Just a necessary recap of the definitions:
For a finite measure space $(\Omega,dx)$ and a function $p \in L_\infty(\Omega),$ we definite the modular $$\rho_{p(\cdot)}(f)= \int_\Omega |f(x)|^{p(x)}dx,$$ where $f:\Omega\rightarrow \mathbb{R}$ is a measurable function. The space $L_{p(\cdot)}$ consists of all measurable functions $f$ such that $\rho_{p(\cdot)}(f/\lambda)<\infty$ for a $\lambda >0.$ This space is endowed with the (Luxemberg) norm $$\|f\|_{L_{p(\cdot)}}= \inf\{\lambda >0: \rho_{p(\cdot)}(f/\lambda)\leq 1 \}.$$ (This is not the most general definition but this is the situation I am in: Finite measure space and $1\leq p(x) <\infty$ f.a.a. $x \in \Omega$).
This is just a particular example of a Orlicz–Musielak space.
I am very interested in any result in concerning the following question:
If $p_k \rightarrow p $ in the norm of $L_\infty,$ what about $$\|f\|_{L_{p_k(\cdot)}} \rightarrow \|f\|_{L_{p(\cdot)}}?$$
In this monograph there are some results in this direction:
Corollary 3.5.4 implies that the modular and the norm are lower semicontinuous.
Lemma 3.5.5 says that if the sequence stays between two functions for which the modular is finite, then the modular is continuous.
Theorem 3.5.7 tells me that if the convergence is monotone, i.e. $p_k(x)\leq p_{k+1}(x) $ then the above will also be correct.
Are there any other results that address this topic?
Can Lemma 3.5.5 be extended to the norm?
Also I would also be very happy about any results in the setting of Orlicz–Musielak spaces if you are aware of any.