The entropy of a convolution over the hypercube
There is no such $C$. Define $g\colon\mathbb{Z}_2^n\to\mathbb{R}$ by
$$g(x_1,\dots,x_n)=\begin{cases}
2^{2n/3}&amp;\text{ if $x_1=\dots=x_n=0$}\\
1&amp;\text{ otherwise.}\end{cases}$$

Then $g*g$ satisfies
$$(g*g)(x_1,\dots,x_n)=\begin{cases}
2^{4n/3}+2^n-1&amp;\text{ if $x_1=\dots=x_n=0$}\\
2^{2n/3}\cdot 2+2^n-2&amp;\text{ otherwise.}\end{cases}$$

Let $f=g/\|g\|_2$. Then $H(f)=H(g/\|g\|_2)$ is $o(1)$ (in fact it's exponentially small in $n$), while $H(g*g/\|g*g\|_2)$ is about $n$.