2
$\begingroup$

(True/False)
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
If $f_n\in L^p(\Omega)$ for $1<p<\infty$ and converges weakly to $f\in L^p$, then $||f||_p\leq \lim\inf_{n\to\infty}||f_n||_p$

This is a practice qual problem.

I'm a bit stuck on how to use the condition that $f_n\to f$ weakly.
$\forall g\in L^q$, $\int_\Omega (f_n - f)g\to 0$
If we let $g = \chi_\Omega$, then $\int_\Omega (f_n -f) \to 0$, but this doesn't imply convergence in $L^1$

If we can show $||f||^p_p\leq \int_\Omega \lim\inf |f_n|^p$, then we would be done by fatou's lemma, but I don't see how weak convergence implies this condition.

$\endgroup$
1
  • 1
    $\begingroup$ A start would be $|\int fg| = \lim |\int f_ng| $ for any $g\in L^q.$ $\endgroup$ – zhw. May 21 '18 at 18:40
2
$\begingroup$

Hint: start with Hölder $$ \left|\int_{\Omega}f_n g\right|\le\|f_n\|_p\|g\|_q,\quad \forall g\in L^q. $$ Now take $\liminf$ of both sides and note that $\liminf=\lim$ if the latter exists. Use the weak convergence and the norm definition.

$\endgroup$
1
  • $\begingroup$ Thanks. This and the above comment was enough for me to solve the problem. $\endgroup$ – RaiRsXOT6L May 21 '18 at 18:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.