Convolution is continuous
Let $f,g\in L^2\left(\mathbb T,\mathbb C\right)$ (Hilbert space of $1$-periodic functions) then $f*g$ should be continuous by Young's inequality (the map is $(f,g)\mapsto f*g$)
$$\big\lvert f*g(x)-f*g(z)\big\rvert=\left\lvert\int_0^1f(y)\left(g(x-y)-g(z-y)\right)dy\right\rvert.$$
Do you have an idea how to apply the inequality now?