Express $|x|^{-p},\;x\in\mathbb{R^+},\;0.5<p<1$ as sum of $L^1(\mathbb{R^+})+L^2(\mathbb{R^+})$ function.
I have been able to show that $|x|^{-p}$ is neither $L^1$ nor $L^2$, but how do I proceed with second part of the question in the attached image 1?
The function $|x|^{-p}$ is integrable on $[0,1]$ when $p<1$.
It is integrable on $[1,\infty)$ when $p>1$. That is why it is not integrable on $[0,\infty)$ for any value of $p$.
Similarly it is in $L^2[0,1]$ when $p<1/2$, and in $L^2[1,\infty)$ when $p>1/2$.
It should now be clear how to divide the function into two parts, one on $[0,1]$, the other on $[1,\infty)$.