Weighted shifted radial primitive

Statement

Let $0<a<b$ , $d>0$ , $e\in\mathbb R$ , and let $f(R,y)$ be smooth and supported in $[a,b]\times\mathbb T^n$ . Put $U=R^d$ and

F(U,y)=\frac{R^e f(R,y)}{dR^{d-1}},\qquad \mathcal T_e f(R,y)=R^{-e}I_\chi F(R^d,y),

where $I_\chi$ is the localized transport primitive and its transition lies in a compact subset of $U>0$ . For

\mathcal D_R=\partial_R+MdR^{d-1}v\cdot\nabla_y,

the exact weighted identity is

(\mathcal D_R+e/R)\mathcal T_e f =f-R^{-e}dR^{d-1}\chi'(R^d)J_MF(R^d,y).

Consequences

This follows by the chain rule and cancellation of the derivative of $R^{-e}$ with $e/R$ . All radial powers and the coordinate change have bounded derivatives on the fixed positive shell. The output is compactly supported, and fixed coefficient derivatives have no loss in powers of $M$ by the fixed-shift formula.

The zero weighted mean condition $\int R^e\langle f\rangle_y\,dR=0$ is exactly $\int\langle F\rangle_y\,dU=0$ . Under a Diophantine drift, it gives arbitrarily high inverse-power bounds for the defect when the source has the required derivative bounds. For an auxiliary-independent source the defect is zero. Parameter derivatives of a moving coordinate map or of the drift require a separate chain-rule calculation.