Averages Along the Primes: Improving and Sparse Bounds

Abstract

Abstract Consider averages along the prime integers ℙ given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo>𝒜</m:mo> </m:mrow> <m:mi>N</m:mi> </m:msub> <m:mi>f</m:mi> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:munder> <m:mo>∑</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mo>𝕇</m:mo> <m:mo>:</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>N</m:mi> </m:mrow> </m:munder> <m:mrow> <m:mo>(</m:mo> <m:mo>log</m:mo> <m:mi>p</m:mi> <m:mo>)</m:mo> <m:mi>f</m:mi> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>p</m:mi> <m:mo>)</m:mo> <m:mo>.</m:mo> </m:mrow> </m:mrow> </m:math> {\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 &lt; p &lt; 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:msup> <m:mi>p</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:msup> <m:msub> <m:mrow> <m:mrow> <m:mrow> <m:mo>‖</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mo>𝒜</m:mo> </m:mrow> <m:mi>N</m:mi> </m:msub> <m:mi>f</m:mi> </m:mrow> <m:mo>‖</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>𝓁</m:mi> <m:msup> <m:mi>p</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:msub> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mi>p</m:mi> </m:msub> <m:msup> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:msub> <m:mrow> <m:mrow> <m:mrow> <m:mo>‖</m:mo> <m:mi>f</m:mi> <m:mo>‖</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>𝓁</m:mi> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mo>.</m:mo> </m:mrow> </m:math> {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 &lt; p &lt; 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.

Locations

• Concrete Operators - View - PDF
• arXiv (Cornell University) - View - PDF