207 research outputs found
Metric Spaces with Linear Extensions Preserving Lipschitz Condition
We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its
finiteness means that Lipschitz functions on an arbitrary subset of M can be
linearly extended to functions on M whose Lipschitz constants are enlarged by a
factor controlled by \lambda(M). We prove that \lambda(M) is finite for several
important classes of metric spaces. These include metric trees of arbitrary
cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain
classes of Riemannian manifolds of bounded geometry and finite direct sums of
arbitrary combinations of these objects. On the other hand we construct an
example of a two-dimensional Riemannian manifold M of bounded geometry for
which \lambda(M)=\infty.Comment: Several new results are added, some important estimates are improve
Extension of Lipschitz Functions Defined on Metric Subspaces of Homogeneous Type
If a metric subspace of an arbitrary metric space carries a
doubling measure , then there is a simultaneous linear extension of all
Lipschitz functions on ranged in a Banach space to those on .
Moreover, the norm of this linear operator is controlled by logarithm of the
doubling constant of .Comment: 12 page
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