Inequality in $L^p(\Omega)$

Question

Let $u_n, u, v \in L^p(\Omega)$, with $u_n, u, v$ bounded a.e. for $x \in \Omega$ and $p \geqslant 1$, where $\Omega$ is a bounded domain. Is it possible to obtain an inequality of the type: $$ \|(u_n- u)v\|_{L^p(\Omega)} \leqslant C\|u_n - u\|_{L^p(\Omega)}\|v\|_{L^p(\Omega)} $$ for some $C > 0$?

I've tried using Holder's inequality, to no avail.

Answer 1

In general the inequality $\|fg\|_p \leq \|f\|_p \|g\|_p$ is false because of scaling issues; for example let $f_n = g_n = n^{1/p}\chi_{[0,1/n]}$. Then the right hand side is 1, but the left hand side is $n$, so you’d need a constant which depends on the functions.

But since your functions are in $L^{\infty}$, in many situations a bound like $\|(u_n-u)v\|_p \leq \|v\|_{\infty}\|u_n-u\|_p$ suffices, but without more context it’s unclear if this works for your purposes.