C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 69, N umb er 1, Pages 847–853 (2020) D O I: 10.31801/cfsuasm as.673191
ISSN 1303–5991 E-ISSN 2618-6470
http://com munications.science.ankara.edu.tr
FOURIER-BESSEL TRANSFORMS OF DINI-LIPSCHITZ
FUNCTIONS ON LEBESGUE SPACES Lp; (Rn+)
ISMAIL EKINCIOGLU, ESRA KAYA, AND S. ELIFNUR EKINCIOGLU
Abstract. In this paper, we prove a generalization of Titchmarsh’s theorem for the Laplace-Bessel di¤erential operator in the space Lp; (Rn+)for functions satisfying the ( ; p)-Laplace-Bessel Lipschitz condition for 1 < p 2 and
> 0.
1. Introduction
Integral transforms and their inverse transforms are widely used to solve various problems in calculus, fourier analysis, mechanics, mathematical physics, and com-putational 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 sin-gular integral equations. Since sinsin-gular 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
Received by the editors: January 10, 2020; Accepted: March 06, 2020. 2010 Mathematics Subject Classi…cation. Primary 42B10; Secondary 26A16.
Key words and phrases. Laplace-Bessel di¤erential operator, generalized shift operator, Laplace-Bessel Lipschitz function.
c 2 0 2 0 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a th e m a t ic s a n d S ta t is t ic s
Lebesgue space L2
;n (see [10]). El Hamma and Daher proved a generalization of
Titchmarsh’s theorem for the Bessel transform in the space L2; (Rn+) (see [1]) .
In this paper we prove a generalization of Titchmarsh’s theorem for the Laplace-Bessel transform in the space Lp; (Rn+), where 1 < p 2 and > 0.
2. Preliminaries Let Rn
+ be the part of the Euclidean space Rn of points x = (x1; :::; xn), de…ned
by the inequality xn > 0. We write x = (x0; xn); x0 = (x1; : : : ; xn 1) 2 Rn 1+ . S+n
denote the unit sphere on Rn+, which can be de…ned as S+n = fx 2 Rn+ : jxj = 1g.
S+ = S(Rn+) be the space of functions which are the restrictions to Rn+ of the test
functions of the Schwartz that are even with respect to xn, decreasing su¢ ciently
rapidly at in…nity, together with all derivatives of the form D = Dx00Bnn = D11:::Dn 1n 1Bnn= @ 1 @x 1 1 : : : ::: @ n 1 @x n 1 n 1 B n n ;
i.e., for all ' 2 S+, sup x2Rn
+
x D ' <1, where = ( 1; :::; n) and = ( 1; :::; n)
are multi-indexes, and x = x 1
1 : : : xnn and Bn = @2 @x2 n + xn @ @xn is the Bessel di¤erential expansion. For 0, we introduce the Bessel normalized function of the …rst kind j de…ned by
j (z) = ( + 1) 1 X n=0 ( 1)n n! (n + + 1) z 2 2n ; (1)
where is the gamma-function (see [9]). Moreover, from (1) we see that lim
z!0
j 1 2 (z) 1
z2 6= 0
by consequence, there exist C > 0 and > 0 satisfying
jzj ) j 1
2 (z) 1 C jzj
2: (2)
The function u = j 1
2 (z) satis…es the di¤erential equation
Bxnu(x; y) = Bynu(x; y)
with the initial conditions u(x; 0) = f (x) and uy(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
F f (x) = Z Rn f (y) e i(x0y0)j 1 2 (xnyn) yndy; F 1f (x) = Cn; F f ( x0; xn);
where (x0; y0) = x
1y1+: : : +xn 1yn 1, j , > 0, is the normalized Bessel function,
and
Cn; = (2 )n 12 1 2(( + 1)=2);
(see [4, 9, 11]). This transform is associated to the Laplace-Bessel di¤erential op-erator = n X i=1 @2 @x2 i + xn @ @xn ; > 0: (3)
The expression (3) is a hybrid of the Hankel transform.
For a …xed parameter > 0, let Lp; = Lp; (Rn+) be the space of measurable
functions with a …nite norm kfkLp; = Z Rn + jf(x)jpxndx !1=p ; 1 p < 1: The space of the essentially bounded measurable function on Rn
+ is denoted by
L1; (Rn
+). For for f 2 Lp; , I.A. Kipriyanov (for n = 1 B.M. Levitan [7, 8])
investigated the generalized convolution ( -convolution) (f g)(x) =
Z
Rn +
f (y) Tyg(x) yndy;
associated with the Laplace-Bessel di¤erential operator, where Tyis the generalized
shift operator ( -shift) de…ned by Tyf (x) = C Z 0 f x0 y0;px2 n 2xnyncos + yn2 sin 1 d ; being C = 12 +1
2 [ 2 ] 1 (see [5, 6, 7, 8]). We note that this convolution
satis…es the property (f g)(x) = (g f )(x) (see [2, 3]). The following relation connect the generalized shift operator and the Fourier-Bessel transform, we have
F [Tyf (x)] = j 1
2 (xnyn)F [f (x)]: (4)
Given 1 < p 2, 1 p+
1
q = 1 and f 2 Lp; , we have the Hausdor¤-Young inequality
kF fkq; Cqkfkp; ; (5)
where and Cq is a positive constant.
