Multipliers spaces, Muckenhoupt weights and pseudo-differential operators

Let σ(x,ξ) be a sufficiently regular function defined on Rd ×Rd . The pseudo-differential operator with symbol σ is defined on the Schwartz class by the formula f →σf (x) = Rd σ(x,ξ)fˆ(ξ)e2πixξ dξ, where fˆ(ξ) = Rd f (x)e−2πixξ dx is the Fourier transform of f . In this paper, we shall consider the regularity of the following type: (a) |∂α ξ σ(x,ξ)| Aα(1 + |ξ |)−|α|, (b) |∂α ξ σ(x +y, ξ) −∂α ξ σ(x,ξ)| Aαω(|y|)(1 + |ξ |)−|α| (α ∈ Nd ), where ω is suitable positive function and we prove boundedness results on multipliers spaces Xr =M(Hr →L2) for pseudo-differential operators whose symbol σ(x,ξ) satisfies the regularity condition on x. © 2005 Elsevier Inc. All rights reserved