Small cap square function estimates

Abstract

We introduce small cap square function estimates for parabola and cone, and prove the sharp estimates. More precisely, we study the inequalities of form \[ \|f\|_p\le C_{α,p}(R) \Big\|(\sum_{γ\inΓ_α(R^{-1})}|f_γ|^2)^{1/2}\Big\|_p, \] where ${\Gamma }_{\alpha }(R^{-1})$ is the set of small caps of width $R^{-\alpha }$. We find the sharp constant $C_{\alpha ,p}(R)$.

24 pages, to appear in JFAA