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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 [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

Citations:

Zbl 0806.42013
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