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Consider the standard Sobolev space $H\equiv H_0^1(\Omega)$, where $\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Let us consider on $H$ the norm $$\|u\|^2=\int |\nabla u|^2,$$

and define $S\subset H$ by $$S=\{u\in H_0^1(\Omega): \|u\|=1\}.$$

Let $1<q<p<2^\star$ where $2^\star$ is the critical Sobolev exponent. We know that $$\tag{1}0<C_p=\inf_{u\in S}\frac{1}{ \|u\|_p},$$

and $$\tag{2}0<C_q=\inf_{u\in S}\frac{1}{\|u\|_q}.$$

So, my question is the following. Let $C_{p,q}$ be defined by $$C_{p,q}=\inf_{u\in S}\frac{1}{\|u\|_q\|u\|_p}.$$

Is it true that $C_{p,q}=C_pC_q$?

My thoughts on the problem: The answer seems to be true to me, because of the inequality $\|u\|_{q}\le C\|u\|_p$, so it seems that when one maximizes the norm $\|u\|_q$ over $S$, the norm $\|u\|_p$ is also maximized over $S$, however, the argument is not that simple.

Any idea is aprreciated.

Tomás
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  • Here $|u|_p$ is the $L^p$ norm of $u$? – Neal Aug 03 '16 at 00:08
  • Yes @neal, and same for $|u|_q$. – Tomás Aug 03 '16 at 00:09
  • One can check that the inf's in (1), (2) and in the definition of $C_{p,q}$ are attained by some functions. Then, your assertion holds if and only if there is a $u$ solving both (1) and (2). This, however, seems to be unlikely. – gerw Aug 03 '16 at 06:36
  • I think it is possible to solve (1) and (2) on an interval. Then one could check whether the solutions coincide. – gerw Aug 03 '16 at 06:39
  • Indeed @gerw, after your comment I saw that an argument with a minimizing sequence shows that we can change infimum to minimum. Now it seems that the answer to my question is negative. – Tomás Aug 03 '16 at 10:23
  • @gerw, because of the inequality $|u|_q\le C|u|_p$, we have that $$\frac{1}{|u|_q|u|_p}\ge \frac{1}{C|u|_p^2}\ge \frac{1}{CC_p},$$ so it seems to me that the infimum is attained in the function that maximizes the norm $|u|_p$ over the sphere. What do you think? – Tomás Aug 04 '16 at 12:32
  • I don't think so. Using this function $u$, the inequality $|u|_q \le C |u|_p$ is (most likely) not sharp. – gerw Aug 04 '16 at 17:55

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