Discrete fractional integral operators with binary quadratic forms as phase polynomials

TL;DR

This paper investigates discrete fractional integral operators associated with binary quadratic forms, providing new estimates for cases lacking translation invariance, which were previously unexplored in the field.

Contribution

It introduces the first estimates for discrete fractional integral operators with non-translation-invariant phase polynomials, expanding the understanding of these operators.

Findings

Established bounds for operators with non-translation-invariant phases

Extended the theory beyond classical translation-invariant cases

Paved the way for further analysis of quadratic form-based operators

Abstract

We give estimates on discrete fractional integral operators along binary quadratic forms. These operators have been studied for 30 years starting with the investigations of Arkhipov and Oskolkov, but efforts have concentrated on cases where the phase polynomial is translation invariant or quasi-translation invariant. This work presents the first results for operators with neither translation invariant nor quasi-translation invariant phase polynomials.