On discrete fractional integral operators and related Diophantine equations

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.

14 pages, (much) revised version

References in corpus (3)

• Vinogradov's mean value theorem via efficient congruencing
• Vinogradov's mean value theorem via efficient congruencing, II
• On discrete fractional integral operators and related Diophantine equations

Cited by in corpus (1)

• On discrete fractional integral operators and related Diophantine equations