Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II

Abstract

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces $$\dot{\textbf{F}}^{\alpha }_{p,q}$$ F ˙ p , q α for the endpoint case of $$p = \infty $$ p = ∞ and the full scale of parameters $$\alpha \in \mathbb {R}$$ α ∈ R and $$q \in (0,\infty ]$$ q ∈ ( 0 , ∞ ] . In particular, a Peetre-type characterization of the anisotropic Besov space $$\dot{\textbf{B}}^{\alpha }_{\infty ,\infty } = \dot{\textbf{F}}^{\alpha }_{\infty ,\infty }$$ B ˙ ∞ , ∞ α = F ˙ ∞ , ∞ α is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $$\dot{\textbf{F}}^{\alpha }_{\infty ,q}$$ F ˙ ∞ , q α .

Classification of anisotropic Triebel-Lizorkin spaces

“…In the proof of Theorem 1.1 , the criterion ( 1.1 ) is used to control the overlap of the Fourier supports of the A -dilates and B -dilates of the analyzing vectors and , respectively, that are used to define the spaces and . Combined with our maximal characterizations of Triebel–Lizorkin spaces obtained in [ 18 , 19 ], this allows to conclude that the analyzing vectors and for A respectively B define the same space .…”

“…For , the space is a dense subspace of . This fact follows easily from the various atomic and molecular decompositions of , see, e.g., [ 4 , 6 , 18 , 19 ].…”

“…The space consists of all such that if , and In [ 6 ] and the introduction of this paper, the spaces , , are alternatively defined using the cube instead of an expansive ellipsoid . However, it is easily seen that both conditions define the same space, see, e.g., [ 19 , Lemma 2.2]. See also Theorem 3.1 below for the equivalent norm on used in the introduction.…”

“…[ 18 , Remark 3.6]). Similarly, assertions (ii) and (iii) are part of [ 19 , Theorem 3.3] (cf. [ 19 , Remark 3.4]) and [ 19 , Theorem 4.1], respectively.…”

“…Lastly, let us mention an application of Theorem 1.1 . In [ 18 , 19 ], we proved continuous maximal characterizations of anisotropic Triebel–Lizorkin spaces and obtained new results on their molecular decomposition. These results were obtained under the additional assumption that the expansive matrix A is exponential, in the sense that for some matrix .…”