70 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
Triangular de Rham Cohomology of Compact Kahler Manifolds
We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with
values in a Lie group G. By definition, this is the quotient of the set of flat
connections in the trivial principle bundle by the so-called gauge
equivalence. We consider the case when M is a compact K\"ahler manifold and G
is a solvable complex linear algebraic group of a special class which contains
the Borel subgroups of all complex classical groups and, in particular, the
group of all triangular matrices. In this case, we get a
description of the set H^1_{DR}(M,G) in terms of the 1-cohomology of M with
values in the (abelian) sheaves of flat sections of certain flat Lie algebra
bundles with fibre (the Lie algebra of G) or, equivalently, in terms
of the harmonic forms on M representing this cohomology
Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates
First we prove a new inequality comparing uniformly the relative volume of a
Borel subset with respect to any given complex euclidean ball \B \sub \C^n
with its relative logarithmic capacity in \C^n with respect to the same ball
\B.
An analoguous comparison inequality for Borel subsets of euclidean balls of
any generic real subspace of \C^n is also proved.
Then we give several interesting applications of these inequalities.
First we obtain sharp uniform estimates on the relative size of \psh
lemniscates associated to the Lelong class of \psh functions of logarithmic
singularities at infinity on \C^n as well as the Cegrell class of
\psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W
\Sub \C^n.
Then we also deduce new results on the global behaviour of both the Lelong
class and the Cegrell class of \psh functions.Comment: 25 page
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