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

Studia Mathematica (1999)

- Volume: 133, Issue: 1, page 19-27
- ISSN: 0039-3223

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topGatto, A., and Vági, Stephen. "On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type." Studia Mathematica 133.1 (1999): 19-27. <http://eudml.org/doc/216601>.

@article{Gatto1999,

abstract = {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.},

author = {Gatto, A., Vági, Stephen},

journal = {Studia Mathematica},

keywords = {Sobolev spaces of fractional order; homogeneous type series; -families of operators; Triebel-Lizorkin spaces},

language = {eng},

number = {1},

pages = {19-27},

title = {On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type},

url = {http://eudml.org/doc/216601},

volume = {133},

year = {1999},

}

TY - JOUR

AU - Gatto, A.

AU - Vági, Stephen

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

JO - Studia Mathematica

PY - 1999

VL - 133

IS - 1

SP - 19

EP - 27

AB - 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.

LA - eng

KW - Sobolev spaces of fractional order; homogeneous type series; -families of operators; Triebel-Lizorkin spaces

UR - http://eudml.org/doc/216601

ER -

## References

top- [1] M. Christ and J. L. Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), 51-80. Zbl0645.42017
- [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. Zbl0921.43005
- [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). Zbl0806.42013

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