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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

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    As I wrote in one of your other questions, this is the wrong direction for this inequality. Restricting to a Sobolev space cannot help you - $H^2_1$ is dense in $L^2$, so if this was true for all $u\in H^2_1$ it would be true for all $u \in L^2$. – 2017-02-20

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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 $(\int_M u^2)^{1/2} \approx \delta^{1/2}$.