If $f \in \mathbf{L}_{2}^{2}(\mathbf{R}^2)$, show that $f$ satisfies the Zygmund class estimate:
$|f(x+2y)-2f(x+y)+f(x)|\leq c|y|$.
I know that $f\in \mathbf{L}_{2}^{2}(\mathbf{R}^2)$ satisfies the Lipschitz class, but how do you show it for the zygmund class?
Appreciate the help!