We know that by Sobolev embedding (the critical case), we have \[ W^{1,n}(\mathbb R^n) \hookrightarrow BMO(\mathbb R^n), \] and we would have to ask the following two questions. **Question 1.** For a function $f\in W^{1,n}(\mathbb R^n)$ and a smooth function $\chi\in \mathcal C(\mathbb R)$ with $\chi(0)=0$, do we have $\chi(f)\in BMO(\mathbb R^n)$? Or what kind of condition should we imposed on $\chi$ to ensure that $\chi(f)\in BMO(\mathbb R^n)$? **Question 2.** For a smooth function $\chi\in \mathcal C(\mathbb R)$ with $\chi(0)=0$, what kind of conditions should we impose on $\chi$ to ensure the following inequality hold: \[ \norm{ \chi(f) g}_{L^p(\mathbb R^n)} \lesssim \norm{\chi(f)}_{BMO(\mathbb R^n)} \norm{ g }_{L^p(\mathbb R^n)},\qquad \forall\, f\in W^{1,n}(\mathbb R^n),\,\forall\,g\in L^p(\mathbb R^n). \] **Remark.** The first interesting case should be when $n=2$.