On Some Multipliers Related to Discrete Fractional Integrals

Abstract

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler's identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy--Littlewood circle method, and a discrete analogue of the Stein--Weiss inequality on product space through implication methods, we establish $\ell^p\rightarrow\ell^q$ bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators.