This question has bothered me for a while, and I hope to have it clarified if possible.
Consider a set of functions $G:=\{ g : (0,1) \rightarrow \mathbb{R}^+ \;|\; (\int_0^1 g(t)^p dt)^{1/p} \leq k \text{ and} \int_0^1g(t)dt=1\}$ for some $k\geq 1$
Is it always possible to find a $g^* \in G$ such that $||g^*||_q=\infty$ where $q>p\geq 1$ or not?
I am not sure if this should be something obvious or not. Does this depend on $k$? If so, any way to understand the dependency?
Let $r$ satisfy $p<r<q$. Set $s=1-\frac{p}{r}$. Note that $0<s<1$. Set $g(t)=s^{1/p}t^{-1/r}$. Now, $$\int_0^1 g(t)^pdt=\int_0^1 st^{-p/r}dt= t^s|_0^1=1$$
Now set $u=1-\frac{q}{r}$, and note that $u<0$. $$\int_0^1 g(t)^qdt=s^{q/p}\int_0^1t^{-q/r}dt=\frac{s^{q/p}}{u}t^u|_0^1=\infty$$