Lemma: Uniform averaged resolvent expansion

For the radial parameters (eq:radial-parameters), a smooth common-support family $F_{\delta ,H,\theta }$, and a smooth compact weight $w_{\delta ,\theta }(H)$, one has after pairing in $H$

The averaged remainder, with spectator derivatives through order $r$, is bounded by

Constants depend on fixed supports and bounded parameter derivatives of the coordinate changes, including a positive lower bound on ${\kappa }_{\delta }$.

Proof. Choose a fixed compact cutoff ${\omega }_0=1$ near zero. Write

The quotient $G$ is smooth, has common compact support, and $\Vert G{\Vert }_{C^1}\le C_N\Vert F{\Vert }_{C^{N+2}}$, also after the displayed parameter derivatives.

Polynomial division gives

Integrate explicitly where ${\omega }_0=1$, and expand the denominator away from zero. This gives

The remainder after removal of the logarithm is smooth near $z=0$. Combining these identities with $z=\delta L_{\delta }$ gives the coefficients in (eq:resolvent-expansion).

To estimate the remaining resolvent, choose $C_0>\sup |L_{\delta }|$, and a smooth ${\chi }_{\mathrm{in}}$ equal one on $[0,2C_0]$, zero on $[4C_0,\infty )$. Split by ${\chi }_{\mathrm{in}}(t/\delta )$.

On the outer support, $|t-\delta L_{\delta }|\ge t/2$, so the integral is $O(1+|\log \delta |)$ in a zeroth-order norm.

For the inner part put $t=\delta x$, and let

The explicit coordinate change is

The quantities $D_{\delta },{\kappa }_{\delta }$ are independent of $B_1,B_2$. The transformed inner test is

It has common compact support. The chain rule gives uniformly bounded first $L$-derivatives. Spectator derivatives of order $r$ can add at most $r$$H$-derivatives. These are covered by (eq:parameter-seminorm) and ${\mathcal{W}}_{N,r}$.

Uniformly for compact $x$-ranges,

Indeed, subtract $\psi (x)$ times a fixed cutoff in the principal-value integral; the remaining numerator is controlled by a first derivative. The delta part is controlled by $|\psi (x)|$. Integrating in $x,\eta$ gives an $O(1)$ inner bound.

The polynomial-division remainder contains ${\delta }^{N+1}$. Combining both pieces proves (eq:resolvent-bound). To differentiate with respect to spectators, use the $L$-coordinate representation first. Its singular factor is fixed; differentiation acts only on the transformed smooth test. This also justifies differentiation through the pairing.