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Let $\|u\|^2_{L^2_\eta}$ be the exponentially weighted norm of the space of functions $u(x)$ for which $u(x)\mathrm{e}^{\eta\cdot x}$ with $\eta\in \mathbb{R}$ is in $L^2(\mathbb{R})$. How can I show that the norms defined in this way are not equivalent for different weights?

Thanks very much in advanced.

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If $u_n=1_{[n,n+1]}$, then for any real $t$ $$\|u_n\|^2_{L^2_t}=g(t)e^{tn},$$ where $g(t):=\frac{e^t-1}t$ if $t\ne0$ and $g(0):=1$.

So, $\|u_n\|_{L^2_t}/\|u_n\|_{L^2_s}\to\infty$ as $n\to\infty$ if $s<t$. So, the norms $\|\cdot\|_{L^2_t}$ are not equivalent to each other for different values of $t$. $\quad\Box$

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  • $\begingroup$ Thanks very much for your answer and all the best $\endgroup$
    – Lars Siemer
    Commented Nov 12 at 18:28
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    $\begingroup$ @LarsSiemer : If you are satisfied with this answer, these guidelines may be relevant now. $\endgroup$
    – Iosif Pinelis
    Commented Nov 12 at 19:26

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