Question on Bochner spaces: $L^1(0,T,L^1(\Gamma))$ bound and convergence of a sequence

Question

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!

Answer 1

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.