Bessel Transform of -Bessel Lipschitz Functions

Using a generalized translation operator, we obtain an analog of Theorem 5.2 in Younis (1986) for the Bessel transform for functions satisfying the -Bessel Lipschitz condition in . 1. Introduction and Preliminaries Younis Theorem 5.2 [1] characterized the set of functions in satisfying the Cauchy Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms; namely, we have the following. Theorem 1 (see [1]). Let . Then the followings are equivalent:(1) , , ,(2) , where stands for the Fourier transform of . In this paper, we obtain a generalization of Theorem 1 for the Bessel transform. For this purpose, we use a generalized translation operator. Assume that ; is the Hilbert space of measurable functions on with finite norm Let be the Bessel differential operator. For , we introduce the Bessel normalized function of the first kind defined by where is the gamma function (see [2]). The function satisfies the differential equation with the initial conditions and . is function infinitely differentiable, even, and, moreover, entirely analytic. Lemma 2. For the following inequality is fulfilled: with , where is a certain constant which depends only on . Proof. Analog of Lemma 2.9 is in [3]. Lemma 3. The following inequalities are valid for Bessel function :(1) , for all , (2) . Proof. See [4]. The Bessel transform we call the integral transform from [2, 5, 6] The inverse Bessel transform is given by the formula We have the Parseval's identity In , consider the generalized translation operator defined by where The following relations connect the generalized translation operator and the Bessel transform; in [7] we have 2. Main Result In this section we give the main result of this paper. We need first to define -Bessel Lipschitz class. Definition 4. Let and . A function is said to be in the -Bessel Lipschitz class, denoted by Lip( , , 2), if Our main result is as follows. Theorem 5. Let . Then the followings are equivalents(1) ？？Lip . (2) . Proof. Assume that Lip( , , 2). Then we have If then and Lemma 2 implies that Then We obtain where is a positive constant. So that where since . This proves that Suppose now that We write where Estimate the summands and from above. It follows from the inequality that To estimate , we use the inequality of Lemma 3. Set Using integration by parts, we obtain where are positive constants and this ends the proof.