Multipliers spaces and pseudo-differential operators

Abstract

Let $σ(x,ξ) $ be a sufficiently regular function defined on $R^d \times R^d.$ The pseudo-differential operator with symbol $σ$ 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 $\hat{f}(ξ)=\int_{R^d} f(x)e^{-2πixξ}dx$ is the Fourier transform of $f.$ In this paper, we shall consider the regularity of the following type : \begin{description} \item[(a)] $| \partial_ξ^ασ(x,ξ) | \leq A_α(1+| ξ|) ^{-| α|},$ \item[(b)] $| \partial_ξ^ασ(x+y,ξ) -\partial_ξ^ασ(x,ξ) | \leq A_αω(| y|) (1+| ξ|) ^{-| α|},$ \end{description} %