Exercise 4.1 in Brezis' Functional analysis

Question

I'm trying to solve exercise 4.1 (page 118) from H. Brezis' "Functional Analysis ...".

Let $\alpha > 0 , \beta > 0$. Set $$f(x) = \{ 1 + |x|^\alpha \}^{-1} \{ 1 + |log|x||^\beta \}^{-1}, x \in \mathbb{R}^n.$$ Under what conditions (on $\alpha, \beta, p$) does f belong to $L^p(\mathbb{R}^n)$?

Can someone, please, give me a hint?

Thank you!

Answer 1

Writing $l(x)$ for $|\log|x||$.

Hints: First, all that matters is what happens for, say, $|x| > 10$ and what happens for $|x|<1/10$.

For $|x|>10$, $1+|x|^\alpha \sim |x|^\alpha$ and $1+l(x)^\beta\sim l(x)^\beta$. After that "substitution", write the integral of the $p$-th power in polar corrdinates; you get $\int_{10}^\infty t^\gamma \log(t)^\delta dt$, where you can figure out what $\gamma$ and $\delta$ are. This converges if $\gamma<-1$ and diverges for $\gamma >-1$. When $\gamma=-1$ you have $\int \log(t)^\delta dt/t$, which you can do by a change of variable.

Similarly for $|x|<1/10$, except then $1+|x|^\alpha\sim 1$.