1. CJM 2015 (vol 68 pp. 129)
 Shiozawa, Yuichi

Lower Escape Rate of Symmetric Jumpdiffusion Processes
We establish an integral test on the lower escape rate
of symmetric jumpdiffusion processes generated by regular Dirichlet
forms.
Using this test, we can find the speed of particles escaping
to infinity.
We apply this test to symmetric jump processes of variable order. We also derive the upper and lower escape rates of time changed
processes
by using those of underlying processes.
Keywords:lower escape rate, Dirichlet form, Markov process, time change Categories:60G17, 31C25, 60J25 

2. CJM 2015 (vol 67 pp. 1358)
3. CJM 2015 (vol 67 pp. 1161)
 Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun

Nontangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights
or in the class of $QC(\mathbb{R}^n)$ weights, and
$L_w:=w^{1}\mathop{\mathrm{div}}(A\nabla)$
the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$,
$n\ge 2$. In this article, the authors establish the nontangential
maximal function characterization
of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for
$p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and
$w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$,
the authors prove that the associated Riesz transform $\nabla L_w^{1/2}$
is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical
Hardy space $H_w^p(\mathbb{R}^n)$.
Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform Categories:42B30, 42B35, 35J70 

4. CJM 2013 (vol 66 pp. 354)
 Kellerhals, Ruth; Kolpakov, Alexander

The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3space
Due to work of W. Parry it is known that the growth
rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$
is a Salem number. This being the arithmetic situation, we prove that the simplex group
(3,5,3) has smallest growth rate among all cocompact hyperbolic
Coxeter groups, and that it is as such unique.
Our approach provides a different proof for
the analog situation in ${\mathbb H^2}$
where E. Hironaka identified Lehmer's number as the minimal growth
rate among all cocompact planar hyperbolic Coxeter groups and showed
that it is (uniquely) achieved by the Coxeter triangle group (3,7).
Keywords:hyperbolic Coxeter group, growth rate, Salem number Categories:20F55, 22E40, 51F15 

5. CJM 2013 (vol 65 pp. 1095)
 Sambou, Diomba

RÃ©sonances prÃ¨s de seuils d'opÃ©rateurs magnÃ©tiques de Pauli et de Dirac
Nous considÃ©rons les perturbations $H := H_{0} + V$ et $D := D_{0} +
V$ des Hamiltoniens libres $H_{0}$ de Pauli et $D_{0}$ de Dirac en
dimension 3 avec champ magnÃ©tique non constant, $V$ Ã©tant un
potentiel Ã©lectrique qui dÃ©croÃ®t superexponentiellement dans la
direction du champ magnÃ©tique. Nous montrons que dans des espaces de
Banach appropriÃ©s, les rÃ©solvantes de $H$ et $D$ dÃ©finies sur le
demiplan supÃ©rieur admettent des prolongements mÃ©romorphes. Nous
dÃ©finissons les rÃ©sonances de $H$ et $D$ comme Ã©tant les pÃ´les de
ces extensions mÃ©romorphes. D'une part, nous Ã©tudions la
rÃ©partition des rÃ©sonances de $H$ prÃ¨s de l'origine $0$ et d'autre
part, celle des rÃ©sonances de $D$ prÃ¨s de $\pm m$ oÃ¹ $m$ est la
masse d'une particule. Dans les deux cas, nous obtenons d'abord des
majorations du nombre de rÃ©sonances dans de petits domaines au
voisinage de $0$ et $\pm m$. Sous des hypothÃ¨ses supplÃ©mentaires,
nous obtenons des dÃ©veloppements asymptotiques du nombre de
rÃ©sonances qui entraÃ®nent leur accumulation prÃ¨s des seuils $0$ et
$\pm m$. En particulier, pour une perturbation $V$ de signe dÃ©fini,
nous obtenons des informations sur la rÃ©partition des valeurs propres
de $H$ et $D$ prÃ¨s de $0$ et $\pm m$ respectivement.
Keywords:opÃ©rateurs magnÃ©tiques de Pauli et de Dirac, rÃ©sonances Categories:35B34, 35P25 

6. CJM 2012 (vol 64 pp. 1395)
 Rodney, Scott

Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients
This article gives an existence theory for weak solutions of second order nonelliptic linear Dirichlet problems of the form
\begin{align*}
\nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta
\\
u&=\varphi\text{ on }\partial \Theta.
\end{align*}
The principal part $\xi'P(x)\xi$ of the above equation is assumed to
be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that
may vanish for nonzero $\xi\in\mathbb{R}^n$. This is achieved using
techniques of functional analysis applied to the degenerate Sobolev
spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and
$QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in
previous works.
Sawyer and Wheeden give a regularity theory
for a subset of the class of equations dealt with here.
Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions Categories:35A01, 35A02, 35D30, 35J70, 35H20 

