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Plancherel-Pólya type inequality on spaces of homogeneous type and its applications. (English) Zbl 0920.42011
The author considers spaces of homogeneous type. He proved a discrete Calderón formula for them in [“Discrete Calderón reproducing formula” (preprint)] which he uses to give a Plancherel-Pólya formula, characterizing $$ \left\{ \sum_{k \in Z} (2^{k \alpha} | | D_k (f)| | _p)^q \right \}^{1/q} $$ where $S_k$ is an approximation to the identity and $D_k = S_k - S_{k-1}$, in terms of the behavior of $f$ on the homogeneous space analogue of dyadic cubes. This is used to extend the definition of Besov spaces $\dot{B}_p^{\alpha q}$ for homogeneous spaces from the range $-\varepsilon < \alpha < \varepsilon, 1 \leq p,q \leq \infty$ given in [{\it Y.-S. Han} and {\it E. T. Sawyer}, Mem. Am. Math. Soc. 530 (1994; Zbl 0806.42013)], to $-\varepsilon < \alpha < \varepsilon, p_0 < p \leq \infty$, and $0 < q \leq \infty$, with $p_0 = \max(\frac 1{1 + \varepsilon}, \frac 1{1+\alpha + \varepsilon})$. He similarly extends the definition of the Triebel-Lizorkin spaces $\dot{F}_p^{\alpha q}$ for homogeneous spaces to $-\varepsilon < \alpha < \varepsilon$, $p_0 < p$, $q < \infty$. He shows that $H^p(X) = \dot{F}^{0,2}_p(X)$ for $\frac 1{1+ \varepsilon} < p \leq 1$, where $H^p(X)$ is the Hardy space on spaces of homogeneous type introduced by Macías and Segovia.

MSC:
42B25Maximal functions, Littlewood-Paley theory
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
43A85Analysis on homogeneous spaces
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