Discrete Fractional Integration Operators Along the Primes

Abstract

We prove that the discrete fractional integration operators along the primes \[ T^λ_{\mathbb{P}}f(x) := \sum_{p} \frac{f(x-p)}{p^λ} \cdot \log p \] are bounded ${\ell }^p\to {\ell }^{p^{\prime }}$ whenever $\frac{1}{p^{\prime }}<\frac{1}{p}-(1-\lambda ),p>1.$ Here, the sum runs only over prime $p$.

References in corpus (2)

• ${\ell }^p\left({\mathbb{Z}}^d\right)$-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates
• ${\ell }^p$-improving inequalities for Discrete Spherical Averages