Multipliers spaces, Muckenhoupt weights and pseudo-differential operators

Let σ(x,ξ)σ(x,ξ) be a sufficiently regular function defined on Rd×RdRd×Rd. The pseudo-differential operator with symbol σ is defined on the Schwartz class by the formulaf→σf(x)=∫Rdσ(x,ξ)fˆ(ξ)e2πixξdξ, where fˆ(ξ)=∫Rdf(x)e−2πixξdx is the Fourier transform of f.In this paper, we shall consider the regularity of the following type:(a)|∂ξασ(x,ξ)|⩽Aα(1+|ξ|)−|α|,(b)|∂ξασ(x+y,ξ)−∂ξασ(x,ξ)|⩽Aαω(|y|)(1+|ξ|)−|α|(α∈Nd)(α∈Nd), where ω   is suitable positive function and we prove boundedness results on multipliers spaces Xr=M(Hr→L2)Xr=M(Hr→L2) for pseudo-differential operators whose symbol σ(x,ξ)σ(x,ξ) satisfies the regularity condition on x.