is, , such that for a sequence of integers satisfying . Recently (see [16]) the description of the Taylor coefficient of analytic functions , where is a lacunary sequence, belonging to the weighted Bergman-Besov space has been achieved under certain conditions on the weight. It corresponds again with certain weighted .
In this paper, we consider families of intervals where for some increasing sequences and such that and we use the notation . We will introduce the spaces given by sequences verifying and the obvious modifications for or .
In particular, for . Also a lacunary sequence corresponds to where with (that is, ) for some .
We will give the necessary and sufficient conditions for a sequence to belong to the multiplier space whenever . We also get some applications to multipliers between certain weighted mixed norm spaces of analytic functions. The paper is organized as follows. Section 2 contains the definitions and first properties of the spaces , studying inclusions between them and conditions for coincidence results . Section 3 contains the main result, which is split into three subsections: the case where intervals in are union of intervals in , to be denoted , the case where for each there exists such that either or , and finally the case where there exists such that and . In Section 4, we include some application to multipliers on spaces of analytic functions and extend some recent result on weighted Bergman-Besov classes.
From now on, we will write whenever there exists such that and, as usual, stands for the cardinal of , for and also denotes a constant that may vary from line to line.
2. Generalized Mixed Norm Spaces
Definition 2. Let and let be a collection of disjoint intervals in , say , where . One sets . One writes for the space of sequences verifying This space becomes a Banach space under the norm with the obvious modifications for or .
Remark 3. Of course . In particular, whenever .
An elementary approach, using Hölder’s inequality, leads to the duality
for and .
Remark 4. It is clear that in the case and also in the case .
Moreover, for ,
Remark 5. Let .(i)If is a subcollection of intervals in , then .(ii)If for two disjoint collections and , then .
We would like to analyze the embedding between and .
Proposition 6. Let be a collection of disjoint intervals in and let with . Then (with equivalent norms) if and only if
In particular, if , then
Proof. ) Assume, for instance, and that for all supported in . Hence, taking , one concludes that for any . Hence .
) Note that and assume . Then
since in .
Proposition 7. Let and let be a collection of disjoint intervals in with .
Then if and only if and .
Proof. ) Assume that there exists such that for all supported in . Hence, taking and , one concludes that . Hence . Let and consider . Applying the above inequality, we obtain . Therefore, .
) Let us denote
Hence the mapping
is an isometric embedding from into . Taking into account that for any Banach space and , we conclude that
Therefore
We would like to analyze the embedding between and for whenever .
Proposition 8. Let and . If , (respect. ), and (respect. ), then
Proof. Proposition 6 gives , and clearly . Then the result follows using whenever .
Let us mention another particular case where they coincide.
Proposition 9. Let be such that with and define Then .
Proof. Note that is again an interval in . Using that for , then
On the other hand, using now for ,
The previous idea is easily generalized using the following definition.
Definition 10. Let and . One says that if the following conditions hold:(i);(ii) for all ;(iii) for all .
Proposition 11. Let and . Then(i) for ;(ii) for .
Moreover, the embeddings above are of norm .
Proof. (i) Case : let and . We know that there is such that . Hence
This gives .
The case : let and . Therefore
The case follows using (9) and the previous one.
(ii) The case : let . Then
To cover the remaining cases, from (9), we simply need to show that for . Now observe that
Theorem 12. Let and with .
(with equivalent norms) if and only if .
Proof. ) Assume that for all finitely supported. Let and define
Then and .
One concludes that which implies, in the case , .
) Case . From Proposition 11, we only need to show . Using now Hölder’s inequality for ,
Therefore, if , we have
Case . Using again Proposition 11, we will show . Using ,
Let us now exhibit an example where neither nor .
Example 13. Let and take as shown in Figure 1 with
Let us see that neither nor .
Taking we have
Then it is enough to consider and . Now and, since , Hence we have and .
We would like to explain a procedure to analyze the general case .
Definition 14. Let and be families of disjoint intervals in with . For each , one uses the notation, as above, which now might be empty. One also defines We write and for the mappings given by
Similarly, interchanging and , we define , , , and .
Definition 15. One defines the “left” and “right” part of the interval by and, denoting and , one has where whenever .
Similarly, interchanging and , we consider , , , and .
With this notation out of the way we can classify intervals in into four different types (according to ). Note that for each interval there are four possibilities: coincides with for some , can be written as a union of at least two intervals in , is strictly contained into some interval , or there exists which overlaps with and its complement .
Therefore we decompose into four disjoint sets defined as follows.
Definition 16. Let and be families of disjoint intervals in with . One introduces
We define the sets , , , and similarly.
Remark 17. Using (38), we can also give a description of the sets above in terms of and :
Using the above decomposition, we can generalize Propositions 8 and 11 and Theorem 12. Note that implies and also that corresponds to the case where or equivalently for any .
Theorem 18. Let and be collections of intervals such that
Proof. ) Arguing as in Theorem 12, for we consider
Hence
Therefore, using that and , we conclude that .
) Denote and let .
Case . If , then
If , we have
This shows .
Case : arguing as in Proposition 11, we simply show that for .
Observe that
Also we have
Finally
Combining the above estimates, we conclude this implication.
Corollary 19. Let and be such that
The next result can be achieved using duality, but we include a direct proof.
Theorem 20. Let and be such that
Proof. ) Repeat the argument presented in the direct implication of Theorem 18.
) Denote again .
Case . Observe first that if , we have
for some . Since for , we obtain
Also if , then where . Note that for some and
Hence
On the other hand,
Combining the previous cases, we get .
Case . Arguing as in Proposition 11, we simply show that for . Consider
Now observe that
Also note, since ,
Finally