It is a simple fact that $L^p$-functions cannot grow arbitrarily fast. More precisely, one has for every $\ell>0$ $$ |\{f\geq\ell\}|\leq \frac{\|f\|_{L^p}^p}{\ell^p} $$ for every $f\in L^p$.
My question is: Can this result be improved if e.g. $f\in W^{1,p}$? From intuition I'd expect something like $ |\{f\geq\ell\}|\leq \frac{\|f\|_{W^{1,p}}^p}{\ell^{p+1}} $...