I have been working through Erdos & Yau's `Linear Boltzmann equation as the weak coupling limit of a random Schrodinger Equation,' (arXiv link: http://arxiv.org/abs/math-ph/9901020), and for an embarrasingly long time have been stuck on a bound they claim is a 'simple estimate.'

To get to the point, they state:

$\sup_{z\in\mathbb{R}^{d}}\sup_{\alpha\in\mathbb{R}}\int_{\mathbb{R}^{d}}\frac{1}{|\alpha - |w|^{2} + i\epsilon|}\frac{1}{[w-z]^{2d}}dw \leq C|\log(\epsilon)|$

where $[x]:=(1+|x|^{2})^{\frac{1}{2}}$ and $d\geq2$ (the standard notation wasn't working so I went with this).

This bound has frustrated all my attempts at proof for too long, which is why I am finally caving and posting here. My starting point was the easy bound:

$\int_{-\epsilon}^{\epsilon}\frac{1}{|r+i\epsilon|}dr \leq C|\log(\epsilon)|$

My plan then, was to split the integral into two pieces:

$\int_{|\alpha-|w|^{2}|<1}$ and $\int_{|\alpha-|w|^{2}|\geq1}$

On the latter, I use an $L^{\infty}$ bound on $\frac{1}{|\alpha-|w|^{2}+i\epsilon|}$, and, because we are assuming $\epsilon<<1$, $1\leq|\log(\epsilon)|$.

On the former, however, I haven't had as much luck. I attemped a dyadic decomposition around the sphere $|w|=\sqrt{\alpha}$ (assuming $\alpha>0$), but because a small region around such a sphere depends on $\alpha$ like $\alpha^{(d-1)/2}$, I couldn't seem to make the estimate independent of $\alpha$.

I have also tried moving to radial coordinates at various stages, but haven't had any luck.

I apologize if the bound is more trivial than I realize, and would appreciate any help/direction in proving it.