Convergence criteria in $L^p$

Question

Show that $f(x)=\frac{1}{(1+\lvert x\rvert^2)^\alpha} \in L^p(\Bbb R^n)$ when $2 \alpha p > n$. I don't really see where does the $n$ come from, I know that we would like that the dominant coefficient of the denominator of the function to be bigger than $1$ if we are only in $\Bbb R$. However, when $n>1$, with Tonelli's theorem, I have something like

$$\int_{\Bbb R^n} \vert f(x) \rvert^p dx=\int_\Bbb R...\int_\Bbb R \frac{1}{(1+x_1^2+x_2^2+\dots x_n^2)^{\alpha p}}dx_1\cdots dx_n$$

Now, integrating with respect to $x_1$ if $2 \alpha p<1$, then I guess I will get something with arctan for the next integral w.r.t to $x_2$ and things will not be easier to integrate... How can I do this properly ? I thought of using polar coordinates so that $\vert x \vert^2=r$ but I don't think it very useful when $n>2$.

Answer 1

Use polar coordinates (coarea formula) in $\mathbb{R}^n$. You get \begin{eqnarray} \int_{\mathbb{R}^n} \frac{d x}{(1 + | x |^2)^{\alpha p}} & = & \int_0^{+ \infty} \int_{\mathbb{S}^{n - 1} (r)} \frac{d \sigma}{(1 + | \sigma |^2)^{\alpha p}} d r \nonumber\\ & = & \int_0^{+ \infty} \frac{1}{(1 + r^2)^{\alpha p}} \left( \int_{\mathbb{S}^{n - 1} (r)} d \sigma \right) d r \nonumber\\ & = & | \mathbb{S}^{n - 1} | \int_0^{+ \infty} \frac{r^{n - 1}}{(1 + r^2)^{\alpha p}} d r. \nonumber \end{eqnarray} Now, the 1d integrand, around $\infty$, is equivalent to $$ \frac{r^{n - 1}}{(1 + r^2)^{\alpha p}} \thickapprox \frac{1}{r^{2 \alpha p - (n - 1)}} . $$ Hence, it converges when $2 \alpha p - (n - 1) > 1$. That is when $2 \alpha p > n$.

Answer 2

A direct way without using polar coordinates is to note the formula, valid for $0< f\le 1$: $$ \int_{\mathbb R^n}f(x)\,dx \sim \sum_{k=0}^\infty 2^{-k}|\{x:f(x)\sim 2^{-k}\}|. $$ Here the notation $A\sim B$ means that $C^{-1}A\le B\le CA$ for a universal constant $C$. We can take $C = 2$ in the equality above, for example. In general, the implied constants may change from line to line, but importantly they always depend at most on $n$. The notation $|E|$ denotes the Lebesgue measure of $E$.

For $k\ge 0$ fixed, the set $E_k = \{x: (1+|x|^2)^{\alpha p}\sim 2^k\}$ has approximately the same measure as the set $F_k = \{x : |x|\sim 2^{k/2\alpha p}\}$, and $|F_k|\sim 2^{nk/2\alpha p}$ by approximating the measure of an annulus from within and from out by cubes of diameter $\sim 2^{k/2\alpha p}$. (This is where $n$ enters the picture—in $\mathbb R^1$, cubes are replaced by intervals in the estimation.) Thus \begin{align*} \int_{\mathbb R^n}\frac{dx}{(1+|x|^2)^{\alpha p}} &\sim \sum_{k=0}^\infty 2^{k(\frac{n}{2\alpha p}-1)}, \end{align*} which is evidently convergent if and only if $$ \frac{n}{2\alpha p} - 1 < 0 \iff 2\alpha p > n. $$