On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type

Gatto, A.; Vági, Stephen

Studia Mathematica, Tome 133 (1999), p. 19-27 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce Sobolev spaces $L_{α}^{p}$ for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in $L^{p}$ with fractional derivative of order α, $D^{α} f$ , as introduced in [2], in $L^{p}$ . We show that for small α, $L_{α}^{p}$ coincides with the continuous version of the Triebel-Lizorkin space $F_{p}^{α, 2}$ as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity operator is replaced by an invertible operator. Then we show that the family $t^{α} D^{α} q (x, y, t)$ is an ε-family of operators in this new sense, where $q (x, y, t) = t \partial / \partial t s (x, y, t)$ , and s(x,y,t) is a Coifman type approximation to the identity.

Publié le : 1999-01-01

Zbl 0934.46030

EUDML-ID : urn:eudml:doc:216601

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     author = {A. Gatto and Stephen V\'agi},
     title = {On Sobolev spaces of fractional order and $\epsilon$-families of operators on spaces of homogeneous type},
     journal = {Studia Mathematica},
     volume = {133},
     year = {1999},
     pages = {19-27},
     zbl = {0934.46030},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv133i1p19bwm}
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Gatto, A.; Vági, Stephen. On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type. Studia Mathematica, Tome 133 (1999) pp. 19-27. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv133i1p19bwm/

Bibliographie

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[00001] [2] A. E. Gatto, C. Segovia and S. Vági, On fractional differentiation and integration on spaces of homogeneous type, Rev. Mat. Iberoamericana 12 (1996), 111-145. | Zbl 0921.43005

[00002] [3] Y. S. Han, B. Jawerth, M. Taibelson and G. Weiss, Littlewood-Paley theory and ε-families of operators, Colloq. Math. 60/61 (1990), 321-359.

[00003] [4] Y. S. Han and E. T. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 530 (1994). | Zbl 0806.42013