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

• Autorzy: A. Eduardo Gatto , Stephen Vági
• 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.

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 26A33: Fractional derivatives and integrals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 133
• Numer: 1
• Strony: 19-27

Daty

wydano

1999

otrzymano

1997-11-17

Twórcy

autor

A. Eduardo Gatto

• Department of the Mathematical Sciences, DePaul University, 2219 North Kenmore Ave., Chicago, Illinois 60614, U.S.A.,

autor

Stephen Vági

• Department of the Mathematical Sciences, DePaul University, 2219 North Kenmore Ave., Chicago, Illinois 60614, U.S.A.,

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).