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 [Y.-S. Han and 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:

 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 43A85 Harmonic analysis on homogeneous spaces

Zbl 0806.42013
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