example of function in Lp for specific p values

Question

Please give an example of a function $f:(0,\infty) \rightarrow \mathbb{C}$ with the following properties:

(a): $f \in L^p(0,\infty)$ for $2 \leq p \leq \infty$, but $f \notin L^p(0,\infty)$ if $1 \leq p < 2$

(b): $f \in L^p(0,\infty)$ for $2 < p < 4$, but not for $p$ outside this range

I'm pretty sure I figured out part (a) by letting $$ f(x) = \sum_{n=1}^\infty \frac{1}{n} {\large \chi}_{[n,n+1)} (x)$$ but I'm not sure about part (b). I'd appreciate any help.

Answer 1

Something like $$ (a)\quad f(x) = \frac{1}{\sqrt{1+x}\log(2+x)}, \qquad (b)\quad f(x) = \frac{1}{x^{1/4}(1+x^{1/4})}\,. $$

Answer 2

You are on the right track with your $f$. One typically uses that \begin{align} g(x)=\chi_{[1,\infty)}\frac{1}{x} \end{align} is in $L^p$ for $p>1$ and that \begin{align} h(x)=\chi_{(0,1)}\frac{1}{x} \end{align} is in $L^p$ for $p<1$.

For b), you could use a combination like $f=g^\frac{1}{2}+h^\frac{1}{4}$. Then, \begin{align} |f|^p=g^\frac{p}{2}+h^\frac{p}{4} \end{align} and one of the addends is not integrable for $p$ outside of $(2,4)$.