Envelope-controlled amplitude

Use the domain and parameters of a weighted coefficient class, and fix an additional pointwise envelope $0<P(x)\le1$ . An envelope-controlled amplitude of order $\alpha$ , denoted here by $a\in\mathcal A^\alpha(w,P)$ , satisfies for every $I$ ,

|\partial_x^I a_{\varepsilon,\lambda}(x)| \le C_I\varepsilon^\alpha\Lambda_\varepsilon^{m_I} \sqrt{w(x)}\,\delta(x)^{-n_I}P(x),

with the same uniformity convention and a common parameter domain. Complex amplitudes are measured by their modulus.

Scope of an amplitude bound

The estimate controls the coefficient before any oscillatory exponential is attached. Differentiating a full oscillatory field also differentiates its phase. The inequality does not assert smoothness of $a/P$ , divisibility by a cutoff, or smooth zero extension across a temporal endpoint. Such properties require their own hypotheses. No derivative of $P$ is taken when reading this pointwise bound.