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 \(\textup{I}_\omega(\mu\ast\sigma) \lesssim 1,\) where \(\mu\) is any Borel measure on \(B(1)\) with \(\textup{I}_t(\mu)\leq 1\) 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 \(s\in [\frac{2}{3}, 1]\).