Multipliers spaces and pseudo-differential operators

Let $\sigma (x,\xi )$ be a sufficiently regular function defined on $R^d \times R^d.$Thepseudo-differentialoperatorwithsymbol$\sigma$ is defined on the Schwartz class by the formula: $$f\to \sigma f(x)={\int }_{R^d}\sigma (x,\xi )\widehat{f}(\xi )e^{2\pi ix\xi }d\xi ,$$ where $\hat{f}(\xi)=\int_{R^d} f(x)e^{-2\pi ix\xi}dx$istheFouriertransformof$f.$ In this paper, we shall consider the regularity of the following type : \begin{description} \item[(a)] $| \partial_{\xi}^{\alpha}\sigma(x,\xi) | \leq A_{\alpha}(1+| \xi|) ^{-| \alpha|},$em[(b)]$| \partial_{\xi}^{\alpha}\sigma(x+y,\xi) -\partial_{\xi}^{\alpha}\sigma(x,\xi) | \leq A_{\alpha}\omega(| y|) (1+| \xi|) ^{-| \alpha|},$ \end{description} %