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Zeitschrift für Analysis und ihre Anwendungen

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Volume 29, Issue 2, 2010, pp. 183–217
DOI: 10.4171/ZAA/1405

Published online: 2010-03-01

Littlewood–Paley Theory for the Differential Operator ∂2/∂x122/∂x22 − ∂2/∂x32

Kwok-Pun Ho[1]

(1) The Education University of Hong Kong, China

Littlewood–Paley theory for the differential operator, ∆D = 2x12x22x3 is developed. This study leads to the introduction of a new class of Triebel–Lizorkin spaces   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    α,qp   (D) to   α−2,qp  (D).

Keywords: Littlewood–Paley theory, Triebel–Lizorkin Spaces

Ho Kwok-Pun: Littlewood–Paley Theory for the Differential Operator ∂2/∂x122/∂x22 − ∂2/∂x32. Z. Anal. Anwend. 29 (2010), 183-217. doi: 10.4171/ZAA/1405