# Discrete Fractional Integration Operators Along the Primes

## TL;DR

This paper establishes boundedness properties of discrete fractional integration operators along primes, extending harmonic analysis techniques to number-theoretic settings with implications for understanding prime-related operators.

## Contribution

It proves boundedness of fractional integration operators along primes in discrete settings, a novel extension of classical harmonic analysis to prime number sequences.

## Key findings

- Operators are bounded on ^p to ^{p'} under specified conditions.
- Provides new bounds for prime-based fractional operators.
- Extends harmonic analysis techniques to prime number sequences.

## Abstract

We prove that the discrete fractional integration operators along the primes \[ T^{\lambda}_{\mathbb{P}}f(x) := \sum_{p} \frac{f(x-p)}{p^{\lambda}} \cdot \log p \] are bounded $\ell^p\to \ell^{p'}$ whenever $ \frac{1}{p'} < \frac{1}{p} - (1-\lambda), \ p > 1.$ Here, the sum runs only over prime $p$.