On bounded energy of convolution of fractal measures

For all $s\in [0,1]$ and $t\in (0,s]\cup [2-s,2)$, we find the supremum of numbers $\omega \in (0,2)$ such that ${\text{I}}_{\omega }(\mu \ast \sigma )\lesssim 1$, where $\mu$ is any Borel measure on $B(1)$ with ${\text{I}}_t(\mu )\le 1$ and $\sigma$ is any $(s,1)$-Frostman measure on a $C^2$-graph with non-zero curvature. As an application, we use this to show the sharp $L^6$-decay of Fourier transform of $\sigma$ when $s\in [\frac{2}{3},1]$.