Lemma: Shifted momentum bounds

Let $r\ge 1$, $\mathbf{\nu }\in S^2$, $\ell \in \{1,2\}$, $k\in \{0,1,2\}$, and $N>k+3$. Then

The constants are uniform in $r,\mathbf{\nu }$. For $0\le r\le 1$, the first integral is uniformly bounded. In particular, all these assertions apply with $N=12$.

Proof. On $|\alpha |\le r/2$, the denominator is at least $(r/2{)}^{\ell }$, and the remaining weight is integrable.

On $|\alpha |>r/2$, first consider $|\alpha -r\mathbf{\nu }|\le r/2$. Here $|\alpha |$ is comparable to $r$, and

This part is bounded by $C{r}^{3-\ell +k-N}$. In the remaining part the denominator gives $C{r}^{-\ell }$, while

This proves the tail bound. Since $N>k+3$, the tail also satisfies the full-integral bound.

For $0\le r\le 1$, split into a fixed-size neighborhood of $\alpha =r\mathbf{\nu }$ and its complement. Local integrability of $|p{|}^{-\ell }$ treats the first region, and the fixed polynomial weight treats the second.