Fourier-Bessel Transforms of Dini-Lipschitz Functions on Lebesgue Spaces $L_{p,\gamma}(\mathbb{R}^{n}_{+})$

In this paper, we prove a generalization of Titchmarsh's theorem for the Laplace-Bessel differential operator in the space $L_{p,\gamma}(\mathbb{R}^{n}_{+})$ for functions satisfying the $(\psi,p)$-Laplace-Bessel Lipschitz condition for $1 0 $.


Introduction
Integral transforms and their inverse transforms are widely used to solve various problems in calculus, fourier analysis, mechanics, mathematical physics, and computational mathematics. Fourier transform is one of the most important integral transforms. Since it was introducted by Fourier in the early 1880s, it has become an important mathematical concept that is at the centre of the highly developed branch of mathematics called Fourier Analysis. It has many application areas. The Fourier transform of the kernel of singular integral operator is very important in applications of singular integral operator theory. The properties of the Fourier transform of the kernel give information about the existence of the solution of singular integral equations. Since singular integrals are convolution type operators, their Fourier transforms are the product of the Fourier transforms of two functions.
As it is well known that if Lipschitz conditions are applied on a function f (x), then these conditions greatly a¤ect the absolute convergence of the Fourier-Bessel series and behaviour of F f Fourier-Bessel transforms of f . In general, if f (x) belongs to a certain function class, then the Lipschitz conditions have bearing as to the dual space to which the Fourier coe¢ cients and Fourier-Bessel transforms of f (x) belong. Younis (see [12]) worked the same phenomena for the wider Dini Lipschitz class for some classes of functions. Daher, El Quadih, Daher and El Hamma proved an analog Younis (see [12,Theorem 2.5]) in for the Fourier-Bessel transform for functions satis…es the Fourier-Bessel Dini Lipschitz condition in the 848 ISM AIL EKINCIOGLU, ESRA KAYA, AND S. ELIFNUR EKINCIOGLU Lebesgue space L 2 ;n (see [10]). El Hamma and Daher proved a generalization of Titchmarsh's theorem for the Bessel transform in the space L 2; (R n + ) (see [1]) . In this paper we prove a generalization of Titchmarsh's theorem for the Laplace-Bessel transform in the space L p; (R n + ), where 1 < p 2 and > 0.

Preliminaries
Let R n + be the part of the Euclidean space R n of points x = (x 1 ; :::; x n ), de…ned by the inequality x n > 0. We write x = (x 0 ; x n ); x 0 = (x 1 ; : : : ; x n 1 ) 2 R n 1 + . S n + denote the unit sphere on R n + , which can be de…ned as S n + = fx 2 R n + : jxj = 1g. S + = S(R n + ) be the space of functions which are the restrictions to R n + of the test functions of the Schwartz that are even with respect to x n , decreasing su¢ ciently rapidly at in…nity, together with all derivatives of the form : : : ::: @ n 1 @x n 1 n 1 B n n ; i.e., for all ' 2 S + , sup are multi-indexes, and x = x 1 1 : : : x n n and B n = @ 2 @x 2 n + x n @ @x n is the Bessel di¤erential expansion. For 0, we introduce the Bessel normalized function of the …rst kind j de…ned by (1) where is the gamma-function (see [9]). Moreover, from (1) we see that The function u = j 1 2 (z) satis…es the di¤erential equation with the initial conditions u(x; 0) = f (x) and u y (x; 0) = 0 is function in…nitely di¤erentiable, even, and, moreover entire analytic. The Fourier-Bessel transformation and its inverse on S + are de…ned by where (x 0 ; y 0 ) = x 1 y 1 +: : :+x n 1 y n 1 , j , > 0, is the normalized Bessel function, and C n; = (2 ) n 1 2 1 2 (( + 1)=2); (see [4,9,11]). This transform is associated to the Laplace-Bessel di¤erential operator The expression (3) is a hybrid of the Hankel transform. For a …xed parameter > 0, let L p; = L p; (R n + ) be the space of measurable functions with a …nite norm The space of the essentially bounded measurable function on R n + is denoted by . For for f 2 L p; , I.A. Kipriyanov (for n = 1 B.M. Levitan [7,8]) investigated the generalized convolution ( -convolution) associated with the Laplace-Bessel di¤erential operator, where T y is the generalized shift operator ( -shift) de…ned by 2x n y n cos + y 2 n sin 1 d ; being C = 1 2

Fourier-Bessel Transforms of Dini-Lipschitz Functions
In this section we give the main result of this paper. We need …rst to de…ne ( ; p)-Laplace Bessel Lipschitz class. where (x) is a continuous increasing function on R n + , (0) = 0, and (xs) = (x) (s) for all x; s 2 R n + . Theorem 2. Let f (x) belong to Lip( ; ; p). Then Z j j jF f ( )j q n d = O( ( q )); as ! +1: Proof. Let f 2 Lip( ; ; p). Then we have Now we consider Fourier-Bessel transform of generalized shift operator. We get . From formulas (4) and (5), we obtain Z It follows from the above consideration that there exists a positive constant C such that In fact, we have Z j j<1 C q ( q ) 1 + (2 q ) + 2 (2 q ) + 3 (2 q ) + : : : : Thus, the proof of theorem is completed.
We can give the following result which is used for many the theorem given above. It is well known that ; i = 1; : : : ; n 1; We can use the mathematical induction method for k = 1, we get @ 2 f (y) @y 2 k + y n @f (y) @y n e ix 0 y 0 j 1 2 (x n y n )y n dy = C n; y n @f (y) @y n e ix 0 y 0 j 1 2 (x n y n )y n dy = I 1 + I 2 : If we apply partial integration to the second term of I 1 and I 2 , then we have F u (x) = C n; Z R n + f (y)e ix 0 y 0 j 1 2 (x n y n ) y n dy: Here, if we use the following equality [8], as ! +1.
There are many examples. Here is one of them and a simple method to produce many more: f (x) = jxj 1 p for 1 < p < 1, where f (0) = 0 is understood. These functions are uniformly continuous on all of R n + . If p = 2, f belongs to the Lipschitz class at R + .