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1. CMB Online first

Awonusika, Richard; Taheri, Ali
 A spectral identity on Jacobi polynomials and its analytic implications The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq j\leq \ell$, $\alpha,\beta\gt -1$) are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt -1$) into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed and a direct trace interpretation of the Maclaurin coefficients is presented. Keywords:Jacobi coefficient, Laplace-Beltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomialCategories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15

2. CMB Online first

Lee, Hao
 Irregular Weight one points with $D_{4}$ Image Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series. Keywords:weight one points, irregular, dihedral image, generalized eigenform, eigencurve, tangent space,Categories:11F33, 11F80

3. CMB Online first

Reijonen, Atte
 Remark on integral means of derivatives of Blaschke products If $B$ is the Blachke product with zeros $\{z_n\}$, then $|B'(z)|\le \Psi_B(z)$, where $$\Psi_B(z)=\sum_n \frac{1-|z_n|^2}{|1-\overline{z}_nz|^2}.$$ Moreover, it is a well-known fact that, for $0\lt p\lt \infty$, $$M_p(r,B')= \left(\frac{1}{2\pi}\int_{0}^{2\pi} |B'(re^{i\t})|^p\,d\t \right)^{1/p}, \quad 0\le r\lt 1,$$ is bounded if and only if $M_p(r,\Psi_B)$ is bounded. We find a Blaschke product $B_0$ such that $M_p(r,B_0')$ and $M_p(r,\Psi_{B_0})$ are not comparable for any $\frac12\lt p\lt \infty$. In addition, it is shown that, if $0\lt p\lt \infty$, $B$ is a Carleson-Newman Blaschke product and a weight $\omega$ satisfies a certain regularity condition, then $$\int_\mathbb{D} |B'(z)|^p\omega(z)\,dA(z)\asymp \int_\mathbb{D} \Psi_B(z)^p\omega(z)\,dA(z),$$ where $dA(z)$ is the Lebesgue area measure on the unit disc. Keywords:Bergman space, Blaschke product, Hardy space, integral meanCategories:30J10, 30H10, 30H20

4. CMB Online first

Marković, Marijan
 Differential-free characterisation of smooth mappings with controlled growth In this paper we give some generalizations and improvements of the PavloviÄ result on the Holland-Walsh type characterization of the Bloch space of continuously differentiable (smooth) functions in the unit ball in $\mathbf{R}^m$. Keywords:Bloch type space, Lipschitz type space, Holland-Walsh characterisation, hyperbolic distance, analytic function, Mobius transformCategories:32A18, 30D45

5. CMB Online first

Rocha, Pablo Alejandro
 A remark on certain integral operators of fractional type For $m, n \in \mathbb{N}$, $1\lt m \leq n$, we write $n = n_1 + \dots + n_m$ where $\{ n_1, \dots, n_m \} \subset \mathbb{N}$. Let $A_1, \dots, A_m$ be $n \times n$ singular real matrices such that $\bigoplus_{i=1}^{m} \bigcap_{1\leq j \neq i \leq m} \mathcal{N}_j = \mathbb{R}^{n},$ where $\mathcal{N}_j = \{ x : A_j x = 0 \}$, $dim(\mathcal{N}_j)=n-n_j$ and $A_1+ \dots+ A_m$ is invertible. In this paper we study integral operators of the form $T_{r}f(x)= \int_{\mathbb{R}^{n}} \, |x-A_1 y|^{-n_1 + \alpha_1} \cdots |x-A_m y|^{-n_m + \alpha_m} f(y) \, dy,$ $n_1 + \dots + n_m = n$, $\frac{\alpha_1}{n_1} = \dots = \frac{\alpha_m}{n_m}=r$, $0 \lt r \lt 1$, and the matrices $A_i$'s are as above. We obtain the $H^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})$ boundedness of $T_r$ for $0\lt p\lt \frac{1}{r}$ and $\frac{1}{q}=\frac{1}{p} - r$. Keywords:integral operator, Hardy spaceCategories:42B20, 42B30

6. CMB Online first

Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan
 Remarks on inner functions and optimal approximants We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions. Keywords:inner function, reproducing Kernel Hilbert Space, operator-theoretic function theoryCategories:46E22, 30J05

