Littlewood–paley theory for the differential operator ∂²/∂x₁² ∂²/∂x₂² - ∂²/∂x₃²

Littlewood–Paley theory for the differential operator, ∆D = ∂2x₁∂2x₂-∂2x₃ is developed. This study leads to the introduction of a new class of Triebel–Lizorkin spaces Ḟ  α,qp   (D) associated with the dilation (x₁, x₂, x₃) → (2 ̌ ¹x₁, 2 ̌ ²x₂,2 ̌ ¹⁺ ̌ ² x₃),(v₁,v₂) ∈ ℤ². The corresponding atomic and molecular decompositions are obtained. A frame generated by modulations, dilations and translations is also studied. Using this result, we show that ∆D is a linear isomorphism from Ḟ   α,qp   (D) to Ḟ  α−2,qp(D). Copyright © 2010 European Mathematical Society.