Square-like functions generated by the Laplace-Bessel differential operator
© Kele¿ and Bayrakç¿; licensee Springer. 2014
Received: 24 June 2014
Accepted: 19 October 2014
Published: 31 October 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.
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.
In the case , we identify with the space of continuous functions vanishing at infinity, and set .
where is the normalized Bessel function, which is also the eigenfunction of the Bessel operator ; and (see ).
- (a)If , then for every ,
- (b)If , , then and
where is independent of f.
being defined by (2.2) and .
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]).
where is the well-known Hardy-Littlewood maximal function.
(see for details ).
where is known as ‘wavelet function’, , and the function is the generalized Gauss-Weierstrass semigroup.
3 Main theorems and proofs
- (a)Let , (), . We have(3.1)
- (b)Let , (). We have(3.2)
where limit can be interpreted in the norm and pointwise for almost all . If , the convergence is uniform on .
- (b)Let , . Applying Fubini’s theorem, 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, , .
and the proof is complete. □
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).
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