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

Gatto, 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 -