The boundedness of Fourier transform on the Herz type amalgams and Besov spaces
© Zhou and Cao; licensee Springer. 2014
Received: 20 January 2014
Accepted: 11 July 2014
Published: 19 August 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.
Here denotes a Beurling-Herz space.
In , the authors give a result as regards the Young theorem for amalgams and Besov spaces.
- (2)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)
Now we give the definitions as regards Herz spaces.
The usual modification should be made when or (see ).
We set for .
For , we denote . Now let us introduce the Herz type Besov spaces (see ).
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)
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 .
we can assume .
where we use the inequality (see ).
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 .
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.
Proof of Proposition 3.6 Let be an even function with . We take , . So . Since (), it follows that . From this we have ().
That is to say, if and only if . forces . From this it follows that , which is equivalent to . □
Here we use (see ); through we have . This completes the proof of Proposition 3.7. □
Project (Grant Nos. 11261055, 11161044) was supported by NSFC, and SF of Xinjiang (Grant Nos. 2011211A005, BS120104).
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