1. CJM Online first
2. CJM 2017 (vol 70 pp. 721)
 Bao, Guanlong; Göğüş, Nihat Gökhan; Pouliasis, Stamatis

On Dirichlet Spaces with a Class of Superharmonic Weights
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the AlexanderTaylorUllman
inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where
equality occurs.
We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via
the balayage of the measure $\mu$.
We show that $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and innerouter factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.
Keywords:Dirichlet space, Hardy space, superharmonic weight Categories:30H10, 31C25, 46E15 

3. 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 

4. CJM 2013 (vol 66 pp. 284)
 Eikrem, Kjersti Solberg

Random Harmonic Functions in Growth Spaces and Blochtype Spaces
Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces
of harmonic functions in the unit disk and multidimensional unit
ball
which admit a twosided radial majorant $v(r)$.
We consider functions $v $ that fulfill a doubling condition. In the
twodimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty
(a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$
where $\xi =\{\xi_{ji}\}$ is a sequence of random
subnormal variables and $a_{ji}$ are
real; in higher dimensions we consider series of spherical harmonics.
We will obtain conditions on the coefficients $a_{ji} $ which imply
that $u$ is in $h^\infty_v(\mathbf B)$ almost surely.
Our estimate improves previous results by Bennett, Stegenga and
Timoney, and we prove that the estimate is sharp.
The results for growth spaces can easily be applied to Blochtype
spaces, and we obtain a similar characterization for these spaces,
which generalizes results by Anderson, Clunie and Pommerenke and by
Guo and Liu.
Keywords:harmonic functions, random series, growth space, Blochtype space Categories:30B20, 31B05, 30H30, 42B05 

5. CJM 2013 (vol 66 pp. 641)
 Grigor'yan, Alexander; Hu, Jiaxin

Heat Kernels and Green Functions on Metric Measure Spaces
We prove that, in a setting of local Dirichlet forms on metric measure
spaces, a twosided subGaussian estimate of the heat kernel is equivalent
to the conjunction of the volume doubling propety, the elliptic Harnack
inequality and a certain estimate of the capacity between concentric balls.
The main technical tool is the equivalence between the capacity estimate and
the estimate of a mean exit time in a ball, that uses twosided estimates of
a Green function in a ball.
Keywords:Dirichlet form, heat kernel, Green function, capacity Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07 

6. CJM 2011 (vol 64 pp. 24)
 Borodachov, S. V.

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves
We consider the problem of minimizing the energy of $N$ points
repelling each other on curves in $\mathbb{R}^d$ with the potential
$xy^{s}$, $s\geq 1$, where $\, \cdot\, $ is
the Euclidean norm. For a sufficiently smooth, simple, closed,
regular curve, we find the next order term in the asymptotics of the
minimal $s$energy. On our way, we also prove that at
least for $s\geq 2$, the minimal pairwise distance in optimal configurations
asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length
of the curve.
Keywords:minimal discrete Riesz energy, lower order term, power law potential, separation radius Categories:31C20, 65D17 

7. CJM 2011 (vol 64 pp. 869)
 Hu, ZeChun; Sun, Wei

Balayage of SemiDirichlet Forms
In this paper we study the balayage of semiDirichlet forms. We
present new results on balayaged functions and balayaged measures
of semiDirichlet
forms. Some of the results are new even in the Dirichlet forms setting.
Keywords:balayage, semiDirichlet form, potential theory Categories:31C25, 60J45 

8. CJM 2011 (vol 64 pp. 183)
 Nowak, Adam; Stempak, Krzysztof

Negative Powers of Laguerre Operators
We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$.
For these operators we prove twoweight $L^pL^q$ estimates with ranges of $q$ depending
on $p$. The case of the harmonic oscillator (Hermite operator) has recently
been treated by Bongioanni and Torrea by using a straightforward
approach of kernel estimates. Here these results are applied in certain Laguerre settings.
The procedure is fairly direct for Laguerre function expansions of
Hermite type,
due to some monotonicity properties of the kernels involved.
The case of Laguerre function expansions of convolution type is less straightforward.
For halfinteger type indices $\alpha$ we transfer the desired results from the Hermite setting
and then apply an interpolation argument based on a device we call the
convexity principle
to cover the continuous range of $\alpha\in[1/2,\infty)^d$. Finally, we investigate negative powers
of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and
isomorphic to $\mathbb Z^d_2$. The two weight $L^pL^q$ estimates we obtain in this setting are essentially
consequences of those for Laguerre function expansions of convolution type.
Keywords:potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillator Categories:47G40, 31C15, 26A33 

