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: \[f\to\sigma f(x)=\int_{R^d} \sigma(x,\xi) \hat{f}(\xi)e^{2\pi ix\xi}d\xi, \] 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)] $| \partial_{\xi}^{\alpha}\sigma(x,\xi) | \leq A_{\alpha}(1+| \xi|) ^{-| \alpha|},$ \item[(b)] $| \partial_{\xi}^{\alpha}\sigma(x+y,\xi) -\partial_{\xi}^{\alpha}\sigma(x,\xi) | \leq A_{\alpha}\omega(| y|) (1+| \xi|) ^{-| \alpha|},$ \end{description} %