Let $\{f_k\}_{k=1}^\infty\in L^{loc}_1(\mathbb R^n)$, show $\{f_k^2\}_{k=1}^\infty$ does not converge in $\mathscr{D}^\prime(\mathbb R^n)$

Question

Let $\{f_k\}_{k=1}^\infty\in L^{loc}_1(\mathbb R^n)$ be a sequence of real value functions such that

$\mbox{supp}(f_k)\subseteq\{|x|\leq k^{-1}\}$, $\quad \int f_k(x)dx=1$ $\quad for\quad all \quad k=1,2........$

show that the sequence $\{f_k^2\}_{k=1}^\infty$ does not converge in $\mathscr{D}^\prime(\mathbb R^n)$ $as \quad k\rightarrow \infty$

Answer 1

$1=\int f_k =\int_{|x|\ \leq 1/k} f_k \leq (\int_{|x|\ \leq 1/k} f_k^{2})^{1/2}m_k^{1/2}$ where $m_k$ is the measure of $\{x:|x|\leq \frac 1k\}$. Note that $m_k \to 0$. It is now obvious that if $\phi$ is a non-negative test function which is $1$ on $\{x:|x|\leq 1\}$ then $\int f_k^{2} \phi \geq \int_{|x|\ \leq 1/k} f_k^{2} \phi$ tends to $\infty$. Hence $(f_k^{2})$ does not converge in $\mathscr D'$