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 \[ \|f\|_p\le C_{\alpha,p}(R) \Big\|(\sum_{\gamma\in\Gamma_\alpha(R^{-1})}|f_\gamma|^2)^{1/2}\Big\|_p, \] where $\Gamma_\alpha(R^{-1})$ is the set of small caps of width $R^{-\alpha}$. We find the sharp constant $C_{\alpha,p}(R)$.