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

Esra Kaya,Şeyma Eki̇nci̇oglu,İsmail Eki̇nci̇oğlu

doi:10.31801/cfsuasmas.673191

Abstract

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 $.

Highlights

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
The Fourier transform of the kernel of singular integral operator is very important in applications of singular integral operator theory

Summary

Introduction

Integral transforms and their inverse transforms are widely used to solve various problems in calculus, fourier analysis, mechanics, mathematical physics, and computational mathematics. 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 If f (x) belongs to a certain function class, the Lipschitz conditions have bearing as to the dual space to which the Fourier coe¢ cients and Fourier-Bessel transforms of f (x) belong. 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. El Hamma and Daher proved a generalization of Titchmarsh’s theorem for the Bessel transform in the space L2; (Rn+) (see [1]). Bessel transform in the space Lp; (Rn+), where 1 < p 2 and > 0

Preliminaries

Fourier-Bessel Transforms of Dini-Lipschitz Functions