Search for: All records

Endpoint Estimates are obtained for the Discrete Prime averages. 

Abstract Consider averages along the prime integers ℙ given by {\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓ p -improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds {N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}. The maximal function 𝒜 * f = sup N |𝒜 N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known. 

Let $ Tf =\sum _{I} \varepsilon _I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $ \lvert \varepsilon _I\rvert =1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, $\displaystyle \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} .$ Above, we use the one-sided $ A_2$ characteristic for the weight $ w$. This is an instance of a one-sided $ A_2$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $ A_2$ theorem do not seem to apply in the one-sided setting.