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# Mathematical Research Letters

## Volume 22 (2015)

### Number 3

### On discrete fractional integral operators and related Diophantine equations

Pages: 841 – 857

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n3.a11

#### Author

#### 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 $\gamma (m) = (m, m^2, \dotsc , m^k)$ 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.