We are given the measure space $(\mathbb{R},\mathcal{M_1},\mathcal{L}^1)$ and two functions $f,g:[0,1]\rightarrow\mathbb{R}$ where $f$ is in $L_p([0,1])$ ($\int_{[0,1]}|f|^p\mathrm{d}\mathcal{L}^1<\infty$) and $g$ is in $L_q([0,1])$ and $p,q\in[1,\infty]$. I want to find out for what values of $p$ and $q$ the product $f\cdot g^2$ is in $L_1([0,1])$. I also wondered if there is a general way to solve this for an arbitrary constant $a$ so $f\cdot g^a$.
I know that if we only have $f\cdot g^1$ we can use Hölder's inequality to get the condition $\frac{1}{p}+\frac{1}{q}=1$.
I think it should be possible to use Hölder's inequality in some form but I do not really see how so I would appreciate some help.