A Note on a Residual Subset of Lipschitz Functions on Metric Spaces
نویسنده
چکیده
Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous of a result proved for real valued Lipschitz maps defined on R by Alberti, Bianchini and Crippa in [1]. Mathematics Subject Classification: 53C23, 30Lxx
منابع مشابه
Compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions
We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.
متن کاملWeighted composition operators between Lipschitz algebras of complex-valued bounded functions
In this paper, we study weighted composition operators between Lipschitz algebras of complex-valued bounded functions on metric spaces, not necessarily compact. We give necessary and sufficient conditions for the injectivity and the surjectivity of these operators. We also obtain sufficient and necessary conditions for a weighted composition operator between these spaces to be compact.
متن کاملQuasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions
We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...
متن کاملWeighted Composition Operators Between Extended Lipschitz Algebras on Compact Metric Spaces
In this paper, we provide a complete description of weighted composition operators between extended Lipschitz algebras on compact metric spaces. We give necessary and sufficient conditions for the injectivity and the sujectivity of these operators. We also obtain some sufficient conditions and some necessary conditions for a weighted composition operator between these spaces to be compact.
متن کاملCompact composition operators on certain analytic Lipschitz spaces
We investigate compact composition operators on ceratin Lipschitzspaces of analytic functions on the closed unit disc of the plane.Our approach also leads to some results about compositionoperators on Zygmund type spaces.
متن کامل