Under what conditions does $f$ belong to $L^{p}(\mathbb{R}^{N} )$?

Question

Set $\alpha >0$ and, $\beta >0$. Set

$$f(x)=(1+|x|^{\alpha})^{-1}(1+|\log|x||^{\beta})^{-1}, x \in \mathbb{R}^N$$

Under what conditions does f belong to $L^{p}(\mathbb{R}^{N})? $

Answer 1

$p>0$, Using Polar Coordinates you get

$$\int_{\Bbb R^N} |f(x)|^p dx = c_n \int_0^\infty \frac{r^{N-1}dr}{(1+r^\alpha)^p(1+|\log r|^\beta)^p}$$

At $r= 0$ we have $$ \frac{r^{N-1}}{(1+r^\alpha)^p(1+|\log r|^\beta)^p} \to 0~~as~~~r\to0.$$

Hence, The $$ \int_0^1 \frac{r^{N-1}dr}{(1+r^\alpha)^p(1+|\log r|^\beta)^p}<\infty$$

At $r =\infty$ We have, $$ \frac{r^{N-1}}{(1+r^\alpha)^p(1+|\log r|^\beta)^p} \sim r^{N-\alpha p-1}\log^{-p\beta}(r) .$$

Then use the Bertrand criteria to conclude that, this converges iff

$$ -N+\alpha p +1>1 ~~~or ~~~ (-N+\alpha p +1= 1 ~~~and~~~ \beta p>1)$$