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The boundedness of Fourier transform on the Herz type amalgams and Besov spaces
Journal of Inequalities and Applications volume 2014, Article number: 296 (2014)
In this paper, using the Young inequality as regards the Herz space, the authors obtain the boundedness of the Fourier transform on the Herz type amalgams and Besov spaces. As corollaries, the authors get the estimates of the Fourier transform on the weighted amalgams and Besov spaces.
One of the central problems in classic harmonic analysis is the study of the boundedness of Fourier transform on a given function space. A basic result is the Hausdorff-Young inequality and its various extensions and generalizations. Here and below we define the Fourier transform of to be
Theorem A If and , then
Theorem B If and , then
Here denotes a Beurling-Herz space.
In , the authors give a result as regards the Young theorem for amalgams and Besov spaces.
Theorem C (1) Let , and . Then
Let , and . Then
Inspired by Theorem B and Theorem C, in this paper, we discuss the boundedness as regards the Fourier transform on the Herz type amalgams and the Besov spaces. In Section 2, we give the definition of function spaces. The main theorems and their proofs are contained in Section 3.
2 Function spaces
In this section, we give the definition of function spaces that we work on. Let and be a Schwartz space and its dual space.
Definition 2.1 (Weighted Lebesgue spaces)
Let be a nonnegative function, , where
Now we give the definitions as regards Herz spaces.
Definition 2.2 Suppose , , . The Herz space is defined by
where , ,
The usual modification should be made when or (see ).
Definition 2.3 Let be even functions satisfying the following condition:
We set for .
For , we denote . Now let us introduce the Herz type Besov spaces (see ).
Definition 2.4 Let , , , and , ; then we define
Obviously, the definition of the Herz-Besov space is independent of the choice of , and , standard Besov spaces. More information as regards the Besov space and Herz type Besov spaces can be found in [7, 9–12].
Definition 2.5 (Herz type amalgam space)
Let , , , , and . Set , the translation of the unit cube. For a Lebesgue locally integrable function f we define
where . is the set of all locally integrable functions f for which the quasi-norm .
3 Main theorems and proofs
In this section we formulate our main theorems and proofs.
Theorem 3.1 Let , , and . Then the Fourier transform is bounded from the Herz type amalgam space to the Herz type Besov space . Here and below, for we write .
Proof of Theorem 3.1 If we note that the lift operator is bounded from to (see ) and the following multiplication operator is an isomorphism:
we can assume .
Let , , where is the Littlewood-Paley function. Using the Young inequality , we obtain
where we use the inequality (see ).
If we put these estimates together, we have
Given , from the definition of the , there are at most three j such that . So we have
This is the desired result. □
Theorem 3.2 Let , , and . Then the Fourier transform is bounded from the Herz type Besov space to the Herz type amalgams space .
Proof of Theorem 3.2 As before, we assume . Let . By the definition of the Herz type amalgam space, we have
When , from the definition of , there are at most three k () such that . If and , we have
Using the above inequality and (see ) we have
This proves Theorem 3.2. □
We note that (see ) and , . Then we have the following.
Corollary 3.3 Let , , , and . Then the Fourier transform is bounded from the weighted amalgam space to the weighted Besov space .
Corollary 3.4 Let , , , and . Then the Fourier transform is bounded from the weighted Besov space to the weighted amalgam space .
If and in Theorem 3.1, we have the following.
Corollary 3.5 Let and . Then the Fourier transform is bounded from the Herz space to the Herz type Besov space .
Next we deduce some necessary condition as regards the boundedness of the Fourier transform.
Proposition 3.6 Let , , , and . If
Proof of Proposition 3.6 Let be an even function with . We take , . So . Since (), it follows that . From this we have ().
Since , we have . But . So
That is to say, if and only if . forces . From this it follows that , which is equivalent to . □
Proposition 3.7 Let , , , and . If
Proof of Proposition 3.7 We estimate and . We have
Here we use (see ); through we have . This completes the proof of Proposition 3.7. □
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Project (Grant Nos. 11261055, 11161044) was supported by NSFC, and SF of Xinjiang (Grant Nos. 2011211A005, BS120104).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Zhou, J., Cao, Y. The boundedness of Fourier transform on the Herz type amalgams and Besov spaces. J Inequal Appl 2014, 296 (2014). https://doi.org/10.1186/1029-242X-2014-296
- Fourier transform
- Herz-Besov space
- Herz-amalgam space