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1. CJM 2013 (vol 66 pp. 284)

Eikrem, Kjersti Solberg
Random Harmonic Functions in Growth Spaces and Bloch-type Spaces
Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v $ that fulfill a doubling condition. In the two-dimensional 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 Bloch-type 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, Bloch-type space
Categories:30B20, 31B05, 30H30, 42B05

2. 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 two-sided sub-Gaussian 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 two-sided estimates of a Green function in a ball.

Keywords:Dirichlet form, heat kernel, Green function, capacity
Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

3. 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 $|x-y|^{-s}$, $s\geq 1$, 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

4. CJM 2011 (vol 64 pp. 869)

Hu, Ze-Chun; Sun, Wei
Balayage of Semi-Dirichlet Forms
In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.

Keywords:balayage, semi-Dirichlet form, potential theory
Categories:31C25, 60J45

5. 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 two-weight $L^p-L^q$ 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 half-integer 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 $L^p-L^q$ 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

6. CJM 2011 (vol 63 pp. 721)

Autin, Aymeric
Isoresonant Complex-valued Potentials and Symmetries
Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian $(\Delta-z)^{-1}$, $z\in\mathbb{C} \setminus \mathbb{R}^+$, has a meromorphic continuation through $\mathbb{R}^+$. The poles of this continuation are called resonances. When $X$ has some symmetries, we construct complex-valued 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

7. CJM 2009 (vol 61 pp. 534)

Chen, Chuan-Zhong; Sun, Wei
Girsanov Transformations for Non-Symmetric Diffusions
Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) 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 semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Keywords:Diffusion, non-symmetric Dirichlet form, Girsanov transformation, $h$-transformation, perturbation of Dirichlet form, generalized Feynman-Kac semigroup
Categories:60J45, 31C25, 60J57

8. CJM 2008 (vol 60 pp. 822)

Kuwae, Kazuhiro
Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms
Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet 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, semi-Dirichlet 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

9. CJM 2008 (vol 60 pp. 572)

Hitrik, Michael; Sj{östrand, Johannes
Non-Selfadjoint 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 non-selfadjoint 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 $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation.

Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularity
Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40

10. 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 post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically 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, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18

11. CJM 2007 (vol 59 pp. 1135)

Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari
Sobolev Extensions of Hölder 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, Hölder continuous, Perron solution, $p$-harmonic, Sobolev extension, Whitney covering
Categories:46E35, 31C45

12. CJM 2005 (vol 57 pp. 1080)

Pritsker, Igor E.
The Gelfond--Schnirelman Method in Prime Number Theory
The original Gelfond--Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on $[0,1]$ to give a Chebyshev-type 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 polynomial-type 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

13. CJM 2004 (vol 56 pp. 529)

Martínez-Finkelshtein, 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 one-dimensional 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, Best-packing on curves
Categories:52A40, 31C20

14. 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 Bernstein-Szeg\"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 Bernstein-Szeg\"o polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside 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

15. 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 (jet-distributions) 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

16. CJM 2002 (vol 54 pp. 225)

Arslan, Bora; Goncharov, Alexander P.; Kocatepe, Mefharet
Spaces of Whitney Functions on Cantor-Type Sets
We introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces $\calE(K)$ can be described for Cantor-type compact sets.

Categories:46E10, 31A15, 46A04

17. 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

18. CJM 1998 (vol 50 pp. 547)

Gauthier, Paul M.
Mittag-Leffler theorems on Riemann surfaces and Riemannian manifolds
Cauchy and Poisson integrals over {\it unbounded\/} sets are employed to prove Mittag-Leffler type theorems with massive singularities as well as approximation theorems for holomorphic and harmonic functions.

Keywords:holomorphic, harmonic, Mittag-Leffler, Runge
Categories:30F99, 31C12

19. 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

20. CJM 1997 (vol 49 pp. 653)

Cascante, Carme; Ortega, Joaquin M.
On $\lowercase{q}$-Carleson measures for spaces of ${\cal M}$-harmonic functions
In this paper we study the $q$-Carleson measures for a space $h_\alpha^p$ of ${\cal M}$-harmonic potentials in the unit ball of $\C^n$, when $q
Categories:32A35, 31C15

21. CJM 1997 (vol 49 pp. 568)

Mateu, Joan
A counterexample in $L^p$ approximation by harmonic functions
For ${n \over {n-2}}\leq p<\infty$ 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

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