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1. CJM Online first

Speissegger, Patrick
Quasianalytic Ilyashenko algebras
I construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation and $\log$-composition; in particular, $\mathcal{F}$ is a Hardy field. Moreover, the field $\mathcal{F} \circ (-\log)$ of germs at $0^+$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

Keywords:generalized series expansion, quasianalyticity, transition map
Categories:41A60, 30E15, 37D99, 03C99

2. CJM Online first

Günther, Christian; Schmidt, Kai-Uwe
$L^q$ norms of Fekete and related polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.

Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial
Categories:11B83, 42A05, 30C10

3. CJM 2016 (vol 68 pp. 876)

Ostrovskii, Mikhail; Randrianantoanina, Beata
Metric Spaces Admitting Low-distortion Embeddings into All $n$-dimensional Banach Spaces
For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that $n$-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension $\log n$. The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension $n$. This partially answers a question of G. Schechtman.

Keywords:basis constant, bilipschitz embedding, diamond graph, distortion, equilateral set, ultrametric
Categories:46B85, 05C12, 30L05, 46B15, 52A21

4. CJM 2015 (vol 67 pp. 1411)

Kawakami, Yu
Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces
We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Keywords:Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem.
Categories:53C42, 30D35, 30F45, 53A10, 53A15

5. CJM 2014 (vol 67 pp. 942)

Roth, Oliver
Pontryagin's Maximum Principle for the Loewner Equation in Higher Dimensions
In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin's maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in $\mathbb{C}^n$.

Keywords:univalent function, Loewner's equation
Categories:32H02, 30C55, 49K15

6. CJM 2014 (vol 67 pp. 848)

Köck, Bernhard; Tait, Joseph
Faithfulness of Actions on Riemann-Roch Spaces
Given a faithful action of a finite group $G$ on an algebraic curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for the induced action of~$G$ on the Riemann-Roch space~$H^0(X,\mathcal{O}_X(D))$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least~$2g_X-2$. This leads to a concise answer to the question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes m})$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we furthermore provide an explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of~$G$ on the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.

Keywords:faithful action, Riemann-Roch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology
Categories:14H30, 30F30, 14L30, 14D15, 11R32

7. 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

8. CJM 2013 (vol 66 pp. 387)

Mashreghi, J.; Shabankhah, M.
Composition of Inner Functions
We study the image of the model subspace $K_\theta$ under the composition operator $C_\varphi$, where $\varphi$ and $\theta$ are inner functions, and find the smallest model subspace which contains the linear manifold $C_\varphi K_\theta$. Then we characterize the case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to the study of the inner functions $\varphi$ and $\psi$ such that the composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of inner functions.

Keywords:composition operators, inner functions, Blaschke products, model subspaces
Categories:30D55, 30D05, 47B33

9. CJM 2013 (vol 65 pp. 1217)

Cruz, Victor; Mateu, Joan; Orobitg, Joan
Beltrami Equation with Coefficient in Sobolev and Besov Spaces
Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.

Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, Calderón-Zygmund operators
Categories:30C62, 35J99, 42B20

10. CJM 2011 (vol 64 pp. 1329)

Izuchi, Kei Ji; Nguyen, Quang Dieu; Ohno, Shûichi
Composition Operators Induced by Analytic Maps to the Polydisk
We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and moreover be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk.

Keywords:composition operators, Hardy spaces, polydisk
Categories:47B33, 32A35, 30H10

11. CJM 2011 (vol 64 pp. 892)

Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong
Boundedness of Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces
Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, Calderón-Zygmund operator, maximal operator
Categories:42B20, 42B25, 30L99

12. CJM 2011 (vol 63 pp. 1025)

Clouâtre, Raphaël
Universal Series on a Riemann Surface
Every holomorphic function on a compact subset of a Riemann surface can be uniformly approximated by partial sums of a given series of functions. Those functions behave locally like the classical fundamental solutions of the Cauchy-Riemann operator in the plane.

Categories:30B60, 30E10, 30F99

13. CJM 2011 (vol 63 pp. 862)

Hosokawa, Takuya; Nieminen, Pekka J.; Ohno, Shûichi
Linear Combinations of Composition Operators on the Bloch Spaces
We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space.

Keywords: composition operator, compactness, Bloch space
Categories:47B33, 30D45, 47B07

14. CJM 2010 (vol 62 pp. 1276)

El Wassouli, Fouzia
A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball
Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$ \|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty. $$

Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle
Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70

15. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for $\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a John--Nirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside $I_0$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$.

Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension
Categories:42B35, 42C10, 30D50, 28A80

16. CJM 2010 (vol 62 pp. 961)

Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
Multiplicative Isometries and Isometric Zero-Divisors
For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlet-type spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zero-divisors
Categories:30H05, 46E15

17. CJM 2009 (vol 62 pp. 646)

Rupp, R.; Sasane, A.
Reducibility in AR(K), CR(K), and A(K)
Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let $A_{\mathbb R}(K)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior $K^\circ$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A(K)$ is $1$. Finally, we also characterize all compact real symmetric sets $K$ such that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass stable rank $1$.

Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibility
Categories:46J15, 19B10, 30H05, 93D15

18. CJM 2009 (vol 61 pp. 503)

Baranov, Anton; Woracek, Harald
Subspaces of de~Branges Spaces Generated by Majorants
For a given de~Branges space $\mc H(E)$ we investigate de~Branges subspaces defined in terms of majorants on the real axis. If $\omega$ is a nonnegative function on $\mathbb R$, we consider the subspace \[ \mc R_\omega(E)=\clos_{\mc H(E)} \big\{F\in\mc H(E): \text{ there exists } C>0: |E^{-1} F|\leq C\omega \mbox{ on }{\mathbb R}\big\} . \] We show that $\mc R_\omega(E)$ is a de~Branges subspace and describe all subspaces of this form. Moreover, we give a criterion for the existence of positive minimal majorants.

Keywords:de~Branges subspace, majorant, Beurling-Malliavin Theorem
Categories:46E20, 30D15, 46E22

19. CJM 2009 (vol 61 pp. 566)

Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Pfaltzgraff, John A.
Convex Subordination Chains in Several Complex Variables
In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$ on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain over $(0,1)$.

Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordination
Categories:32H02, 30C45

20. CJM 2009 (vol 61 pp. 373)

McKee, Mark
An Infinite Order Whittaker Function
In this paper we construct a flat smooth section of an induced space $I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function is not of finite order. An asymptotic method of classical analysis is used.

Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15

21. CJM 2009 (vol 61 pp. 282)

Bouya, Brahim
Closed Ideals in Some Algebras of Analytic Functions
We obtain a complete description of closed ideals of the algebra $\cD\cap \cL$, $0<\alpha\leq\frac{1}{2}$, where $\cD$ is the Dirichlet space and $\cL$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha$.

Categories:46E20, 30H05, 47A15

22. CJM 2008 (vol 60 pp. 960)

Stahl, Saul
Erratum: On the Zeros of Some Genus Polynomials
No abstract.

Categories:05C10, 05A15, 30C15, 26C10

23. CJM 2008 (vol 60 pp. 958)

Chen, Yichao
A Note on a Conjecture of S. Stahl
S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640) conjectured that the zeros of genus polynomial are real. L. Liu and Y. Wang disproved this conjecture on the basis of Example 6.7. In this note, it is pointed out that there is an error in this example and a new generating matrix and initial vector are provided.

Keywords:genus polynomial, zeros, real
Categories:05C10, 05A15, 30C15, 26C10

24. CJM 2006 (vol 58 pp. 1026)

Handelman, David
Karamata Renewed and Local Limit Results
Connections between behaviour of real analytic functions (with no negative Maclaurin series coefficients and radius of convergence one) on the open unit interval, and to a lesser extent on arcs of the unit circle, are explored, beginning with Karamata's approach. We develop conditions under which the asymptotics of the coefficients are related to the values of the function near $1$; specifically, $a(n)\sim f(1-1/n)/ \alpha n$ (for some positive constant $\alpha$), where $f(t)=\sum a(n)t^n$. In particular, if $F=\sum c(n) t^n$ where $c(n) \geq 0$ and $\sum c(n)=1$, then $f$ defined as $(1-F)^{-1}$ (the renewal or Green's function for $F$) satisfies this condition if $F'$ does (and a minor additional condition is satisfied). In come cases, we can show that the absolute sum of the differences of consecutive Maclaurin coefficients converges. We also investigate situations in which less precise asymptotics are available.

Categories:30B10, 30E15, 41A60, 60J35, 05A16

25. CJM 2004 (vol 56 pp. 1190)

Frank, Günter; Hua, Xinhou; Vaillancourt, Rémi
Meromorphic Functions Sharing the Same Zeros and Poles
In this paper, Hinkkanen's problem (1984) is completely solved, {\em i.e.,} it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of $f^{(j)}$ for $j=1,2,3,4$

Keywords:Uniqueness, meromorphic functions, Nevanlinna theory
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