Pome

Lemma 18.39.7BBA6C67F74CF

Lemma 18.39.7.BBA6C67F74CF Let $r\geq 1$ and $\ell \geq 0$ be integers, let $p,b_1,\ldots ,b_ r$ be positive integers, and let $P>0$. The function

\[ |z|^{2\ell }\left(|z|^{2p}+\sum _{j=1}^ r|w_ j|^{2b_ j}\right)^{-P} \]

of $(z,w)\in \mathbb C\times \mathbb C^ r$ is locally integrable near the origin if and only if

\[ \frac{\ell +1}{p}+\sum _{j=1}^ r\frac1{b_ j}>P. \]

In the case of equality, the integral diverges logarithmically.

Proof. After integrating over the angles, substitute $x_0=|z|^{2p}$ and $x_ j=|w_ j|^{2b_ j}$. Up to a positive constant, the integral becomes

\[ \int (x_0+\cdots +x_ r)^{-P}x_0^{(\ell +1)/p-1} \prod _{j=1}^ r x_ j^{1/b_ j-1}\, \mathrm dx_0\cdots \mathrm dx_ r \]

over a box near the origin. This box contains a small simplex $\{ \sum _ jx_ j<\delta \} $ and is contained in a larger one. On either simplex use the coordinates $\rho =\sum _ jx_ j$ and $x_ j=\rho y_ j$. The integral over $y$ is finite and positive, because all its exponents exceed $-1$, and the integral over $\rho $ is

\[ \int _0^\delta \rho ^{(\ell +1)/p+\sum b_ j^{-1}-P-1}\, \mathrm d\rho . \]

This gives the threshold and the logarithmic divergence at equality. Bounded factors in the weight, holomorphic units and additional tangential variables do not affect this local test. $\square$

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