3. 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.
De…nition 1. A function f 2 Lp; (Rn+) is said to be in the ( ; p)-Laplace Bessel
Lipschitz class, denoted by Lip( ; ; p), if
where (x) is a continuous increasing function on Rn
+, (0) = 0, and (xs) =
(x) (s) for all x; s 2 Rn +.
Theorem 2. Let f (x) belong to Lip( ; ; p). Then Z
j j jF f( )j q
nd = O( ( q)); as ! +1:
Proof. Let f 2 Lip( ; ; p). Then we have
kTyf (x) f (x)kp; = O( (y)) as y ! 0:
Now we consider Fourier-Bessel transform of generalized shift operator. We get F [Tyf (x)]( ) = Z Rn + Tyf (x)j 1 2 (xn n)xndx = Z Rn + Ty j 1 2 (xn n) f (x)xndx = Z Rn + j 1 2 (xn n)j 1 2 (yn n)f (x)xndx = j 1 2 (yn n) Z R+ n f (x)j 1 2 (xn n)xndx = j 1 2 (yn n)F (f )( ); where Ty(j p( p x)) = jp( p y)jp( p
x). From formulas (4) and (5), we obtain Z Rn + F jTyf (x) f (x)jqxndx = Z Rn + jF Tyf (x) F f (x)jqxndx = Z Rn + jj 1 2 ( y)F f ( ) F f ( )j q nd = Z Rn + jF f( )[1 j 1 2 ( y)]j q nd = Z Rn + j1 j 1 2 ( y)j q jF f( )jq nd Cq Z Rn + jTyf (x) f (x)jq nd CqkTyf (x) f (x)kqp; : From (2), we have Z 1 h j j 2 h jF f( )jq nd = Cq Z 1 h j j 2 h j1 j 1 2 ( h)j q jF f( )jq nd Cqjhj 1 Z 1 h j j h2 jF f( )jq nd ;
0 < h 1. It follows from the above consideration that there exists a positive constant C such that
Z 1 h j j 2 h jF f( )jq nd C q(h) = C (hq): Therefore, we get Z j j 2 jF f( )j q nd C ( q): In fact, we have Z j j<1jF f( )j q nd = 1 X k=1 Z 2k 1 j j<2k jF f( )j q nd Cq ( q) + Cq (2 ) q + Cq (22 ) q + : : : Cq ( q) 1 + (2 q) + 2(2 q) + 3(2 q) + : : : :
Thus, we can write Z
j j<1jF f( )j q
nd C1 ( q);
where C1= Cq(1 (2 q)) 1 since 2 q < 1. Finally, we get
Z
j j jF f( )j q
nd = O( ( q)) as ! 1:
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
F (B n n f ) (x) = ( x2n) nF f (x); (6) F D2 i i f (x) = ( x2i) iF f (x); i = 1; : : : ; n 1; (7) F ( f ) (x) = jxj2F f (x) and F (f g) = F f F g; (8) F D2x00Bnnf (x) = ( 1)j jx2 F f (x) (9)
We can use the mathematical induction method for k = 1, we get F f (x) = Cn; Z Rn + f (y)e ix0y0j 1 2 (xnyn)yndy = Cn; Z Rn + n X k=1 @2f (y) @y2 k + yn @f (y) @yn e ix0y0j 1 2 (xnyn)yndy
= Cn; Z Rn + n X k=1 @2f (y) @y2 k e ix0y0j 1 2 (xnyn)yndy Cn; Z Rn + n X k=1yn @f (y) @yn e ix0y0j 1 2 (xnyn)yndy = I1+ I2:
If we apply partial integration to the second term of I1and I2, then we have
F u (x) = Cn; Z Rn + f (y)e ix0y0 j 1 2 (xnyn) yndy:
Here, if we use the following equality [8], Z 1 0 f (y) j 1 2 (xy)y dy = jxj 2 Z 1 0 f (y)j 1 2 (xy)y dy then we have F ( f ) (x) = jxj2F f (x): Since f 2 Lip( ; ; p), it is clear that
jjF ( f ) jjLq; (j j ) Cn; O( (
q))
as ! +1.
There are many examples. Here is one of them and a simple method to produce many more: f (x) = jxj1p for 1 < p < 1, where f(0) = 0 is understood. These
functions are uniformly continuous on all of Rn
+. If p = 2, f belongs to the Lipschitz
class at R+.
References
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Current address : Ismail Ekincioglu: Department of Mathematics, Dumlupinar University, Kutahya, Turkey.
E-mail address : ismail.ekincioglu@dpu.edu.tr
ORCID Address: https://orcid.org/0000-0002-5636-1214
Current address : Esra Kaya: Department of Mathematics, Dumlupinar University, Kutahya, Turkey.
E-mail address : kayaesra.e.k@gmail.com
ORCID Address: https://orcid.org/0000-0002-6971-0452
Current address : S. Elifnur Ekincioglu: Department of Mathematics, Dumlupinar University, Kutahya, Turkey.
E-mail address : ekinciogluelifnur@gmail.com
