Smooth molecular decompositions of functions and singular integral operators (Q2783394)

Under minimal assumptions on a function \(\psi\) the authors obtain wavelet-type frames of the form NEWLINE\[NEWLINE \psi_{j,k}(x) = r^{n j/2} \psi(r^jx -sk),\;j \in Z,\;k\in Z^n, NEWLINE\]NEWLINE for some \(r>1\) and \(s>0\). This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Hardy and Sobolev spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows the authors to decompose a broad range of singular integral operators in terms of smooth molecules.