BMO, boundedness of affine operators, and frames (Q1765948)

For real numbers \(r>1, s>0\), define the dilation operator \(D_rf(x)=r^{1/2}f(rx)\) and the translation operator \(T_sf(x)=f(x-s)\). For \(\phi,\psi \in L^2(R)\), define (formally) the operator \(D_{\phi,\psi}^{r,s}f= \sum_{k,l\in Z} \langle f,D_r^kT_s^l \phi\rangle D_r^kT_s^l \psi\). Several conditions for \(D_{\phi,\psi}^{r,s}\) being bounded on \(l^2(R)\) are presented, and related to the vanishing moments of \(\phi\) and \(\psi\). For example, if \(\phi\) and \(\psi\) are smooth and have a vanishing moment, then the associated operator is automatically bounded; this might no longer hold if only one of the functions has a vanishing moment. The results in the paper are obtained via the theory for singular integral operators.