Anisotropic Triebel-Lizorkin spaces with doubling measures
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We introduce and study anisotropic Triebel-Lizorkin spaces associated with general expansive dilations and doubling measures on ℝn with the use of wavelet transforms. This work generalizes the isotropic methods of dyadic ϕ-transforms of Frazier and Jawerth to nonisotropic settings.
We extend results involving boundedness of wavelet transforms, almost diagonality, smooth atomic and molecular decompositions to the setting of doubling measures. We also develop localization techniques in the endpoint case of p = ∞, where the usual definition of Triebel-Lizorkin spaces is replaced by its localized version. Finally, we establish nonsmooth atomic decompositions in the range of 0 < p ≤ 1, which is analogous to the usual Hardy space atomic decompositions.