A framework of Besov–Triebel–Lizorkin type spaces via ball quasi-Banach function sequence spaces I: real-variable characterizations

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2024年4月16日

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Jingsong Sun,
Dachun Yang,
Wen Yuan

摘要

This article is devoted to developing a new and general framework of Besov–Triebel–Lizorkin type spaces based on ball quasi-Banach function sequence spaces. The authors introduce the ball quasi-Banach function sequence space E, the corresponding sequence space ${E}_m$ $E_{m}$ ( $m\in \mathbb {Z}$ $m \in Z$ ), as well as its related inhomogeneous space Y(E) of Schwartz distributions, which contains various Besov–Triebel–Lizorkin type spaces as special cases. Via assuming that the discrete Peetre-type operator is bounded on ${E}_0$ $E_{0}$ , the mth iteration of the discrete left shift is bounded from ${E}_0$ $E_{0}$ to ${E}_m$ $E_{m}$ , and the mth iteration of the discrete right shift is bounded from ${E}_0$ $E_{0}$ to ${E}_{-m}$ $E_{- m}$ , the authors introduce, accordingly, the critical Peetre decay index $J_{{E}}$ $J_{E}$ , the lower critical smoothness index $s_{-}({E})$ $s_{-} (E)$ , and the upper critical smoothness index $s_{+}({E})$ $s_{+} (E)$ . Using these important critical indices and further assuming that there exists a sufficiently large positive integer $\gamma $ $γ$ such that the $\gamma $ $γ$ th iteration of the discrete right shift is bounded from ${E}_{\gamma }$ $E_{γ}$ to ${E}_0$ $E_{0}$ , the authors then establish the $\varphi $ $φ$ -transform characterization of Y(E) and the boundedness of almost diagonal operators on a variant of ${E}_0$ $E_{0}$ , from which the authors further deduce various equivalent characterizations of Y(E), respectively, in terms of smooth molecules, smooth atoms, maximal functions, Littlewood–Paley functions, and wavelets and, as applications, the authors finally obtain the boundedness of pseudo-differential operators and generalized Calderón–Zygmund operators on Y(E). The main novelty exists in that these framework results are of wide generality, the aforementioned three critical indices enable the authors to get rid of the dependence on the explicit expression of both $\Vert \cdot \Vert _{{E}}$ ${‖ \cdot ‖}_{E}$ and $\Vert \cdot \Vert _{Y({E})}$ ${‖ \cdot ‖}_{Y (E)}$ and the Fefferman–Stein vector-valued inequality, and the aforementioned boundedness on the discrete Peetre-type operator and the mth iterations of both the discrete left and the discrete right shifts are proved to be also necessary to guarantee the boundedness of almost diagonal operators.