show an integral is bounded by a constant independent of a parameter

Question

This is a question in Treves.

Suppose $a>1$ and $\tau \in \mathbb R $,

(i) show that for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge(\tau ^2+|\xi|^2+a^2)^{1/2}$

(ii) show that if $k>n/2$,

$\iint [(\tau-ia)^2-|\xi|^2]^{-1}(1+a^2+(\tau-ia)^2+|\xi|^2)^{-k}\,d {\tau}\, d {\xi}\quad$

is bounded by a constant independent of $a>1$

I've figured out the first question, and I guess one has to use (i) to prove (ii), but I really don't have an idea of integrating such a double integral. Everything seems to be messy to me. I just don't dare to try, and actually I don't know where to start. This really embarrassed me. Can anyone give me some ideas? I'll really appreciate it.

Answer 1

You could perhaps begin with $a>1$ which gives $$|(\tau-ia)^2+|\xi|^2 +1+a^2|> (\tau^2+|\xi|^2+1)^{1/2}$$ implying $$|(\tau-ia)^2+|\xi|^2 +1+a^2|^k> (\tau^2+|\xi|^2+1)^{k/2}$$ and using (i) $$ |(\tau-ia)^2-|\xi|^2|^{-1}|1+a^2+(\tau-ia)^2+|\xi|^2|^{-k} <(\tau^2+|\xi|^2+1)^{-(k+1)/2}. $$ The latter function has a bounded integral over $\mathbb R^{n+1}$ taking into account the condition on $k$ ...