Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity

We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear Calderón-Zygmund operators. 1. Bilinear pseudodifferential operators with mild regularity Let us motivate our main result on bilinear pseudodifferential operators (ΨDOs) by revisiting some facts from the linear theory. A sufficiently regular function σ(x, ξ) defined on Rn × Rn has an associated ΨDO Tσ defined by Tσ(f)(x) = ∫ Rn σ(x, ξ)f̂(ξ)eix·ξ dξ x ∈ R, f ∈ S(R). Here S(Rn) is the Schwartz class and f̂ denotes the Fourier transform of f , f̂(ξ) = ∫ Rn e−ix·ξf(x) dx, ξ ∈ R. For m ∈ R, 0 ≤ δ, ρ ≤ 1, the symbol σ(x, ξ) belongs to Hörmander’s class Sm ρ,δ if (1.1) |∂ x ∂ β ξ σ(x, ξ)| ≤ Cα,β(1 + |ξ|) m+δ|α|−ρ|β|, x, ξ ∈ R, where α, β ∈ Zn and |α|, |β| depend on the context. The exploration of classes of smooth symbols, in particular the classes Sm ρ,δ, appears to be predominant in the ΨDO literature. However, as diverse problems in Analysis and PDEs demand, the case in which the symbol has mild or no regularity in x has received considerable attention, see for instance [33], [34], [35], [42], [44], [45], and references therein. For ω,Ω : [0,∞) → [0,∞), m ∈ R and ρ ∈ (0, 1), we write σ ∈ Sm ρ,ω,Ω (this notation is not standard, we introduce it for the sake of presentation) if (1.2) |∂ ξ σ(x, ξ)| ≤ Cβ(1 + |ξ|) m−ρ|β|, x, ξ ∈ R, and (1.3) |∂ ξ σ(x+ h, ξ)− ∂ β ξ σ(x, ξ)| ≤ Cβω(|h|)Ω(|ξ|)(1 + |ξ|) m−ρ|β|, x, ξ ∈ R. Again, the number of derivatives |β| depends on the context. Date: March 4, 2008. 2000 Mathematics Subject Classification. 42B25, 42B20, 47G30.