Bounding the product of lipschitz function [closed]

Question

Assume we know that $f(x,y): \mathbb{R}^{a+b} \to \mathbb{R}^d$ is lispchitz w.r.t. $y$, i.e.

$$\|f(x,y) - f(x,y')\| \leq L \|y-y'\|.$$

Is there a way to bound the product $f(x,y)\cdot f(x,y')^T \in\mathbb{R}^{d\times d}$?

Answer 1

Say $f(x,y)=y$. Then $f(x,y)^2=y^2$, which is not Lipschitz in $y$. So the answer is negative.