Littlewood–Paley Theory for the Differential Operator ∂²/∂<var>x</var>₁² ∂²</sup>/∂<var>x</var>₂² &#8722; ∂²/∂<var>x</var>₃²

Abstract

.ions { line-height: 1.8; } .ions subb { margin-left: -1ex; font-size:80%; } .ions supp { vertical-align: 1.2ex; font-size:80%; } .ionss { line-height: 1.8; } .ionss subbb { margin-left: -3ex; font-size:80%; } .ionss suppp { vertical-align: 1.4ex; font-size:80%; } .ionsss { line-height: 1.8; } .ionsss subbbb { margin-left: -5.5ex; font-size:80%; } .ionsss supppp { vertical-align: 1.4ex; font-size:80%; } 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) → (2ν1x1,2ν2x2,2ν1+ν2x3), (ν1,ν2) ∈ 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).