Wavelet characterization of H\"ormander symbol class $S^m_{\rho,\delta}$ and applications

In this paper, we characterize the symbol in H\"ormander symbol class $S^{m}_{\rho,\delta} (m\in R, \rho,\delta\geq 0)$ by its wavelet coefficients. Consequently, we analyse the kernel-distribution property for the symbol in the symbol class $S^{m}_{\rho,\delta} (m\in R, \rho>0, \delta\geq 0)$ which is more general than known results; for non-regular symbol operators, we establish sharp $L^{2}$-continuity which is better than Calder\'on and Vaillancourt's result, and establish $L^{p} (1\leq p\leq\infty)$ continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator's continuity on the basis of the wavelets coefficients in phase space.