Is $L^p(X^2) \cong L^p(X) \otimes L^p(X)$ true?

Question

I was wondering if for $X$ a compact set of $\mathbb{R}^d$ with $d \in \mathbb{N}$: $p \in [1,+\infty]$ (especially for the case $p=2$ !) one have $$L^p(X^2) \cong L^p(X) \otimes L^p(X)$$