Smooth Molecular Decompositions of Functions and Singular Integral Operators

Under minimal assumptions on a function $\psi$ we obtain wavelet-type frames of the form $$\psi_{j,k}(x) = r^{(1/2)n j} \psi(r^j x - sk), \qquad j \in \mathbb{Z}, k \in \mathbb{Z}^n,$$ 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, Sobolev and Hardy 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 us to decompose a broad range of singular integral operators in terms of smooth molecules.