Just a quick question: consider the multiple integral$$ \int_{\mathbb{R}^d}\frac{1}{(|x|^{2k}+1)^p}\,dx. $$ What is the necessary condition on $(k,p)$ so that the previous integral is finite? Is $2kp>d$ enough?
Thank you!
2026-04-10 17:40:57.1775842857
Integrability of function in $\mathbb{R}^d$
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\begin{align*} \int_{{\bf{R}}^{d}}\dfrac{1}{(|x|^{2k}+1)^{p}}dx&=\omega_{n-1}\int_{0}^{\infty}\dfrac{r^{d-1}}{(r^{2k}+1)^{p}}dr\\ &\leq C\omega_{n-1}\int_{1}^{\infty}\dfrac{r^{d-1}}{r^{2kp}}dr\\ &<\infty, \end{align*} if $2kp>d$.