Let $\mathbb{T}^n := \bigl(\mathbb{R}/\mathbb{Z} \bigr)^n$ be $n$-dimensional torus.
Then, I wonder if $(-\Delta)^{1/2}$ and $\nabla$ are "equivalent" in $L^p(\mathbb{T}^n)$ for each $p \in (1,\infty)$.
That is, for any $f \in W^{1,p}(\mathbb{T}^n)$, do we have some constants $C_1,C_2>0$ depending only on $p$ and $n$ such that \begin{equation} C_1\lVert (-\Delta)^{1/2} f \rVert_p \geq \lVert \nabla f \rVert_p \end{equation} and \begin{equation} C_2 \lVert \nabla f \rVert_p \geq \lVert (-\Delta)^{1/2} f \rVert_p \end{equation}
For $p=2$, I see that $\lVert (-\Delta)^{1/2} f \rVert_2=\lVert \nabla f \rVert_2$. I wonder what happens in general case, except for endpoints.