Determine whether a function is in $L^p$ - specific examples

Question

I am given the two functions $f(x)= \frac{1}{1+|x|^2}$ and $g(x)= \frac{1}{\text{log}(1+|x|)}$ for $x \in \mathbb{R}^n$.

The question is for which $p \in [1, \infty ]$ $f$ and $g$ are in $$L^p(\mathbb{R}^n)= \{f: \mathbb{R}^n \rightarrow \mathbb{R} \text{ measurable } | \| f\|_{L^p(\mathbb{R}^n)} < \infty \},$$ where $\| . \|_{L^p(\mathbb{R}^n)}$ is defined as $( \int_{\mathbb{R^n}} |f(x)|^p dx)^{1/p}$ for $ p < \infty$ and as $\underset{N \subseteq \mathbb{R}^n}{inf} \underset{x \in \mathbb{R}^n \setminus N}{sup} |f(x)|$ for $p= \infty$. Thus, I think that I have to find out whether the $L^p$ norm of these functions is smaller than $\infty$.

For $f$ (I used that $f >0 ~\forall x \in \mathbb{R}^n$):

In the case $p= \infty$, I have $\| f\| = \underset{N \subseteq \mathbb{R}^n}{inf} \underset{x \in \mathbb{R}^n \setminus N}{sup} |\frac{1}{1+|x|^2}| =\underset{N \subseteq \mathbb{R}^n}{inf} \underset{x \in \mathbb{R}^n \setminus N}{sup} \frac{1}{1+|x|^2} $ which is always $\leq 1$ because for $x=0$ $f$ is $1$ and for $x \rightarrow \pm \infty$ $f$ goes to zero. Thus, $f \in L^\infty$. For $p \leq \infty$, I have $\|f \|= ( \int_{\mathbb{R^n}} |\frac{1}{1+|x|^2}|^p dx)^{1/p} = ( \int_{\mathbb{R^n}} (\frac{1}{1+|x|^2})^p dx)^{1/p}$ and then I'm stuck as don't know how to go on for general $p$. I thought that maybe $f(x) \leq 1$ could be useful.

For $g$: In the case $p= \infty$, I argued that the critical point is $x=0$ and if I chose $N$ as the empty set, my norm would not be smaller than $\infty$, hence $g \not \in L^\infty$. For $p < \infty$: $$ \|g \| = ( \int_{\mathbb{R^n}} |\frac{1}{\text{log}(1+|x|)}|^p dx)^{1/p} = ( \text{lim}\int_{-r}^r (\frac{1}{\text{log}(1+|x|)})^p dx)^{1/p}= ( \text{lim}\int_{0}^r (2 \frac{1}{\text{log}(1+x)})^p dx)^{1/p}$$ for $r \rightarrow \infty$. Again I don't know how to proceed here.

It would be great to know whether my solution is correct so far and I would be happy about any information on how to go on with this task!

Answer 1

Here are some facts that are generally helpful to determine whether a given function is in $L^p$.

1. If a function is in $L^p$ and $L^q$ for some $p<q$, then it is also in $L^r$ for all $p\le r\le q$ (this is called interpolation of $L^p$ functions and follows from Hölder's inequality).

2. The function $|x|^{\alpha}$ on $\mathbb{R}^n$ is

   • locally integrable at $0$ if and only if $\alpha>-n$ (it's not allowed to blow up too much),
   • locally integrable at $\infty$ if and only if $\alpha<-n$ (it needs to decay fast enough)

3. $\log$ grows slower than any positive power: for every $\epsilon>0$ there exists $C_\epsilon$ such that $\log(1+|x|)\le C_\epsilon |x|^\epsilon$.

Can you proceed from here?