Weighted coefficient class

Fix an open domain $U$ , weights $0<w(x)\le1$ , $0<\delta(x)\le1$ , and a small parameter $0<\varepsilon<\varepsilon_*<1$ . Write $\Lambda_\varepsilon=1+|\log\varepsilon|$ . A smooth family $f_{\varepsilon,\lambda}(x)$ belongs to the weighted coefficient class $\mathcal C^\alpha(w)$ if for every multi-index $I$ there are $C_I,m_I,n_I\ge0$ with

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

The constants and exponents are independent of $\varepsilon,\lambda,x$ ; the parameter domain is common to all derivative orders. This defines one useful convention for a uniform weighted class, not a universal meaning of that notation.

What the definition controls

Each derivative bound is an assumption. A bound on $f$ alone gives no bounds on its derivatives or those of $f/w$ . Additional parameter derivatives can be included by enlarging $x$ ; otherwise the label $\lambda$ is held fixed. Support, periodicity, and smooth extension conditions must be specified separately. The weights themselves need not be smooth.