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

100%
Eduardo Gatto A., Vági S.
Studia Mathematica
|
1999
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tom 133
|
nr 1
19-27
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.
2
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On molecules and fractional integrals on spaces of homogeneous type with finite measure

100%
Eduardo Gatto A., Vági S.
Studia Mathematica
|
1992
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tom 103
|
nr 1
25-39
EN
In this paper we prove the continuity of fractional integrals acting on nonhomogeneous function spaces defined on spaces of homogeneous type with finite measure. A definition of the molecules which are used in the $H^p$ theory is given. Results are proved for $L^p$, $H^p$, BMO, and Lipschitz spaces.
3
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On the exponential integrability of fractional integrals on spaces of homogeneous type

100%
Eduardo Gatto A., Vági S.
Colloquium Mathematicum
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1993
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tom 64
|
nr 1
121-127
EN
In this paper we show that the fractional integral of order α on spaces of homogeneous type embeds $L^{1/α}$ into a certain Orlicz space. This extends results of Trudinger [T], Hedberg [H], and Adams-Bagby [AB].
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