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

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!

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.
• 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 $(*)$? – kaithkolesidou Feb 6 at 15:37
• You would get weak-star convergence in a dual space to a space of continuous function, i.e., in $C([0,T]\times \Gamma)^*$. – daw Feb 6 at 20:06