Showing that a function lies in $L^p(\mathbb{R}^d)$

Question

In an exam today I had the following problem: \begin{align} f_s(x) = \exp(-\vert x \vert^2)\vert x \vert^s \end{align} for $x\in \mathbb{R}^d$ and $s\geq 0 $ or for $x\in \mathbb{R}^d\setminus\{0\} $ and $s < 0 $. When is $f_s$ in $L^p(\mathbb{R}^d)$? I really have no idea how to show that a specific function lies in $L^p(\mathbb{R}^d)$ for general $p$...

Answer 1

Since $f$ is a radial function, we have $$ \int_{\Bbb R^d}|f(x)|^p\,dx=\int_{\Bbb R^d}e^{-p|x|^2}|x|^{ps}\,dx=C_d\int_0^\infty e^{-pr^2}r^{ps+d-1}\,dr. $$ Can you finish now?

Answer 2

$\exp(-|x|^2)$ belongs to the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ for every $n$, hence there is no integrability issue far from the origin and the given function belongs to $L^p(\mathbb{R}^d)$ as soon as $|x|^{sd}$ is integrable in a neighbourhood of the origin. On the other hand, integrating along shells, $$ \int_{|x|\leq 1}|x|^{ps}\,d\mu = \int_{0}^{1}\rho^{ps} A_{d}\rho^{d-1}\,d\rho$$ where $A_d$ is the surface area of the unit ball in $\mathbb{R}^d$, namely $\frac{2\pi^{d/2}}{\Gamma(d/2)}$.
The last integral is convergent as soon as $ps+d>0$ i.e. as soon as $\color{red}{s>-\frac{d}{p}}$.