Wavelets frame associated to accretive function (Q1876712)

A bounded function on \(R^n\) is said to be strongly para-accretive if there exists a constant \(C>0\) such that \(| \frac1{| Q| } \int_Q b(x)\,dx | \geq C\) for every cube \(Q\subset R^n\). For \(f,g\in L^2(R^n)\), let \(\langle f,g\rangle_b= \int f(x) \overline{g(x)}b(x)\,dx\). A family of functions \(\{\phi_\lambda\}_{\lambda\in \Lambda}\) is in the present paper called a frame w.r.t. the function \(b\) if there are constants \(0<C_1\leq C_2\) such that \(C_1\| g\| ^2 \leq \sum | \langle g, \phi_\lambda \rangle_b| ^2 \leq C_2\| g\| ^2\) for all \(g\in L^2(R^n)\). Based on molecules constructed by Coifman, a frame w.r.t. a certain function \(b\) is constructed. The frame is used to prove the \(T(b)\) theorem.