On bounded energy of convolution of fractal measures

For all \(s∈[0,1]\) and \(t∈(0,s]∪ [2-s,2)\), we find the supremum of numbers \(ω∈(0,2)\) such that \({I}_ω(μσ) 1,\) where \(μ\) is any Borel measure on \(B(1)\) with \({I}_t(μ)≤ 1\) and \(σ\) 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 \(σ\) when \(s∈ [2/3, 1]\).