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Let $\Gamma$ be a compact $C^2$ manifold and suppose that $f_n$ is a non-negative sequence of functions such that:

$(*)\forall t\in (0,T)$ it holds: $\int_{0}^t \int_{\Gamma} f_n \le C_1+C_2t$ for some positive constants $C_1,C_2$.

From $(*)$ we can deduce that ${\vert \vert f_n \vert \vert}_{L^1(0,t,L^1(\Gamma))} \le C_1+C_2T \;\;\forall t\in (0,T)$.

I'm interested in obtaining convergence of $f_n$

Unfortunately, this $L^1(0,t,L^1(\Gamma))$ bound doesn't seem much helpful. If instead I had a bound of $f_n$ in $L^{\infty}(0,t,L^1(\Gamma))$, I could then deduce some convergence using duality arguments. But even in the simpler case in which a sequence is bounded in $L^1(\Gamma)$, then a convergent subsequence may not exist.

MY QUESTIONS:

  1. Do I miss any theorem in Bochner spaces that provides convergence for sequences bounded in $L^1(0,t,L^1(\Gamma))$?
  2. Is there any other way to take advantage of the inequality in $(*)$ in order to deduce a better bound?

Any help is much appreciated.Thanks in advance!

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No, this $L^1$-bound does not give any convergence in $L^1$. Think of the sequence $$ f_n = n^2 \chi_{[0,1/n]^2}, $$ which is uniformly bounded in $L^1(0,1;L^1(0,1))$. It converges in a weak-star sense to the Dirac functional at the origin, and this limit point cannot be represented by an integrable function.

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  • $\begingroup$ First of all, thanks a lot for your answer. To be honest, I wasn't expecting $f_n$ to converge in $L^1$ but maybe in some vector-valued measure space (I don't know if I use the right term here and I don't even know if such a space exist). Is there any possibility for $f_n$ to converge utilizing $(*)$? $\endgroup$ – kaithkolesidou Feb 6 at 15:37
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    $\begingroup$ You would get weak-star convergence in a dual space to a space of continuous function, i.e., in $C([0,T]\times \Gamma)^*$. $\endgroup$ – daw Feb 6 at 20:06

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