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## Studia Mathematica

1999 | 133 | 1 | 19-27
Tytuł artykułu

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

Autorzy
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Języki publikacji
EN
Abstrakty
EN
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 $q(x,y,t) = t ∂/∂t s(x,y,t)$, and s(x,y,t) is a Coifman type approximation to the identity.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
19-27
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-11-17
Twórcy
autor
• Department of the Mathematical Sciences, DePaul University, 2219 North Kenmore Ave., Chicago, Illinois 60614, U.S.A., aegatto@condor.depaul.edu
autor
• Department of the Mathematical Sciences, DePaul University, 2219 North Kenmore Ave., Chicago, Illinois 60614, U.S.A., svagi@condor.depaul.edu
Bibliografia
• [1] M. Christ and J. L. Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), 51-80.
• [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.
• [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.
• [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).
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