Small Cap Square Function Estimates

We introduce and prove small cap square function estimates for the unit parabola and the truncated light cone. More precisely, we study inequalities of the form \(\begin{aligned} \Vert f\Vert _p\le C_{\alpha ,p}(R) \Big \Vert \Big (\sum _{\gamma \in \Gamma _\alpha (R^{-1})}|f_\gamma |^2\Big )^{1/2}\Big \Vert _p, \end{aligned}\) $$\begin{array}{c}\hfill {\Vert f\Vert }_p\le C_{\alpha ,p}(R)\Vert (\sum _{\gamma \in {\mathrm{\Gamma }}_{\alpha }(R^{-1})}{|f_{\gamma }|}^2{)}^{1/2}{\Vert }_p,\end{array}$$ where \(\Gamma _\alpha (R^{-1})\) ${\mathrm{\Gamma }}_{\alpha }(R^{-1})$ is the set of small caps of width \(R^{-\alpha }\) $R^{-\alpha }$ . We find sharp upper and lower bounds of the constant \(C_{\alpha ,p}(R)\) $C_{\alpha ,p}(R)$ .