Here's the statement of the problem:
Let $p \in (1, \infty)$. Prove that there exists $C > 0$ such that $\forall f \in L^p(\mathbb{R}^d)$ \begin{eqnarray} \frac{1}{C}\sum_{j=1}^d \|\partial_j f\|_p \le \|\sqrt{-\Delta}f \|_p \le C\sum_{j=1}^d\| \partial_j f \|_p \end{eqnarray}
Here's the idea, but I cannot fill out the details. Any helps or suggestions will be appreciated.
First, we consider $d = 1$.
Next,for $d \ge 1$, consider symbol $m_j(\xi)$ = $\frac{\xi_j}{\lvert \xi \rvert}$, and $T_jf = \check{m} \ast f$, check that $T_j \in B(L^p(\mathbb{R}^d))$.
Show that $T_j \sim \frac{\partial_j}{\sqrt{-\Delta}}$.
Then, need Riesz transform(some explanation about this transform will be great). \begin{eqnarray} \sqrt{-\Delta}f = (\xi \mapsto \lvert\xi\rvert\hat{f}(\xi))\check{} \end{eqnarray}
If possible, need some information about how to start such a question, and how to find the constant C. Thank you in advance.