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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!

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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 $x \in \mathbb{R}^n$.

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