1. 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 

2. CJM 2012 (vol 65 pp. 879)
 Kawabe, Hiroko

A Space of Harmonic Maps from the Sphere into the Complex Projective Space
GuestOhnita and Crawford have shown the pathconnectedness of the
space of harmonic maps from $S^2$ to $\mathbf{C} P^n$
of a fixed degree and energy.It is wellknown that the $\partial$ transform is defined on this space.
In this paper,we will show that the space is decomposed into mutually disjoint connected subspaces on which
$\partial$ is homeomorphic.
Keywords:harmonic maps, harmonic sequences, gluing Categories:58E20, 58D15 

3. CJM 2012 (vol 66 pp. 197)
 Harris, Adam; Kolář, Martin

On Hyperbolicity of Domains with Strictly Pseudoconvex Ends
This article establishes a sufficient condition for Kobayashi
hyperbolicity of unbounded domains in terms of curvature.
Specifically, when $\Omega\subset{\mathbb C}^{n}$ corresponds to a
sublevel set of a smooth, realvalued function $\Psi$, such that the
form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is KÃ¤hler and
has bounded curvature outside a bounded subset, then this domain
admits a hermitian metric of strictly negative holomorphic sectional
curvature.
Keywords:Kobayashihyperbolicity, KÃ¤hler metric, plurisubharmonic function Categories:32Q45, 32Q35 

4. 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 

5. 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 

6. 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 

7. CJM 2007 (vol 59 pp. 225)
8. CJM 2005 (vol 57 pp. 506)
 Gross, Leonard; Grothaus, Martin

Reverse Hypercontractivity for Subharmonic Functions
Contractivity and hypercontractivity properties of semigroups
are now well understood when the generator, $A$, is a Dirichlet form
operator.
It has been shown that in some holomorphic function spaces the
semigroup operators, $e^{tA}$, can be bounded {\it below} from
$L^p$ to $L^q$ when $p,q$ and $t$ are suitably related.
We will show that such lower boundedness occurs also in spaces
of subharmonic functions.
Keywords:Reverse hypercontractivity, subharmonic Categories:58J35, 47D03, 47D07, 32Q99, 60J35 

9. CJM 2004 (vol 56 pp. 225)
 Blower, Gordon; Ransford, Thomas

Complex Uniform Convexity and Riesz Measure
The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~NeumannSchatten trace ideals. Banach spaces that are $q$uniformly
$\PL$convex in the sense of Davis, Garling and TomczakJaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$uniformly $\PL$convex for
$1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals Categories:46B20, 46L52 

10. CJM 1999 (vol 51 pp. 673)
 Barlow, Martin T.; Bass, Richard F.

Brownian Motion and Harmonic Analysis on Sierpinski Carpets
We consider a class of fractal subsets of $\R^d$ formed in a manner
analogous to the construction of the Sierpinski carpet. We prove a
uniform Harnack inequality for positive harmonic functions; study
the heat equation, and obtain upper and lower bounds on the heat
kernel which are, up to constants, the best possible; construct a
locally isotropic diffusion $X$ and determine its basic properties;
and extend some classical Sobolev and Poincar\'e inequalities to
this setting.
Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutions Categories:60J60, 60B05, 60J35 

11. CJM 1999 (vol 51 pp. 470)
 Bshouty, D.; Hengartner, W.

Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations
In this article we characterize the univalent harmonic mappings from
the exterior of the unit disk, $\Delta$, onto a simply connected
domain $\Omega$ containing infinity and which are solutions of the system
of elliptic partial differential equations $\fzbb = a(z)f_z(z)$
where the second dilatation function $a(z)$ is a finite Blaschke
product. At the end of this article, we apply our results to
nonparametric minimal surfaces having the property that the image
of its Gauss map is the upper halfsphere covered once or twice.
Keywords:harmonic mappings, minimal surfaces Categories:30C55, 30C62, 49Q05 

12. CJM 1998 (vol 50 pp. 1119)
 Anand, Christopher Kumar

Ward's solitons II: exact solutions
In a previous paper, we gave a correspondence between certain exact
solutions to a \((2+1)\)dimensional integrable Chiral Model and
holomorphic bundles on a compact surface. In this paper, we use
algebraic geometry to derive a closedform expression for those
solutions and show by way of examples how the algebraic data which
parametrise the solution space dictates the behaviour of the
solutions.
Dans un article pr\'{e}c\'{e}dent, nous avons d\'{e}montr\'{e} que
les solutions d'un mod\`{e}le chiral int\'{e}grable en dimension \(
(2+1) \) correspondent aux fibr\'{e}s vectoriels holomorphes sur
une surface compacte. Ici, nous employons la g\'{e}om\'{e}trie
alg\'{e}brique dans une construction explicite des solutions. Nous
donnons une formule matricielle et illustrons avec trois exemples
la signification des invariants alg\'{e}briques pour le
comportement physique des solutions.
Keywords:integrable system, chiral field, sigma model, soliton, monad, uniton, harmonic map Category:35Q51 

13. CJM 1998 (vol 50 pp. 547)
14. CJM 1998 (vol 50 pp. 193)
 Xu, Yuan

Intertwining operator and $h$harmonics associated with reflection groups
We study the intertwining operator and $h$harmonics in
Dunkl's theory on $h$harmonics associated with reflection groups. Based
on a biorthogonality between the ordinary harmonics and the action of the
intertwining operator $V$ on the harmonics, the main result provides a
method to compute the action of the intertwining operator $V$ on polynomials
and to construct an orthonormal basis for the space of $h$harmonics.
Keywords:$h$harmonics, intertwining operator, reflection group Categories:33C50, 33C45 

15. CJM 1997 (vol 49 pp. 175)
 Xu, Yuan

Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres
Based on the theory of spherical harmonics for measures invariant
under a finite reflection group developed by Dunkl recently, we study
orthogonal polynomials with respect to the weight functions
$x_1^{\alpha_1}\cdots x_d^{\alpha_d}$ on the unit sphere $S^{d1}$ in
$\RR^d$. The results include explicit formulae for orthonormal polynomials,
reproducing and Poisson kernel, as well as intertwining operator.
Keywords:Orthogonal polynomials in several variables, sphere, hharmonics Categories:33C50, 33C45, 42C10 
