On Some Multipliers Related to Discrete Fractional Integrals
Jinhua Cheng

TL;DR
This paper investigates multipliers related to discrete fractional integrals, uncovering their connections with number theory and establishing bounds using harmonic analysis techniques.
Contribution
It introduces new bounds for discrete fractional integral multipliers and explores their deep links with number theory and harmonic analysis.
Findings
Established $ ext{ell}^p o ext{ell}^q$ bounds for the operators
Revealed connections with Dirichlet characters and Dedekind zeta functions
Enhanced understanding of the interplay between number theory and harmonic analysis
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 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.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Holomorphic and Operator Theory · Mathematical functions and polynomials
