Suppose $q \in [1, \infty)$, I'm trying to find a sequence of functions $g_n \in L^p$ (for all $p \in [1,\infty]$) such that $g_n$ converges in $L^p$ when $p \in [1, q]$ but doesn't converge into $L^p$ when $p \in (q, \infty]$.
I tried the sequence of functions $g_n(x) = \displaystyle{\frac{x^{1/p}}{n}}$, over the measure space $([0,1], \mathcal{B}([0,1]), \lambda)$, where $\mathcal{B}([0,1])$ are the Boreal sets over the interval $[0,1]$. If $p \leqslant q$ this clearly converges to $0 \in L^p$, however this is still true for $p > q$.
Any help would be greatly appreciated.
On the measure space $([0,1], \mathcal{B}([0,1]),\lambda)$, take $$g_n = n^{1/q} \mathbf{1}_{[0,(n\log(n))^{-1}]}. $$ Then $$\int g_n^p \, d \lambda = \frac{n^{p/q-1}}{\log(n)} \to 0\tag{1}$$ if and only if $p \leq q$. Therefore $g_n \to 0$ in $L^p$ for every $p \leq q$. Moreover, $g_n$ does not converge in $L^p$ for any $p>q$. Indeed, if it did converge, the limit would necessarily be $0$ since convergence in $L^p$ for any $p>q$ implies convergence in $L^q$ (the measure $\lambda$ being finite). This is impossible due to $(1)$.
Bonus: if you remove the term $\log(n)$ from the definition of $g_n$, you get a sequence converging in $L^p$ for every $p<q$ but not in $L^p$ for $p\geq q$.