Geometric structure for $OpS_{1,1}^m$

Abstract

With a symbol in $OpS_{1,1}^{m}$ or its kernel-distribution, one has a great difficulty to study the continuity or other properties, here one use the wavelets bases which come from the Beylkin-Coifman-Rokhlin (B-C-R) algorithm to study such operators. Each operator in $OpS_{1,1}^{m}$ corresponds to its wavelet coefficients; withth is idea, one characterizes $OpS_{1,1}^m$ with a discrete space, and characterizes $OpS_{1,1}^{0}$ with a kernel-distribution space. As an application, theorem 1 of chapter 9 in tome II of [8] is a corollary of two theorems of this paper.