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Square-like functions generated by the Laplace-Bessel differential operator
Advances in Difference Equations volume 2014, Article number: 281 (2014)
We introduce a wavelet-type transform associated with the Laplace-Bessel differential operator and the relevant square-like functions. An analogue of the Calderón reproducing formula and the boundedness of the square-like functions are obtained.
MSC:47G10, 42C40, 44A35.
The classical square functions , where , is the Schwartz test function space and , , , play important role in harmonic analysis and its applications; see Stein . There are a lot of diverse variants of square functions and their applications; see Daly and Phillips , Jones et al. , Pipher , Kim . Square-like functions generated by a composite wavelet transform and its estimates are proved by Aliev and Bayrakci .
Note that the Laplace-Bessel differential operator is known as an important operator in analysis and its applications. The relevant harmonic analysis, known as Fourier-Bessel harmonic analysis associated with the Bessel differential operator , has been the research area for many mathematicians such as Levitan, Muckenhoupt, Stein, Kipriyanov, Klyuchantsev, Löfström, Peetre, Gadjiev, Aliev, Guliev, Triméche, Rubin and others (see [7–14]). Moreover, a lot of mathematicians studied a Calderón reproducing formula. For example, Amri and Rachdi , Guliyev and Ibrahimov , Kamoun and Mohamed , Pathak and Pandey , Mourou and Trimèche [19, 20] and others.
In this paper, firstly we introduce a wavelet-like transform associated with the Laplace-Bessel differential operator,
and then the relevant square-like function. The plan of the paper is as follows. Some necessary definitions and auxiliary facts are given in Section 2. In Section 3 we prove a Calderón-type reproducing formula and the boundedness of the square-like functions.
and let be the Schwartz space of infinitely differentiable and rapidly decreasing functions.
(, ; ) space is defined as the class of measurable functions f on for which
In the case , we identify with the space of continuous functions vanishing at infinity, and set .
The Fourier-Bessel transform and its inverse are defined by
where is the normalized Bessel function, which is also the eigenfunction of the Bessel operator ; and (see ).
Denote by () the generalized translation operator acting according to the law:
The generalized convolution ‘B-convolution’ associated with the generalized translation operator is for which
where and . Moreover, the following inequalities are satisfied (see for details ).
If , then for every ,
where is independent of f.
If , , then and
where is independent of f.
Furthermore, if , , then
Now, we will need the generalized Gauss-Weierstrass kernel defined as
being defined by (2.2) and .
The kernel possesses the following properties:
Given a function , the generalized Gauss-Weierstrass semigroup, is defined as
This semigroup is well known and arises in the context of stable random processes in probability, in pseudo-differential parabolic equations and in integral geometry; see Koldobsky, Landkof, Fedorjuk, Aliev, Rubin, Sezer and Uyhan (see [23–26]).
Lemma 2.1 If , (), then
The limit is understood in norm and pointwise almost all . If , then the limit is uniform on .
where is the well-known Hardy-Littlewood maximal function.
Moreover, let be an absolutely continuous function on and
If we denote , we have from (2.13)
(see for details ).
Now, we define the following wavelet-like transform:
where is known as ‘wavelet function’, , and the function is the generalized Gauss-Weierstrass semigroup.
Using wavelet-like transform (2.15), we define the following square-like functions:
3 Main theorems and proofs
Let , (), . We have(3.1)
where , , .
Let , (). We have(3.2)
where limit can be interpreted in the norm and pointwise for almost all . If , the convergence is uniform on .
Theorem 3.2 If , then
Proof of Theorem 3.1 (a) By using the Minkowski inequality, we have
Taking into account the following equality for , , (see [, p.370])
in one dimension. By using this equality, we get
So we have , and then inequality (3.1).
Let , . Applying Fubini’s theorem, we get
By Theorem 1.15 in [, p.3], if (), then
Therefore, by the Minkowski inequality and the Lebesgue dominated convergence theorem, taking into account Lemma 2.1, we have
Finally, for (), we get
Proof of Theorem 3.2 Firstly, let . By making use of the Fubini and Plancherel (for Fourier-Bessel transform) theorems, we get
Now, by using Fubini’s theorem, we have
Since , , , it follows that
Finally, we get
For arbitrary , the result follows by density of the class in . Namely, let be a sequence of functions in , which converge to f in -norm. That is, , .
From the ‘triangle inequality’ (), we have
This shows that the sequence converges to Sf in -norm. Thus
and the proof is complete. □
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The authors would like to thank the referees for their valuable comments. This work was supported by the Scientific Research Project Administration Unit of the Akdeniz University (Turkey).
The authors declare that they have no competing interests.
All authors contributed equally to this work. All authors read and approved the final manuscript.
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Keleş, Ş., Bayrakçı, S. Square-like functions generated by the Laplace-Bessel differential operator. Adv Differ Equ 2014, 281 (2014). https://doi.org/10.1186/1687-1847-2014-281
- square functions
- generalized translation
- wavelet transform
- Calderón reproducing formula