A note on small cap square function and decoupling estimates for the parabola

In this paper, we prove small cap square function and decoupling estimates for the parabola, where the small caps are essentially axis-parallel rectangles of dimensions $\delta \times {\delta }^{\beta }$ for $0\le \beta \le 1$. Our estimates complement the known results for $1\le \beta \le 2$ and are sharp up to polylogarithmic factors.