Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to find an $L_2$ - bound on a certain class of operators, and on my way I produced an estimate for which I need to show that \begin{equation} \sup_{x \in \mathbb{R}^n} \, \int_{\mathbb{R}^n} \frac{1}{(1 + |x - y|)^{n+1}} \,dy \end{equation} is finite. Here I am struggeling with the argument. What I understand is that the integral is finite, but how do I know that it is bounded as a function of $x$ ?

Many thanks for your help!

share|cite|improve this question
up vote 4 down vote accepted

If $f$ is integrable on $\mathbb{R}^n$ then $\int f(y-x)\,dy = \int f(y)\,dy$ for all $x \in \mathbb{R}^n$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.