Theorem: All-region internal-frequency estimates

Fix $j+|\beta |\le 4$, a frequency multiindex $\nu$, and a compact set $B\subset {\mathbb{R}}^5$. There is $C_{B,j,\beta ,\nu }$ such that

In particular, the estimate covers every compact normalized annulus

including its part where $\kappa =0$ and $\theta \ne 0$. It also gives a fixed finite-order bound on low-frequency compact sets.

For finite smooth coefficient arrays, coefficient derivatives through order $r$ multiply the right-hand side by their order-$r$ compact-set seminorm. The degree-at-most-three part satisfies

uniformly for $w$ in a fixed compact parameter set $H$.

Proof. Lemma lem:origin, applied after integration in the compact $\theta$-variable, bounds $A_{j\beta }$ by a first-derivative test seminorm. The polynomial factors are bounded with their needed derivatives on $B$. Transferring $\nu$ derivatives gives the order $1+|\nu |$ bound.

Homogeneity is exact:

This proves (eq:frequency-bound).

For clarity, a compact normalized annulus has a finite cover consisting of a neighborhood of $\kappa =0$, the two nonzero-sheet neighborhoods, and a noncharacteristic remainder. For example, first separate $|\kappa |<2a$ with a fixed $a>0$. On its complement, use sheet neighborhoods $|\omega -s\rho |<c\rho$; outside smaller such neighborhoods the quantity $|p(\kappa )|/|\kappa {|}^2$ has a positive lower bound. The pole formula controls the sheet pieces, the smooth multiplier $-1/p$ controls the remainder, and Lemma lem:origin controls the first region. Thus no nonzero-sheet formula is substituted for the origin region. The global estimate already controls the whole cover with one finite test order.

Smooth parameter differentiation acts only on coefficients. Finally, degree $d\le 3$ scales as $L^{d-2}$, and division by $L^2$ costs at most $L^{-1}$. The finite sum gives the last bound.