Give an example function $f \in L_p(\mathbb {R}) \forall \: p > 7,$ but $f \notin L_p(\mathbb {R}) \: p \leq 7$

Question

Give an example function

$f \in L_p(\mathbb {R}) \forall \: p > 7,$ but $f \notin L_p(\mathbb {R}) \: p \leq 7$

Answer 1

Hint: consider $$f_q(x) = \begin{cases} x^{-q} & \text{ if }\,\,\, |x|>1\\ 0 & \text{ else}\end{cases}$$

For which values of $p$ and $q$ is this in $L^p$?