Let $1<p_0<\infty$ Find a sequence $\{f_k\}$ such that $f_k \in L^p$ for $1 \leq p < \infty,$ $f_k \rightarrow 0$ in $L^p$ for $1 \leq p <p_0$, but $f_k$ does not converge in $L^{p_0}$
I tried using functions like $f_k=k^\alpha \chi_{(0,k^{-1})},\chi$ is characteristic function and also used some polynomial like ${1\over x^\alpha}$ But failed. Please help. Thank you!
The idea to use $f_k=k^\alpha \chi_{(0,k^{-1})}$ is good. The sequence converges to $0$ a.e., so if there is any $L^p$ limit at all, it has to be zero (recall that $L^p$ convergence implies a.e. convergence for a subsequence). Therefore, if the sequence of norms stays constant, the sequence does not have a limit in $L^p$.
Choose $\alpha$ so that these functions have constant norm in $L^{p_0}$. This means $k^{\alpha p_0}k^{-1}$ being independent of $k$, so $\alpha p_0=1$.
If $p<p_0$, the $L^p$ norm tends to $0$.