Let $\Omega \subset R^n$ ($n \geq 3$) be a bounded domain with smooth boundary. Let $n>p\geq 2$ and $u \in W^{1,p}(\Omega)$. Suppose that exists a natural number $j \geq 1$ such that $(|u| - j)^+ \in W^{1,p}_{0}(\Omega).$ Define the set $A_j = \{ x \in \Omega; |u(x) | > j\} $ and consider a number $q > n$ and a function $f \in L^{q}(\Omega).$
I am reading a paper and the author says: applying Holder inequality and the Gagliardo-Nirenberg inequality we have
$$ \int_{A_j} |f| (|u| - j)^+ \leq \| f\|_{L^{\frac{p}{p-1}}(A_j)} \| (|u| - j)^{+}\|_{L^{p}(A_j)} \leq \| f\|_{L^{q}(A_j)}|A_j|^{1 - \frac{1}{q} - \frac{1}{p^*}} \| \nabla u \|_{L^{p}(A_j)}, $$
The first one I obtained by a direct aplication of Holder inequality. With respect to the second inequality, I can obtain the inequality if the inequality below is true
$$ \| (|u| - j)^{+}\|_{L^{p}(A_j)} \leq |A_j|^{\frac{1}{n}} \| \nabla u \|_{L^{p}(A_j)}.$$
The above inequality remembers the Gagliardo-Nirenberg inequality, but I don't know how to obtain such inequality.
Someone could help me to obtain the second inequality?
You need a constant in there: $$\| (|u| - j)^{+}\|_{L^{p}(A_j)} \leq C|A_j|^{\frac{1}{n}} \| \nabla u \|_{L^{p}(A_j)}$$ Let $q =(1/p-1/n)^{-1}$. By the Sobolev-Poincaré inequality (using the fact that the function $w:=(|u| - j)^{+}$ has zero trace), we have $$\| w\|_{L^{q}(A_j)} \leq C \| \nabla w \|_{L^{p}(A_j)} \le C \| \nabla u \|_{L^{p}(A_j)}$$ By Hölder's inequality, $$\| w\|_{L^{p}(A_j)} \leq |A_j|^{1/n}\| w\|_{L^{q}(A_j)}$$