Smooth Molecular Decompositions Of Functions And Singular Integral Operators

Under minimal assumptions on a function $/psi$ we obtain wavelet-type frames of the form $/psi_{j,k}(x) = r^{(1/2)n j} /psi(r^j x - sk), j /in /integer, k /in /integer^n,$ for some $r > 1$ and $s > 0$. This collection is shown to be a frame forMoreUnder minimal assumptions on a function $/psi$ we obtain wavelet-type frames of the form $/psi_{j,k}(x) = r^{(1/2)n j} /psi(r^j x - sk), j /in /integer, k /in /integer^n,$ 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.