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1. CJM 2007 (vol 59 pp. 1135)
Sobolev Extensions of HÃ¶lder Continuous and Characteristic Functions on Metric Spaces We study when characteristic and H\"older continuous functions
are traces of Sobolev functions on doubling metric measure spaces.
We provide analytic and geometric conditions sufficient for extending
characteristic and H\"older continuous functions into globally defined
Sobolev functions.
Keywords:characteristic function, Newtonian function, metric space, resolutivity, HÃ¶lder continuous, Perron solution, $p$-harmonic, Sobolev extension, Whitney covering Categories:46E35, 31C45 |
2. CJM 2007 (vol 59 pp. 828)
Non-Backtracking Random Walks and Cogrowth of Graphs Let $X$ be a locally finite, connected graph without vertices of
degree $1$. Non-backtracking random walk moves at each step with equal
probability to one of the ``forward'' neighbours of the actual state,
\emph{i.e.,} it does not go back along
the preceding edge to the preceding
state. This is not a Markov chain, but can be turned into a Markov
chain whose state space is the set of oriented edges of $X$. Thus we
obtain for infinite $X$ that the $n$-step non-backtracking transition
probabilities tend to zero, and we can also compute their limit when
$X$ is finite. This provides a short proof of old results concerning
cogrowth of groups, and makes the extension of that result to
arbitrary regular graphs rigorous. Even when $X$ is non-regular, but
\emph{small cycles are dense in} $X$, we show that the graph $X$ is
non-amenable if and only if the non-backtracking $n$-step transition
probabilities decay exponentially fast. This is a partial
generalization of the cogrowth criterion for regular graphs which
comprises the original cogrowth criterion for finitely generated
groups of Grigorchuk and Cohen.
Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenability Categories:05C75, 60G50, 20F69 |
3. CJM 2005 (vol 57 pp. 471)
Small Coverings with Smooth Functions under the Covering Property Axiom In the paper we formulate a Covering Property Axiom, \psmP,
which holds in the iterated perfect set model,
and show that it implies the following facts,
of which (a) and (b) are the generalizations
of results of J. Stepr\={a}ns.
\begin{compactenum}[\rm(a)~~]
\item There exists a family $\F$ of less than continuum many $\C^1$
functions from $\real$ to $\real$ such that $\real^2$ is covered
by functions from $\F$, in the sense that for every $\la
x,y\ra\in\real^2$ there exists an $f\in\F$ such that either
$f(x)=y$ or $f(y)=x$.
\item For every Borel function $f\colon\real\to\real$ there exists a
family $\F$ of less than continuum many ``$\C^1$'' functions ({\em
i.e.,} differentiable functions with continuous derivatives, where
derivative can be infinite) whose graphs cover the graph of $f$.
\item For every $n>0$ and
a $D^n$ function $f\colon\real\to\real$ there exists
a family $\F$ of less than continuum many $\C^n$ functions
whose graphs cover the graph of $f$.
\end{compactenum}
We also provide the examples showing that in the above properties
the smoothness conditions are the best possible. Parts (b), (c),
and the examples are closely related to work of
A. Olevski\v{\i}.
Keywords:continuous, smooth, covering Categories:26A24, 03E35 |