Tricky $L^p$ Spaces Inclusions

Question

In sampling theory, $L^p$ spaces are very important mathematical constructs. To this, we are currently studying some of their properties in class. In this regard, one exercise that I just cannot wrap my head around is to find examples for complex functions on $\mathbb{R}$ which show that

1. neither $L^1(\mathbb{R}) \subset L^{\infty}(\mathbb{R})$ nor $L^{\infty}(\mathbb{R}) \subset L^1(\mathbb{R})$
2. neither $L^1(\mathbb{R}) \subset L^{2}(\mathbb{R})$ nor $L^2(\mathbb{R}) \subset L^1(\mathbb{R})$

I am currently stuck in finding examples for such functions $f$. As a refresher, the $p$ norm is defined as

$$ \begin{align*} \left\|f\right\|_p = \begin{cases} \left ( \int_{\mathbb{R}} |f(t)|^p dt \right )^{\frac{1}{p}} & \text{, $1 \leq p < \infty$} \\ \text{sup}_{t \in \mathbb{R}} |f(t)| & \text{, $p = \infty$} \end{cases} \end{align*} $$

and we want to have that $\left\|f\right\|_p < \infty$ (bounded).

I would be very happy if you could help me! Maybe I am just a bit overworked right now :-)

Answer 1

$\mathbf{1}_{\Bbb{R}}$ is $L^{\infty}$ but not $L^{1}$.

$\frac{1}{\sqrt{x}}\mathbf{1}_{(0,1)}$ is $L^{1}$ but not $L^{\infty}$.

$\frac{1}{\sqrt{x}}\mathbf{1}_{(0,1)}$ is $L^{1}$ but not $L^{2}$.

$\frac{1}{x}\mathbf{1}_{[1,\infty)}$ is $L^{2}$ but not $L^{1}$.

However for finite measures $\mu$ you have that $L^{q}(\mu)\subset L^{p}(\mu)$ if $1\leq p\leq q\leq \infty$ . Eg if you replace $\Bbb{R}$ with say a finite interval like $(0,1)$. Then $L^{\infty}((0,1))\subset L^{2}((0,1))\subset L^{1}((0,1))$ with the Lebesgue measure.