Multipliers spaces and pseudo-differential operators

Abstract

Let $\sigma (x,\xi )$ be a sufficiently regular function defined on $R^d\times R^d.$ The pseudo-differential operator with symbol $\sigma$ is defined on the Schwartz class by the formula: \[f\toσf(x)=\int_{R^d} σ(x,ξ) \hat{f}(ξ)e^{2πixξ}dξ, \] where $\hat{f}(\xi )={\int }_{R^d}f(x)e^{-2\pi ix\xi }dx$ is the Fourier transform of $f.$ In this paper, we shall consider the regularity of the following type : \begin{description} \item[(a)] $|{\partial }_{\xi }^{\alpha }\sigma (x,\xi )|\le A_{\alpha }(1+|\xi |{)}^{-|\alpha |},$ \item[(b)] $|{\partial }_{\xi }^{\alpha }\sigma (x+y,\xi )-{\partial }_{\xi }^{\alpha }\sigma (x,\xi )|\le A_{\alpha }\omega (|y|)(1+|\xi |{)}^{-|\alpha |},$ \end{description} %