Good morning This question was asked before, but now I precise it to avoid confusion. In fact I have a problem to prove the following inequality: $$\left(\int_Mu^2\right)^{1/2}\leq C\int_M |u|$$ for any $u\in H_1^2(M)$ which staisfies $\int_Mu=0$;
P.S: $C$ is a constant; $H_1^2(M)=\{f\in L^2(M):|\nabla f|\in L^2(M)\}$; and $M$ is a compact Riemannian manifold
Thank you
Not true. For example, let $A$ and $B$ be disjoint closed subsets of $M$, both of measure $\delta/2$, and $u$ a suitable smoothing of $1_A - 1_B$ (the difference of the indicator functions of $A$ and $B$). Then $\int_M |u| \approx \delta$ while $(\int_M u^2)^{1/2} \approx \delta^{1/2}$.