7. CMB 2017 (vol 60 pp. 673)

Abtahi, Fatemeh; Azizi, Mohsen; Rejali, Ali
 Character Amenability of the Intersection of Lipschitz Algebras Let $(X,d)$ be a metric space and $J\subseteq [0,\infty)$ be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras, and define a special Banach subalgebra of $\bigcap_{\gamma\in J}\operatorname{Lip}_\gamma X$, denoted by $\operatorname{ILip}_J X$. Mainly, we investigate $C$-character amenability of $\operatorname{ILip}_J X$, in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap, and obtain a necessary and sufficient condition for $C$-character amenability of $\operatorname{ILip}_J X$, specially Lipschitz algebras, under an additional assumption. Keywords:amenability, character amenability, Lipschitz algebra, metric spaceCategories:46H05, 46J10, 11J83

8. CMB Online first

Bu, Shangquan; Cai, Gang
 Periodic solutions of second order degenerate differential equations with delay in Banach spaces We give necessary and sufficient conditions of the $L^p$-well-posedness (resp. $B_{p,q}^s$-well-posedness) for the second order degenerate differential equation with finite delays: $(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where $A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from $L^p([-2\pi,0];X)$ (resp. $B_{p,q}^s([-2\pi,0];X)$) into $X$. Keywords:second order degenerate differential equation, Fourier multiplier theorem, well-posedness, Lebesgue-Bochner space, Besov spaceCategories:34G10, 34K30, 43A15, 47D06

9. CMB Online first

Cui, Xiaohui; Wang, Chunjie; Zhu, Kehe
 Area Integral Means of Analytic Functions in the Unit Disk For an analytic function $f$ on the unit disk $\mathbb D$ we show that the $L^2$ integral mean of $f$ on $c\lt |z|\lt r$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$ is a logarithmically convex function of $r$ on $(c,1)$, where $-3\le\alpha\le0$ and $c\in[0,1)$. Moreover, the range $[-3,0]$ for $\alpha$ is best possible. When $c=0$, our arguments here also simplify the proof for several results we obtained in earlier papers. Keywords:logarithmic convexity, area integral mean, Bergman space, Hardy spaceCategories:30H10, 30H20

10. CMB Online first

Figiel, Tadeusz; Johnson, William
 Quotients of Essentially Euclidean Spaces A precise quantitative version of the following qualitative statement is proved: If a finite dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property. Keywords:essentially euclidean spaceCategories:46B20, 46B07, 46B99

11. CMB 2017 (vol 60 pp. 571)

Li, Ji; Wick, Brett D.
 Weak Factorizations of the Hardy space $H^1(\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms This paper provides a constructive proof of the weak factorization of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms. Keywords:Hardy space, BMO space, multilinear Riesz transform, weak factorizationCategories:42B35, 42B20

12. CMB Online first

Wang, Li-an Daniel
 A Multiplier Theorem on Anisotropic Hardy Spaces We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator $T_m : H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss. Keywords:anisotropic Hardy space, multiplier, Fourier transformCategories:42B30, 42B25, 42B35

13. CMB Online first

Bu, Shangquan; Cai, Gang
 HÃ¶lder continuous solutions of degenerate differential equations with finite delay Using known operator-valued Fourier multiplier results on vector-valued HÃ¶lder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed. Keywords:well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, HÃ¶lder continuous function spaceCategories:34N05, 34G10, 47D06, 47A10, 34K30

14. CMB 2017 (vol 60 pp. 690)

Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis
 $\mathcal{Q}_p$ Spaces and Dirichlet Type Spaces In this paper, we show that the MÃ¶bius invariant function space $\mathcal {Q}_p$ can be generated by variant Dirichlet type spaces $\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures $\mu$ on the open unit disk. A criterion for the equality between the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type space $\mathcal {D}_p$ is given. We obtain a sufficient condition to construct different $\mathcal{D}_{\mu, p}$ spaces and we provide examples. We establish decomposition theorems for $\mathcal{D}_{\mu, p}$ spaces, and prove that the non-Hilbert space $\mathcal {Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu, p}$. As an application of the relation between $\mathcal {Q}_p$ and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick to prove the existence without constructing examples. Keywords:$\mathcal {Q}_p$ space, Dirichlet type space, MÃ¶bius invariant function spaceCategories:30H25, 31C25, 46E15

