Let $\Omega$ be a region, and $f\in L^2(\Omega)$, for simplicity we suppose
that $\Omega$ is compact and $f$ is smooth.
If we have the inequaliy
$$\|E_n\|_{L^2}\leq \frac{C}{\sqrt n}\|f\|_{L^2},$$
where $E_n$ is a smooth function and $Supp(E_n)\subset Supp(f)$.
Q
Can we give a point-wise norm up-bound for $|f|^2|E_n|^2$, e.g. $|f|^2|E_n|^2\leq \frac{C_1}{n}|f|^2|f|^2$?
I think $E_n\to0$, is this true?
PS: 1. If $E_n=E_n( f)=\alpha_f$, here $\alpha_f$ is linear for $f$. Could we get the above inequality.
- I think we can get the $|g|^2|E_n|^2\leq C_0|g|^2|f|^2$ for any fixed number $C_0<1$ and $n$ large enough.
The proof is as follows.
If there exists some point $x$ such that the inequality violates, then $|g|^2|E_{n_i}|^2> C_0|g(x)|^2|f(x)|^2>0$ for a subsequence $\{n_i\}\to\infty$. But by the hypothesis, $E_{n_i}$ is smooth, some near a small neighborhood of $x$, the inequality still holds, i.e. $|g(y)|^2|E_{n_i}(y)|^2> C_0|g(x)|^2|f(x)|^2>0$ for any $y$ in this neighborhood. So, $\int_\Omega|g|^2|E_{n_i}|^2>\delta>0$ as $n_i\to\infty$, but $\int_\Omega|g|^2|E_{n_i}|^2\to0$. A contradiction.