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Boundedness of localization operators on Lorentz mixed-normed modulation spaces
Journal of Inequalities and Applications volume 2014, Article number: 430 (2014)
In this work we study certain boundedness properties for localization operators on Lorentz mixed-normed modulation spaces, when the operator symbols belong to appropriate modulation spaces, Wiener amalgam spaces, and Lorentz spaces with mixed norms.
In this paper we will work on with Lebesgue measure dx. We denote by the space of complex-valued continuous functions on rapidly decreasing at infinity. For any function , the translation and modulation operator are defined as and for , respectively. For , we write the Lebesgue spaces .
Let be the usual scalar product on . The Fourier transform (or ) of is defined to be
For a fixed nonzero the short-time Fourier transform (STFT) of a function with respect to the window g is defined as
for . Then the localization operator with symbol a and windows , is defined to be
If and , then the localization operator is a well-defined continuous operator from to . Moreover, it is to be interpreted in a weak sense as
Fix a nonzero window and . Then the modulation space consists of all tempered distributions such that the short-time Fourier transform is in the mixed-norm space . The norm on is . If , then we write instead of . Modulation spaces are Banach spaces whose definitions are independent of the choice of the window g (see [2, 3]).
spaces are function spaces that are closely related to spaces. We consider complex-valued measurable functions f defined on a measure space . The measure μ is assumed to be nonnegative. We assume that the functions f are finite valued a.e. and some , , where . Then, for ,
is the distribution function of f. The rearrangement of f is given by
for . The average function of f is also defined by
Note that , , and are nonincreasing and right continuous functions on . If is continuous and strictly decreasing then is the inverse function of . The most important property of is that it has the same distribution function as f. It follows that
The Lorentz space denoted by (shortly ) is defined to be vector space of all (equivalence classes) of measurable functions f such that , where
By (1.1), it follows that and so . Also, is a normed space with the norm
For any one of the cases ; or and , the Lorentz space is a Banach space with respect to the norm . It is also well known that if , we have
Let X and Y be two measure spaces with σ-finite measures μ and ν, respectively, and let f be a complex-valued measurable function on , , and . The Lorentz mixed norm space is defined by
Fix a window function , , and . We let denote the subspace of tempered distributions consisting of such that the Gabor transform of f is in the Lorentz mixed norm space . We endow it with the norm , where is the norm of the Lorentz mixed norm space. It is well known that is a Banach space and different windows yield equivalent norms. If and , then the space is the standard modulation space , and if and , in this case (see [8, 9]), where the space is Lorentz type modulation space (see ). Furthermore, the space was generalized to by taking weighted Lorentz space rather than Lorentz space (see [11, 12]).
In this paper, we will denote the Lorentz space by , the Lorentz mixed norm space by , the standard modulation space by , the Lorentz type modulation space by , and the Lorentz mixed-normed modulation space by .
Let . Fix a compact with nonempty interior. Then the Wiener amalgam space with local component and global component is defined as the space of all measurable functions such that for each compact subset , for which the norm
is finite, where is the characteristic function of K and
In this paper, we consider boundedness properties for localization operators acting on Lorentz mixed-normed modulation spaces for the symbols in appropriate function spaces like modulation spaces, Wiener amalgam spaces, and Lorentz spaces with mixed norms. Our results extend some results in [1, 12] to the Lorentz mixed-normed modulation spaces.
2 Boundedness of localization operators on Lorentz mixed normed modulation spaces
We start with the following lemma, which will be used later on.
Lemma 2.1 Let , , , . Then and
with the norm inequality
where , , .
Proof It is well known that there are convolution relations between Lorentz spaces and
Let be a test function such that . Let be a translation invariant Banach space of functions with the property that . In the spirit of [13, 16], the Wiener amalgam space with local component X and global component is defined as the space of all functions or distributions for which the norm
is finite, where , . Moreover, different choices of yield equivalent norms and give the same space.
The boundedness of for is established by our next theorem. The proof is similar to Lemma 4.1 in  but let us provide the details anyway, for completeness’ sake.
Let , . If and , then with
Let , . If and , then with
Proof (i) Let and set . By using the equality , we write
This completes the proof.
Using Lemma 2.1 and (2.3), we have
Theorem 2.2 Let , . If , , then is bounded on for every with
Thus by using (2.5) we have
Hence we get
Theorem 2.3 Let be a window function. If , , and , then
is bounded for every , where , , and , and the operator norm satisfies the estimate
Proof Let , , and . Then we have and . Since , then . By using the equality (3.6) in , we get
Hence we have . Similarly, . By the Hölder inequality for Lorentz spaces with mixed norm and (2.6) we have
Since , then for every . Also since , then we have
By using (2.7), (2.8), and applying again the Hölder inequality, we get
If , then by Theorem 8 in . Thus we have from (2.9) that
Hence is bounded. Also we have
Theorem 2.4 Let , where , . If and then
is bounded for every , with
for some .
Proof Since , then for every . Also since , there exists such that . Then and
for all . Let be the space of the bounded linear operators from into . Also let T be an operator from into by . Take any and . Assume that . By the Hölder inequality we get
Hence by (2.11)
Thus the operator
is bounded. Now let . Take any and . Then , . Applying the Hölder inequality
By using (2.14) we write
Hence by (2.15)
That means the operator
is bounded. Combining (2.13) and (2.16) we obtain
is bounded by interpolation theorem for . That means the localization operator
is bounded for . Hence there exists such that
This implies that it is also true for . From (2.10) and (2.17) we write
Proposition 2.1 Let , where , . If and then
Proof Let and let . Then for every . Since , there exists a number such that . Hence by (2.12),
Then the localization operator from into is bounded for . □
Proposition 2.2 Let , where , . If and then the localization operator
is bounded, where , .
Proof Let . Then for every with . Take any and . Applying the Hölder inequality we have by (2.11)
Similarly to (2.12), we get
Then the localization operator from into is bounded. □
Corollary 2.1 It is known by Proposition 2 in that for . Then . So, Theorem 2.4, Propositions 2.1 and 2.2 are still true under the same hypotheses for them if .
Corollary 2.2 It is known that if and , then Lorentz mixed-normed modulation space is the Lorentz type modulation space . Therefore our theorems hold for a Lorentz type modulation space rather than for a Lorentz mixed-normed modulation space.
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The author declares that she has no competing interests.
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Sandıkçı, A. Boundedness of localization operators on Lorentz mixed-normed modulation spaces. J Inequal Appl 2014, 430 (2014). https://doi.org/10.1186/1029-242X-2014-430
- localization operator
- Lorentz spaces
- Lorentz mixed normed spaces
- Lorentz mixed-normed modulation spaces
- Wiener amalgam spaces