The Banach space $L^p(\mathbb{R})$ is the space of all functions $f:\mathbb{R}\to \mathbb{C}$ for which $\left(\int_{\mathbb{R}}|f(x)|^{p}dx\right)^{\frac{1}{p}}<\infty$
Showing that $L^2(\mathbb{R})\setminus L^3(\mathbb{R})$ is non-empty is very simple since a function like $\displaystyle f(x)=\frac{\chi_{(0,1]}}{\sqrt[3]{x}}$ (here is $\displaystyle\chi$ is the characteristic/indicator function) clearly works, but the other way round seems harder. So I need to find a function that is in $L^3(\mathbb{R})$ but not in $L^2(\mathbb{R})$. A function like $\displaystyle\frac{\chi_{(0,\infty)}}{\sqrt{\sinh(x)}}$ seems to work, but the integral becomes too hard to evaluate or estimate when $p=3$. Can someone give an easy example?
How about something which decays to $0$ like $x^{-\frac{5}{12}}\chi_{(1,\infty)}(x)$? because $$\int_1^\infty (x^{-\frac{5}{12}})^3\text{d}x=[-4x^{-\frac{1}{4}}]_1^\infty=4$$ $$\int_1^\infty (x^{-\frac{5}{12}})^2\text{d}x=[6x^{+\frac{1}{6}}]_1^\infty=\infty$$ In general any function of the form $\frac{1}{x^\alpha}$ where $3\alpha>1$ and $2\alpha <1 $ would work.