On discrete fractional integral operators and related Diophantine equations

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 $\gamma (m) = (m, m^2, \dotsc , m^k)$ to Vinogradov’s mean value theorem. Sharp $l^p \to l^q$ estimates of the discrete fractional integral along the hyperbolic paraboloid in $\mathbb{Z}^3$ are also obtained except for endpoints.

Full Text (PDF format)

Published 20 May 2015