Lemma: Uniform anisotropic moment expansion

The supports of $F_j$, $j=0,1$, are uniformly compact when $\alpha ,h$ range over a compact subset of $(0,\infty {)}^2$. For $0<\delta \le 1$,

uniformly on those parameter sets.

Proof. Set

Then

Compact profile supports bound $Q$ and $s$, uniformly for the stated parameters.

The functions $F_j$ are radial in $X$. Odd transverse moments vanish, and

Taylor-expand in $\xi$ through cubic order at $(\delta s,0)$. The odd terms integrate to zero; the fourth-order remainder is bounded by $C{\delta }^2|X{|}^4\Vert \psi {\Vert }_4$. Expand the zero-transverse term to first order in $v$, and replace the transverse Hessian at $v=\delta s$ by its value at zero. These errors are also $O({\delta }^2\Vert \psi {\Vert }_4)$.