9. CJM 2011 (vol 63 pp. 721)
 Autin, Aymeric

Isoresonant Complexvalued Potentials and Symmetries
Let $X$ be a connected Riemannian manifold such that the resolvent of
the free Laplacian $(\Deltaz)^{1}$, $z\in\mathbb{C} \setminus
\mathbb{R}^+$, has a meromorphic continuation
through $\mathbb{R}^+$. The poles of this continuation are called
resonances. When $X$ has some symmetries, we construct complexvalued
potentials, $V$, such that the resolvent of $\Delta+V$, which has also
a meromorphic continuation, has the same resonances with
multiplicities as the free Laplacian.
Categories:31C12, 58J50 

10. CJM 2009 (vol 61 pp. 534)
 Chen, ChuanZhong; Sun, Wei

Girsanov Transformations for NonSymmetric Diffusions
Let $X$ be a diffusion process, which is assumed to be
associated with a (nonsymmetric) strongly local Dirichlet form
$(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For
$u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet
space, we investigate some properties of the Girsanov transformed
process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of
$X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$.
We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to
be in duality with respect to the measure $e^{2u}m$. We also
construct a counterexample, which shows that this condition may
not be satisfied and hence $(Y,\widehat{Y})$ may not be dual
processes. Then we present a sufficient condition under which $Y$
is associated with a semiDirichlet form. Moreover, we give an
explicit representation of the semiDirichlet form.
Keywords:Diffusion, nonsymmetric Dirichlet form, Girsanov transformation, $h$transformation, perturbation of Dirichlet form, generalized FeynmanKac semigroup Categories:60J45, 31C25, 60J57 

11. CJM 2008 (vol 60 pp. 822)
 Kuwae, Kazuhiro

Maximum Principles for Subharmonic Functions Via Local SemiDirichlet Forms
Maximum principles for subharmonic
functions in the framework of quasiregular local semiDirichlet
forms admitting lower bounds are presented.
As applications, we give
weak and strong maximum principles
for (local) subsolutions of a second order elliptic
differential operator on the domain of Euclidean space under conditions on coefficients,
which partially generalize the results by Stampacchia.
Keywords:positivity preserving form, semiDirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition Categories:31C25, 35B50, 60J45, 35J, 53C, 58 

12. CJM 2008 (vol 60 pp. 572)
 Hitrik, Michael; Sj{östrand, Johannes

NonSelfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point
This is the third in a series of works devoted to spectral
asymptotics for nonselfadjoint
perturbations of selfadjoint $h$pseudodifferential operators in dimension 2, having a
periodic classical flow. Assuming that the strength $\epsilon$
of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$
(and may sometimes reach even smaller values), we
get an asymptotic description of the eigenvalues in rectangles
$[1/C,1/C]+i\epsilon [F_01/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point
value of the flow average of the leading perturbation.
Keywords:nonselfadjoint, eigenvalue, periodic flow, branching singularity Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40 

13. CJM 2008 (vol 60 pp. 457)
 Teplyaev, Alexander

Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are
generalizations of postcrit8cally finite selfsimilar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local selfsimilarity, and allow countably many cells
connected at each junction point.
In particular, we consider postcritically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, selfsimilarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 

14. CJM 2007 (vol 59 pp. 1135)
 Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari

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 

15. CJM 2005 (vol 57 pp. 1080)
 Pritsker, Igor E.

The GelfondSchnirelman Method in Prime Number Theory
The original GelfondSchnirelman method, proposed in 1936, uses
polynomials with integer coefficients and small norms on $[0,1]$
to give a Chebyshevtype lower bound in prime number theory. We
study a generalization of this method for polynomials in many
variables. Our main result is a lower bound for the integral of
Chebyshev's $\psi$function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and
connects the subject to the potential theory with external fields
generated by polynomialtype weights. We also solve the
corresponding potential theoretic problem, by finding the extremal
measure and its support.
Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials Categories:11N05, 31A15, 11C08 

