On a sequence $f_k$ in $L^{2+\frac{1}{k}}$

Question

Suppose that $f_k\in L^{2+\frac{1}{k}}(\Omega)$ with the property that $\|f_k\|_{L^{2+\frac{1}{k}}(\Omega)} = 1$ for all $k\ge 1$. $\Omega$ is a bounded domain in $\mathbb{R}^n$. Can such a sequence of functions strongly converge to ZERO in $L^2(\Omega)$, is it possible?

Answer 1

I think the answer would be yes. Have you tried taking a characteristic function over a set of size $n^{-1}$ with height $n^{k / (2k + 1)}$. The $(2 + 1/k)$-norm will be $1$ but the (square of the) $L^2$-norm will be $$\frac1{n^{1- \frac{2k}{2k + 1}}}.$$ When fix the $n$ large enough in terms of $k$ to make that expression as small as you want. For instance: $$ n = 2^{k (2k + 1)}. $$ would make it.