7. CJM 2012 (vol 65 pp. 544)
 Deitmar, Anton; Horozov, Ivan

Iterated Integrals and Higher Order Invariants
We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The nonuniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 

8. CJM 2011 (vol 63 pp. 648)
 Ngai, SzeMan

Spectral Asymptotics of Laplacians Associated with Onedimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of onedimensional
selfsimilar measures that are defined by iterated function systems
with overlaps and satisfy a family of secondorder selfsimilar
identities. As applications of our result we obtain the spectral dimension
of important measures such as the infinite Bernoulli convolution
associated with the golden ratio and convolutions of Cantortype measures.
The main novelty of our result is that the iterated function systems
we consider are not postcritically finite and do not satisfy the
wellknown open set condition.
Keywords:spectral dimension, fractal, Laplacian, selfsimilar measure, iterated function system with overlaps, secondorder selfsimilar identities Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75 

9. CJM 2010 (vol 62 pp. 737)
 Ditzian, Z.; Prymak, A.

Approximation by Dilated Averages and KFunctionals
For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$
normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of
$f( \mathbf{x})$ is given by \[ A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}t\mathbf{u})d\mu(\mathbf{u}). \] It
will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has
the equivalence \[ \A_tff\_B\approx \inf \{
(\fg\_B+t^2 \P(D)g\_B): P(D)g\in B\}\quad\text{for }t>0, \]
where $\varphi(t)\approx \psi(t)$ means
$c^{1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions
for which translations are continuous isometries and $P(D)$ is an
elliptic differential operator induced by $\mu$. Many applications are
given, notable among which is the averaging operator with $d\mu(\mathbf{u})=
\frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set
in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of
$S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate
of approximation by averages on the boundary of a convex set under
more restrictive conditions is also shown to be equivalent to an
appropriate $K$functional.
Keywords:rate of approximation, Kfunctionals, strong converse inequality Categories:41A27, 41A35, 41A63 

10. CJM 2010 (vol 62 pp. 1116)
 Jin, Yongyang; Zhang, Genkai

Degenerate pLaplacian Operators and Hardy Type Inequalities on
HType Groups
Let $\mathbb G$ be a steptwo nilpotent group of Htype with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$Laplacian operator $L_{p,k} u= \operatorname{div}_X (\nabla_{X} u^{p2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.
Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, Htype groups Categories:35H30, 26D10, 22E25 

11. CJM 2009 (vol 62 pp. 74)
 Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre

Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces
We present the explicit formulas for the projectors on the generalized
eigenspaces associated with some eigenvalues for linear neutral functional
differential equations (NFDE) in $L^{p}$ spaces by using integrated
semigroup theory. The analysis is based on the main result
established elsewhere by the authors and results by Magal and Ruan
on nondensely defined Cauchy problem.
We formulate the NFDE as a nondensely defined Cauchy problem and obtain
some spectral properties from which we then derive explicit formulas for
the projectors on the generalized eigenspaces associated with some
eigenvalues. Such explicit formulas are important in studying bifurcations
in some semilinear problems.
Keywords:neutral functional differential equations, semilinear problem, integrated semigroup, spectrum, projectors Categories:34K05, 35K57, 47A56, 47H20 

12. CJM 2004 (vol 56 pp. 209)
 Schmuland, Byron; Sun, Wei

A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations
In \cite{P69}, Walter Philipp wrote that ``\dots the law of the
iterated logarithm holds for any process for which the BorelCantelli
Lemma, the central limit theorem with a reasonably good remainder and
a certain maximal inequality are valid.'' Many authors \cite{DW80},
\cite{I68}, \cite{N91}, \cite{OY71}, \cite{Y79} have followed this
plan in proving the law of the iterated logarithm for sequences (or
fields) of dependent random variables.
We carry on this tradition by proving the law of the iterated
logarithm for a random field whose correlations satisfy an exponential
decay condition like the one obtained by Spohn \cite{Sp86} for
certain Gibbs measures. These do not fall into the $\phi$mixing or
strong mixing cases established in the literature, but are needed for
our investigations \cite{SS01} into diffusions on configuration
space.
The proofs are all obtained by patching together standard results from
\cite{OY71}, \cite{Y79} while keeping a careful eye on the
correlations.
Keywords:law of the iterated logarithm Categories:60F99, 60G60 
