Wavelet Expansions on Spaces of Homogeneous Type

Wavelet Expansions on Spaces of Homogeneous Type Up to now, we succeeded in building a Littlewood-Paley analysis on a space of homogeneous type developed by G. David, J. L. Journé and S. Semmes, and in proving the boundedness of a class of Calderón-Zygmund operators on wavelet spaces. Now we aim at bridging the gap between operator theory and wavelets. We will use the theory of Calderón-Zygmund operators to derive a wavelet expansion from a Littlewood-Paley analysis. For doing it we recall the Littlewood-Paley analysis developed by G. David, J. L. Journé and S. Semmes, which has been described in Chapter 1. Let S _k and D _k be operators defined by Coifman's construction. G. David, J. L. Journé and S. Semmes provided the following identity: For f G L ², $f = \sum\limits_k {\left( {T_N } \right)} ^{ - 1} D_k^N D_k \left( f \right)$ where the operator T _N is invertible on L ² for large N and (T _N)−¹ is bounded on L ² uniformly for large N.