Lebesgue integration of 1/{|x|^{\alpha}}

Question

I would like to know for which values of $\alpha$, the following statement holds: $$ \int_{\mathbb{R}^n}\dfrac{1}{\Vert x\Vert^{\alpha}}<\infty. $$ Thanks!

There is no $\alpha$. Either the integral diverges at $0$ or at $\infty$. — Jochen, Sep 27 '18 at 08:02

score 1 · Accepted Answer · answered Sep 27 '18 at 08:03

1

For no $\alpha$! Either integral diverges on $\{\|x\| \leq 1\}$ or the integral diverges on $\{\|x\| > 1\}$

answered Sep 27 '18 at 08:03

Kavi Rama Murthy

311,013

When is it convergent in the unit ball? Thx. – Alex Sep 27 '18 at 08:25
1

The integral over the unit ball is finite iff $\alpha <n$. – Kavi Rama Murthy Sep 27 '18 at 08:34

Lebesgue integration of 1/{|x|^{\alpha}}

1 Answers1