How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where $\mathbb{T}=[0,1)$ and $f,g\in L^2(\mathbb{T})$
Many thanks