Theorem: Whole-support scalar expansion

For every $N\ge 0$,

with

Spectator derivatives through order $r$ are controlled by $\Vert A{\Vert }_{4N+24+2r}$, on fixed compact supports.

If $\nu =D_H\rho$ for a homogeneous constant-coefficient operator of order $q$, then ${u_n}^{\nu }=0$ for $n<q$. Every ${u_n}^{\nu }[A]$ depends only on jets of $A$ of order at most $2n$ at $(v,\xi )=0$. For $n<q$, the same locality holds for ${v_n}^{\nu }[A]$. For $q>0$,

Proof. The radial identity and resolvent lemma give an expansion with smooth $U_j,V_j$. Smoothness of the logarithmic pairing follows by changing $H$ to $(L,\eta )$, leaving the fixed locally integrable function $\log (-L+i0)$. Taylor expansion through order $N-j$ gives (eq:scalar-coefficients) and the stated remainder.

The derivative ledger is as follows. Radial order $N+2$ costs at most $2N+4$ amplitude derivatives. Parameter Taylor differentiation costs at most $N+1$ more. The $H$-test order costs at most $N+2+r$, and spectator differentiation costs at most $r$. Thus $4N+7+2r$ is sufficient for this part of the proof. The larger $4N+24+2r$ bound conservatively includes all fixed-support quotient, coordinate, and tail estimates.

The scaling relations

show that the coefficient of ${\delta }^{n-j}$ in ${L_{\delta }}^j{f}_{j,\delta ,H}/e_{\delta }$ is a homogeneous polynomial of degree $n$ in $H$. A profile derivative of order $q>n$ annihilates it.

A radial jet of order $j$ uses at most $2j$ amplitude derivatives at the shifted radial origin. Taking $n-j$ parameter derivatives and setting $\delta =0$ uses jets of order at most $2j+n-j\le 2n$ at zero. This proves locality of every $u_n$. For $n<q$, the nonlocal finite-part portion of $v_n$ is again polynomial of smoothing degree $n$ and vanishes. Its remaining logarithmic-profile portion uses those same finite radial jets.

At $\delta =0$,

The finite part is independent of $H$. The zero mass $\int \nu =0$ removes it and the constant logarithm of $4\alpha d/\beta$, giving (eq:v0).