On discrete fractional integral operators and related Diophantine equations

TL;DR

This paper investigates discrete fractional integral operators along curves and surfaces, establishing $l^p o l^q$ bounds by connecting them to solutions of Diophantine equations, notably relating to Vinogradov's mean value theorem.

Contribution

It provides new sharp $l^p o l^q$ estimates for discrete fractional integrals along specific curves and surfaces, linking harmonic analysis with number theory.

Findings

Bounds for fractional integral operators along polynomial curves.

Connection to Vinogradov's mean value theorem.

Sharp estimates for hyperbolic paraboloid in $ extbf{Z}^3$.

Abstract

We study discrete versions of fractional integral operators along curves and surfaces. $l^p \to l^q$ estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the discrete fractional integral along the curve $\gamma(m) = (m,m^2,...,m^k)$ to Vinogradov's mean value theorem. Sharp $l^p \to l^q$ estimates of the discrete fractional integral along the hyperbolic paraboloid in $\mathbf{Z}^3$ are also obtained except for endpoints.