Smooth Molecular Decompositions of Functions and Singular Integral Operators

• Author(s): John E. Gilbert,
• Publisher: American Mathematical Soc.
• Pages: 74
• ISBN_10: 0821827723
  ISBN_13: 9780821827727
  
• Language: en
• Categories: Mathematics / General , Mathematics / Calculus , Mathematics / Probability & Statistics / General , Mathematics / Functional Analysis ,

Description:... Under minimal assumptions on a function $\psi$ the authors obtain wavelet-type frames of the form $\psi_{j, k}(x) = r DEGREES{(1/2)n j} \psi(r DEGREESj x - sk), j \in \integer, k \in \integer DEGREESn, $ 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 ter