Discrete Fractional Integration Operators Along the Primes

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 $p$.