Small cap square function estimates

We introduce small cap square function estimates for parabola and cone, and prove the sharp estimates. More precisely, we study the inequalities of form $$\mathrm{\parallel }f{\mathrm{\parallel }}_p\le C_{\alpha ,p}(R)\parallel (\sum _{\gamma \in {\mathrm{\Gamma }}_{\alpha }(R^{-1})}\mathrm{\mid }{f}_{\gamma }{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}{\parallel }_p,$$ where ${\mathrm{\Gamma }}_{\alpha }(R^{-1})$ is the set of small caps of width $R^{-\alpha }$. We find the sharp constant $C_{\alpha ,p}(R)$.