15. CMB 2016 (vol 60 pp. 655)

Zhuo, Ciqiang; Sickel, Winfried; Yang, Dachun; Yuan, Wen
 Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces via the sequence $\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between $f$ and the ball average $B_{\ell,2^{-k}}f$. These results give a way to introduce Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any smoothness order on metric measure spaces. As special cases, the authors obtain a new characterization of Morrey-Sobolev spaces and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent interest. Keywords:Besov space, Triebel-Lizorkin space, ball average, CalderÃ³n reproducing formulaCategories:42B25, 46E35, 42B35

16. CMB 2016 (vol 60 pp. 712)

Chen, Chung-Chuan
 Disjoint Hypercyclicity and Weighted Translations on Discrete Groups Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space $\ell^p(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained. Keywords:disjoint hypercyclicity, topological transitivity, weighted translation, $\ell^p$-spaceCategories:47A16, 47B38, 43A15

17. CMB 2016 (vol 60 pp. 131)

Gürbüz, Ferit
 Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively. Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey spaceCategories:42B20, 42B25

18. CMB 2016 (vol 60 pp. 217)

Wang, Yuanyi
 Condition $C'_{\wedge}$ of Operator Spaces In this paper, we study condition $C'_{\wedge}$ which is a projective tensor product analogue of condition $C'$. We show that the finite-dimensional OLLP operator spaces have condition $C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property. Keywords:operator space, local theory, tensor productCategory:46L07

19. CMB 2016 (vol 60 pp. 104)

Diestel, Geoff
 An Extension of Nikishin's Factorization Theorem A Nikishin-Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow. Keywords:factorization, type, cotype, Banach spacesCategories:46E30, 28A25

20. CMB 2016 (vol 60 pp. 522)

Iena, Oleksandr; Leytem, Alain
 On the Singular Sheaves in the Fine Simpson Moduli Spaces of $1$-dimensional Sheaves In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm-1$ on a projective plane we study the closed subvariety $M'$ of sheaves that are not locally free on their support. We show that for $d\ge 4$ it is a singular subvariety of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted as a (partial) modification of $M\setminus M'$ by line bundles (on support). Keywords:Simpson moduli spaces, coherent sheaves, vector bundles on curves, singular sheavesCategory:14D20

21. CMB 2016 (vol 60 pp. 586)

Liu, Feng; Wu, Huoxiong
 Endpoint Regularity of Multisublinear Fractional Maximal Functions In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy-Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on vector-valued function $\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions. Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variationCategories:42B25, 46E35

22. CMB 2016 (vol 59 pp. 813)

Kaimakamis, George; Panagiotidou, Konstantina; Pérez, Juan de Dios
 A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of Their generalized Tanaka-Webster Lie Derivative On a real hypersurface $M$ in a non-flat complex space form there exist the Levi-Civita and the k-th generalized Tanaka-Webster connections. The aim of the present paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operator with respect to the Levi-Civita connections coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in directions of any vecro field orthogonal to the structure vector field. Keywords:$k$-th generalized Tanaka-Webster connection, non-flat complex space form, real hypersurface, Lie derivative, structure Jacobi operatorCategories:53C15, 53B25

23. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
 Nilpotent Group C*-algebras as Compact Quantum Metric Spaces Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups. Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growthCategories:46L87, 20F65, 22D15, 53C23, 58B34

24. CMB 2016 (vol 60 pp. 546)

Karzhemanov, Ilya
 On Polarized K3 Surfaces of Genus 33 We prove that the moduli space of smooth primitively polarized $\mathrm{K3}$ surfaces of genus $33$ is unirational. Keywords:K3 surface, moduli space, unirationalityCategories:14J28, 14J15, 14M20

25. CMB 2016 (vol 59 pp. 834)

Liao, Fanghui; Liu, Zongguang
 Some Properties of Triebel-Lizorkin and Besov Spaces Associated with Zygmund Dilations In this paper, using CalderÃ³n's reproducing formula and almost orthogonality estimates, we prove the lifting property and the embedding theorem of the Triebel-Lizorkin and Besov spaces associated with Zygmund dilations. Keywords:Triebel-Lizorkin and Besov spaces, Riesz potential, CalderÃ³n's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theoremCategories:42B20, 42B35
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