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Does this inequality holds for Poincaré Inequality?

$$||v||_{L^2} \leqslant C_p |v|_{H^1}$$

and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm.

I'm really confused with norms and semi norms in $H^1$ and $L^2$.


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Can you please clarify what norms and seminorms are this? – Tomás Dec 11 '12 at 13:52
I edited my post. I think should be clear now. – BRabbit27 Dec 11 '12 at 17:41
What is the definition of $|v|_{H^1}$? What is $v'$ and what is that $p$ in $C_p$? – Tomás Dec 11 '12 at 17:47
Well, certainly I don't know what's the definition of $|v|_{H^1}$ I just know that is a semi norm --- $v'$ is just the first derivative of a function $v$ (any function) --- That $p$ is just a letter to say that $C_p$ = Poincaré Constant. – BRabbit27 Dec 12 '12 at 15:13
Are your functions defined in $\mathbb{R}$? – Tomás Dec 12 '12 at 16:18
up vote 1 down vote accepted

The answer is no, which you can verify by calculating $\|v\|_{L^2}$ and $\|v'\|_{L^2}$ for the function $v_n(x)=\min(1,\max(0,n-|x|))$. As $n\to\infty$, one of the norms grows indefinitely while the other remains constant.

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