16. CJM 2004 (vol 56 pp. 529)
 MartínezFinkelshtein, A.; Maymeskul, V.; Rakhmanov, E. A.; Saff, E. B.

Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$
We consider the $s$energy
$$
E(\ZZ_n;s)=\sum_{i \neq j} K(\z_{i,n}z_{j,n}\;s)
$$
for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets
$\Ga$ in $\R^d$ having finite onedimensional Hausdorff measure, where
$$
K(t;s)=
\begin{cases}
t^{s} ,& \mbox{if } s>0, \\
\ln t, & \mbox{if } s=0,
\end{cases}
$$
is the Riesz kernel. Asymptotics for the minimum $s$energy and the
distribution of minimizing sequences of points is studied. In
particular, we prove that, for $s\geq 1$, the minimizing nodes for a
rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with
respect to arclength as $n\to\infty$.
Keywords:Riesz energy, Minimal discrete energy,, Rectifiable curves, Bestpacking on curves Categories:52A40, 31C20 

17. CJM 2003 (vol 55 pp. 576)
 Lukashov, A. L.; Peherstorfer, F.

Automorphic Orthogonal and Extremal Polynomials
It is well known that many polynomials which solve extremal problems
on a single interval as the Chebyshev or the BernsteinSzeg\"o
polynomials can be represented by trigonometric functions and their
inverses. On two intervals one has elliptic instead of trigonometric
functions. In this paper we show that the counterparts of the Chebyshev
and BernsteinSzeg\"o polynomials for several intervals can be represented
with the help of automorphic functions, socalled SchottkyBurnside
functions. Based on this representation and using the SchottkyBurnside
automorphic functions as a tool several extremal properties of such
polynomials as orthogonality properties, extremal properties with
respect to the maximum norm, behaviour of zeros and recurrence
coefficients {\it etc.} are derived.
Categories:42C05, 30F35, 31A15, 41A21, 41A50 

18. CJM 2002 (vol 54 pp. 945)
 Boivin, André; Gauthier, Paul M.; Paramonov, Petr V.

Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications
Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jetdistributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$meromorphic and $L$analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 

19. CJM 2002 (vol 54 pp. 225)
20. CJM 2000 (vol 52 pp. 815)
 Lubinsky, D. S.

On the Maximum and Minimum Modulus of Rational Functions
We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function
with numerator, denominator of degree $\leq m$, $n$, respectively, then there
exists a set $\mathcal{S}\subset [0,1] $ of linear measure $\geq
\frac{1}{4}\exp (\frac{13}{\log \lambda })$ such that for $r\in
\mathcal{S}$,
\[
\max_{z =r} R(z) / \min_{z =r}  R(z) \leq \lambda ^{m+n}.
\]
Here, one may not replace $\frac{1}{4}\exp ( \frac{13}{\log \lambda })$
by $\exp (\frac{2\varepsilon }{\log \lambda })$, for any $\varepsilon >0$.
As our motivating application, we prove a convergence result for diagonal
Pad\'{e} approximants for functions meromorphic in the unit ball.
Categories:30E10, 30C15, 31A15, 41A21 

21. CJM 1998 (vol 50 pp. 547)
22. CJM 1998 (vol 50 pp. 487)
 Barlow, Martin T.

On the Liouville property for divergence form operators
In this paper we construct a bounded strictly positive
function $\sigma$ such that the Liouville property fails for the
divergence form operator $L=\nabla (\sigma^2 \nabla)$. Since in
addition $\Delta \sigma/\sigma$ is bounded, this example also gives a
negative answer to a problem of Berestycki, Caffarelli and Nirenberg
concerning linear Schr\"odinger operators.
Categories:31C05, 60H10, 35J10 

23. CJM 1997 (vol 49 pp. 653)
24. CJM 1997 (vol 49 pp. 568)
 Mateu, Joan

A counterexample in $L^p$ approximation by harmonic functions
For ${n \over {n2}}\leq p<\infty$ we show that the
conditions $C_{2,q}(G\setminus \dox)=C_{2,q}(G \setminus
X)$ for all open sets $G$, $C_{2,q}$ denoting Bessel capacity, are not
sufficient to characterize the compact
sets $X$ with the property that each function harmonic on $\dox$
and in $L^p(X)$ is the limit in the $L^p$ norm of a sequence
of functions which are harmonic on neighbourhoods of $X$.
Categories:41A30, 31C15 
