Both internal placements of K 5

For every compact interval $J$, there are a slightly larger compact interval $J^{\prime }$ and a constant $C_J$ such that

The same type of finite-order estimate holds after any fixed number of derivatives of $B_{-}$.

Proof. Choose a smooth even function equal to one near zero. In the principal-value integral subtract $\psi (0)$ times that function. Near zero the remaining numerator is bounded by $|\lambda |$ times a first-derivative bound. Away from zero the denominator is bounded away from zero on the remaining compact integration set. The delta part is bounded by $|\psi (0)|$. This gives the estimate.

For a derivative, transfer that derivative to the test function and apply the same argument.

Let $K$ be a compact subset of a normalized frequency annulus in $(\theta ,\kappa )$ contained in one of the pole charts of Proposition prop:pole. The compact set may meet ${\lambda }_s=0$.

If $d=j+|\beta |$, then

as distributions. For every fixed frequency multiindex $\nu$, there are $N$ and $C$ such that

For amplitudes (eq:amodule), these estimates are uniform after any fixed number of coefficient derivatives on a compact subset of $V$. No separation from the pole is required.

Proof. Lemma lem:multiplier gives degree $-2$ for $M_{\mathrm{R}}$. The polynomial has degree $d$, giving (eq:homogeneity). A distributional frequency derivative lowers the homogeneity by one.

It remains to verify finite seminorm control on $K$, including its intersection with the pole. In the coordinates $({\lambda }_s,\xi )$, (eq:pole) is a smooth homogeneous coefficient times $B_{-}({\lambda }_s)$ plus a smooth term. The change of frequency coordinates is regular where $\rho >0$. On $K$ all its coefficients and the required finitely many derivatives are bounded.

Apply Lemma lem:PVbound in ${\lambda }_s$, integrate over the remaining compact variables, and include the polynomial factors and any fixed derivatives. This gives a finite-order distribution bound. The homogeneity then makes its constant independent of $L$.

For parameter-dependent coefficients, the same argument is multiplied by their finite smooth seminorms on a compact subset of $V$.

These are estimates for internal frequency distributions. They are not normal rescaling estimates of ${\mathcal{T}}_{\mathrm{form}}$ at $N$.

finite remainder