International Journal of Nonlinear Analysis and Applications
Bessel transforms of Dini-Lipschitz functions on Lebesgue spaces $L_{p,\gamma}(\mathbb{R}^{n}_{+})$
Document Type : Research Paper
Authors
Department of Mathematics, Kutahya Dumlupinar University, Kutahya, Turkey
Abstract
In this paper, we obtain a generalization of Titchmarsh's theorem for the Bessel transform for functions satisfying the $(\psi,p)$-Bessel Lipschitz condition in the Lebesgue space $L_{p,\gamma}(\mathbb{R}^{n}_{+})$ for $1< p\leq2$, $\gamma>0$.
Keywords
[1] I. Ekincioglu, I. K. Ozkin, On high order Riesz transformations generated by a generalized shift operator, Tr. J. Math. 21 (1997) 51–60.
[2] I. Ekincioglu, A. Serbetci, On the singular integral operators generated by the generalized shift operator, Int. J. App. Math. 1 (1999) 29–38.
[3] M. El Hamma, R. Daher, Generalization of Titchmarsh’s Theorem for the Bessel transform, Rom. J. Math. Comput. Sci. 2 (2012) 17–22.
[4] S. El Quadih, R. Daher and M. El Hamma, Generalization of Titchmarsh’s Theorem for the Generalized FourierBessel Transform in the Space L2 α,n, International Journal of Mathematical Modelling Computations, 6 (2016) 253–260.
[5] I. A. Kipriyanov, Fourier-Bessel transformations and imbedding theorems, Trudy Math. Inst. Steklov. 89 (1967) 130–213.
[6] I. A. Kipriyanov, Singular Elliptic Boundary Value Problems, Nauka, Moscow, Russia, 1997.
[7] I. A. Kipriyanov, M. I. Klyuchantsev, On singular integrals generated by the generalized shift operator II, Sib. Mat. Zh. 11 (1970) 1060–1083. English translation: Sib. Math. J. 11 (1970) 787–804.
[8] B. M. Levitan, Bessel function expansions in series and Fourier integrals, Uspekhi Mat. Nauk 6. 42 (1951) 102–143.
[9] B. M. Levitan, Expansion in Fourier series and integrals over Bessel functions, Uspekhi Mat. Nauk 6. 2 (1951) 102–143.
[10] K. Trim`eche, Transmutation operators and mean-periodic functions associated with differential operators, Math. Rep. 4 (1988) 1–282.
[11] M. S. Younis, Fourier transforms of Dini-Lipschitz functions, Internat. J. Math. & Math. Sci. 9 (1986) 301–312.
[2] I. Ekincioglu, A. Serbetci, On the singular integral operators generated by the generalized shift operator, Int. J. App. Math. 1 (1999) 29–38.
[3] M. El Hamma, R. Daher, Generalization of Titchmarsh’s Theorem for the Bessel transform, Rom. J. Math. Comput. Sci. 2 (2012) 17–22.
[4] S. El Quadih, R. Daher and M. El Hamma, Generalization of Titchmarsh’s Theorem for the Generalized FourierBessel Transform in the Space L2 α,n, International Journal of Mathematical Modelling Computations, 6 (2016) 253–260.
[5] I. A. Kipriyanov, Fourier-Bessel transformations and imbedding theorems, Trudy Math. Inst. Steklov. 89 (1967) 130–213.
[6] I. A. Kipriyanov, Singular Elliptic Boundary Value Problems, Nauka, Moscow, Russia, 1997.
[7] I. A. Kipriyanov, M. I. Klyuchantsev, On singular integrals generated by the generalized shift operator II, Sib. Mat. Zh. 11 (1970) 1060–1083. English translation: Sib. Math. J. 11 (1970) 787–804.
[8] B. M. Levitan, Bessel function expansions in series and Fourier integrals, Uspekhi Mat. Nauk 6. 42 (1951) 102–143.
[9] B. M. Levitan, Expansion in Fourier series and integrals over Bessel functions, Uspekhi Mat. Nauk 6. 2 (1951) 102–143.
[10] K. Trim`eche, Transmutation operators and mean-periodic functions associated with differential operators, Math. Rep. 4 (1988) 1–282.
[11] M. S. Younis, Fourier transforms of Dini-Lipschitz functions, Internat. J. Math. & Math. Sci. 9 (1986) 301–312.
)
Volume 12, Issue 2
November 2021Pages 563-568
- Receive Date: 01 October 2019
- Accept Date: 12 April 2021