Inequalities related to the third Jackson q-Bessel function of order zero
© Elmonser et al.; licensee Springer 2013
Received: 24 October 2012
Accepted: 20 May 2013
Published: 6 June 2013
In Bettaibi and Bouzeffour (J. Math. Anal. Appl. 342:1203-1219, 2008), some properties of the third Jackson q-Bessel function of order zero were established. This paper is devoted to studying the q-convolution product by using a q-integral representation of the related q-translation.
The central part of this work is first to study the related q-heat semi-group and its hypercontractivity and second to specify the q-analogue of the Wiener algebra.
1 Introduction: notations and preliminaries
In harmonic analysis the positivity of the translation operator is crucial. It plays a central role in establishing some useful results such as the properties of the convolution product.
In contrast to the classical theory, the positivity of the translation operator associated to the normalized q-Bessel function of order α is not clear at this stage. In fact it is still an open conjecture to find and α which assure the positivity of the related translation. For , it was proved that the q-translation is not positive for all (see ). However, for , the authors proved (see ) that the q-translation is positive for all . This fact helps us to study the harmonic analysis associated to the third Jackson q-Bessel function of order zero and to establish some important inequalities.
This paper is organized as follows. In Section 2 we begin by summarizing some statements concerning the q-translation operator studied in . Then we prove some facts about the positivity and the x-continuity of for an appropriate space and we give an integral representation.
In Section 3, we recall some basic properties of the q-convolution product cited in . Then we establish some results of density.
In Section 4, we study the q-Fourier Bessel transform : after recalling some results in  and by the use of the inversion formula, we prove that we can extend the definition of to and by density to , .
Sections 5 and 6 are reserved to study the q-analogue of some well-known results associated to the heat semi-group and the Wiener algebra.
1.2 Notations and preliminaries
and provided exists.
provided the sums converge absolutely.
- (1)For all , we have(6)
- (2)For and , we have(7)
- (3)Both sides of (6) are majorized by(8)
where , satisfy , and . We have the following behavior (see ).
For all , we have as .
In literature, some authors (see ) developed some elements of q-harmonic analysis related to the normalized q- function using a transmutation operator.
In this paper, we are concerned with , the third Jackson q-Bessel function of order zero. We construct a product formula for this function leading to a positive q-translation which is necessary and constructive for some applications.
We need the following spaces and sets:
, and .
the space of restrictions on of even functions f such that for all , we have and for all , we have as in .
the space of restrictions on of even functions with bounded support such that for all , we have as in .
the space of restrictions on of even functions, for which as in and as in , equipped with the norm
the space of restrictions on of even functions for which as in and(13)
, , the set of all functions defined on such that(14)
, the set of all functions defined on such that(15)
2 A q-generalized translation
provided the sum converges.
The kernel K satisfies the following properties.
It was shown in  that the generalized q-translation satisfies the following results.
The q-generalized translation is positive.
For , , , .
Using the proprieties of the kernel K and the definition of the generalized q-translation, one can state the following results.
- (4)For , we have(29)
The following result is useful for the remainder.
- (1)For , and , we have and(30)
- (2)For and , we have and(31)
which achieves the proof. □
Proof The result follows from the previous proposition, the properties of the q-generalized translation and the Lebesgue theorem. □
3 q-Bessel Fourier transform
In the following propositions, we summarize some of its properties (see ).
- (1)For , we have(35)
- (2)For , we have(37)
- (3)If and as in , then(38)
Theorem 1 (Plancherel formula)
Using this result and the relation (37), one can state the following proposition.
which can be extended for .
where and , .
Proposition 8 For , is dense in .
Proof It suffices to consider functions with compact supports on . □
4 q-convolution product
It satisfies the following properties (see ).
In this section, we shall prove that the notion of q-convolution product can be extended to functions in space. We begin by the following result.
- (2)The function and(42)
Let be such that . For a bounded subset E of , we note the characteristic function of E.
Then the function is integrable on with respect to the measure . From Fubini’s theorem we deduce that for all , the mapping belongs to , and the mapping belongs also to . Then the function is measurable.
This completes the proof. □
Proof From the Holder inequality and Proposition 4, we have, for , and . The continuity of at 0 follows from Corollary 1. □
In the same way, we have the following result.
which gives the result by the use of Corollary 1. □
Now, let us adopt the following notation:
- (i)For all , , we have(47)
- (ii)For all , we have(48)
- (ii)We have, for all ,
5 The q-analogue of the heat semi-group
In this section, we are concerned with the q-analogue of the heat semi-group associated with the third Jackson q-Bessel function of order zero and we define it on by the following.
- (2)By the use of Plancherel formula and Theorem 1, we obtain
Using this second equality and equalizing terms by terms, we obtain the following lemma.
Now, we are in a situation to state some properties of the heat semi-group .
- (1)For all ,(56)
- (2)For all such that and , we have(57)
Follows from the positivity of the q-generalized translation and the fact that . □
Since , then Proposition 10 implies that can be extended to , and we have the following.
So, the result follows by using this equality and Proposition 10. □
where if and (see ).
Using this equality and Proposition 14, one can state the following result.
From the relation (59) and the fact that is a q-analogue of the classical exponential function, we can see that as a q-analogue of the classical heat semi-group.
- (2)For two formal -commuting variables t and (), we have
which proves that is a q-analogue of the classical heat semi-group.
6 The q-analogue of the Wiener algebra
We begin by the following results which are useful in the sequel.
where is defined in (45).
and the following lemma. □
Finally, the dominated convergence theorem achieves the proof. □
In the following result, we summarize some of its density properties.
Proof Let , then and .
- (1)Let . For , there exists an with compact support in such that . By using Theorem 3, we have
with C is some constant.
Let . For , there exists an with compact support in such that .
If tends to , then
If tends to 0, then
Since the sequence is increasing, the use of Fatou Beppo-Levi theorem achieves the result. □
is a linear isomorphism with norm 1.
is with norm bounded by .
Finally, the use of the Riesz-Thorin theorem gives the result (see ). □
where , and are defined by (72).
If and , we consider two sequences and in which converge to f and g respectively in and . From i) we have .
We deduce that the sequence converges to in . Theorem 6 implies that the sequence converges to in .
From the last proposition we deduce the hypercontractivity of the q-analogue of the heat semi-group .
where , , is given by (50) and .
The result follows from the fact that and . □
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