Recent zbMATH articles in MSC 30https://zbmath.org/atom/cc/302022-11-17T18:59:28.764376ZWerkzeugRelation between matrices and the suborbital graphs by the special number sequenceshttps://zbmath.org/1496.110192022-11-17T18:59:28.764376Z"Akbaba, Ümmügülsün"https://zbmath.org/authors/?q=ai:akbaba.ummugulsun"Değer, Ali Hikmet"https://zbmath.org/authors/?q=ai:deger.ali-hikmetSummary: Continued fractions and their matrix connections have been used in many studies to generate new identities. On the other hand, many examinations have been made in the suborbital graphs under circuit and forest conditions. Special number sequences and special vertex values of minimal length paths in suborbital graphs have been associated in our previous studies. In these associations, matrix connections of the special continued fractions \(\mathcal{K}(-1/-k)\), where \(k\in\mathbb{Z}^+\), \(k\geq 2\) with the values of the special number sequences are used and new identities are obtained. In this study, by producing new matrices, new identities related to Fibonacci, Lucas, Pell, and Pell-Lucas number sequences are found by using both recurrence relations and matrix connections of the continued fractions. In addition, the farthest vertex values of the minimal length path in the suborbital graph \(\mathbf{F}_{u, N}\) and these number sequences are associated.Towards a fractal cohomology: spectra of Polya-Hilbert operators, regularized determinants and Riemann zeroshttps://zbmath.org/1496.111202022-11-17T18:59:28.764376Z"Cobler, Tim"https://zbmath.org/authors/?q=ai:cobler.tim"Lapidus, Michel L."https://zbmath.org/authors/?q=ai:lapidus.michel-lSummary: Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil in the case of curves and eventually, by Deligne in the case of varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this paper we study the properties of the derivative operator \(D = \frac{d} {dz}\) on a particular family of weighted Bergman spaces of entire functions on \(\mathbb{C}\). The operator \(D\) can be naturally viewed as the ``infinitesimal shift of the complex plane'' since it generates the group of translations of \(\mathbb{C}\). Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated with any given meromorphic function. With this construction, we show that for a wide class of meromorphic functions, the function can be recovered by using a regularized determinant involving the operator constructed from the meromorphic function. This is illustrated in some important special cases: rational functions, zeta functions of algebraic curves (or, more generally, varieties) over finite fields, the Riemann zeta function, and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed. Our construction is motivated in part by work of Herichi and the second author on the infinitesimal shift of the real line (instead of the complex plane) and the associated spectral operator, as well as by earlier work and conjectures of Deninger on the role of cohomology in analytic number theory, and a conjectural ``fractal cohomology theory'' envisioned in work of the second author and of Lapidus and van Frankenhuijsen on complex fractal dimensions.
For the entire collection see [Zbl 1381.11005].The Chern classes and Euler characteristic of the moduli spaces of abelian differentialshttps://zbmath.org/1496.140232022-11-17T18:59:28.764376Z"Costantini, Matteo"https://zbmath.org/authors/?q=ai:costantini.matteo"Möller, Martin"https://zbmath.org/authors/?q=ai:moller.martin"Zachhuber, Jonathan"https://zbmath.org/authors/?q=ai:zachhuber.jonathanLes strates de différentielles abéliennes \(\Omega\mathcal{M}_{g}(m_{1},\cdots,m_{n})\) paramétrisent les paires \((X,\omega)\) où \(X\) est une surface de Riemann de genre \(g\) et \(\omega\) une différentielle dont les zéros sont d'ordres \((m_{1},\cdots,m_{n})\). Les composantes connexes des strates ont été obtenues dans [\textit{M. Kontsevich} and \textit{A. Zorich}, Invent. Math. 153, No. 3, 631--678 (2003; Zbl 1087.32010)]. Jusqu'à présent, c'était le seul invariant topologique connu pour toute les strates (voir toutefois [\textit{A. Calderon} and \textit{N. Salter}, ``Framed mapping class groups and the monodromy of strata of abelian differentials'', Preprint, \url{arXiv:2002.02472}] pour des avancés concernant le groupe fondamental des strates). Cet article donne une formule pour la caractéristique d'Euler (orbifold) des strates.
Je vais maintenant décrire plus en détails cette formule et la stratégie de la preuve. Contrairement à ce que l'on pourrait penser, la stratégie ne suit pas celle de [\textit{J. Harer} and \textit{D. Zagier}, Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] où la caractéristique d'Euler de l'espace des modules des surfaces de Riemann est calculé. La stratégie des auteurs repose sur l'existence de la compactification lisse des strates introduite dans [\textit{M. Bainbridge} et al., ``The moduli space of multi-scale differentials'', Preprint, \url{arXiv:1910.13492}] et l'utilisation de la suite exacte d'Euler.
Les auteurs calculent la première classe de Chern et pour le caractère de Chern du fibré cotangent logarithmique de la projectivisation de cette compactification. La première classe de Chern s'exprime comme somme de la première classe de Chern et des diviseurs au bord de cette compactification. Le caractère de Chern et la caractéristique d'Euler s'expriment de manière similaire avec les objets correspondants. Les auteurs pensent pouvoir adapter ces formules à d'autres sous-variétés linéaires des strates.
Le texte est dense mais contient les rappels, illustrations et exemples suffisants pour permettre une bonne lecture. Il constitue donc un travail important dont l'étude approfondie devrait constituer un très bon investissement pour le lecteur ou la lectrice.
Reviewer: Quentin Gendron (Guanajuato)Approximation in the mean on rational curveshttps://zbmath.org/1496.140592022-11-17T18:59:28.764376Z"Biswas, Shibananda"https://zbmath.org/authors/?q=ai:biswas.shibananda"Putinar, Mihai"https://zbmath.org/authors/?q=ai:putinar.mihaiFor a positive Borel measure \(\mu\), compactly supported on the complex plane and without point masses, a theorem of Thomson, subsequently generalized to rational functions by Brennan, states that the closure \(P^2(\mu)\) in \(L^2(\mu)\) of the polynomials in one complex variable is different from \(L^2(\mu)\) if and only if there exist analytic bounded point evaluations. Now, given a complex affine curve \(\mathcal{V} \in \mathbb{C}^n\) and a positive Borel measure \(\mu\) supported by a compact subset \(K\) of \(\mathcal{V}\), the goal of the paper is to establish conditions that ensure the validity of Thomson's Theorem on algebraic curves \(\mathcal{V}\), thus relating the density of polynomials in Lebesgue \(L^2\)-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan on rational curves are also obtained.
Reviewer: Carlos Hermoso Ortíz (Madrid)Regular functions of a quaternionic variablehttps://zbmath.org/1496.300012022-11-17T18:59:28.764376Z"Gentili, Graziano"https://zbmath.org/authors/?q=ai:gentili.graziano"Stoppato, Caterina"https://zbmath.org/authors/?q=ai:stoppato.caterina"Struppa, Daniele C."https://zbmath.org/authors/?q=ai:struppa.daniele-carloPublisher's description: This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications.
As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension \(n>2\). (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four.
This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.
See the review of the first edition in [Zbl 1269.30001].A quantitative Gauss-Lucas theoremhttps://zbmath.org/1496.300022022-11-17T18:59:28.764376Z"Totik, Vilmos"https://zbmath.org/authors/?q=ai:totik.vilmosLet \(K\) be a convex subset of the complex plane and let \(K_{\varepsilon}\) be the \(\varepsilon\)-neighborhood of \(K\). For a degree-\(n\) polynomial \(P_n\), the Gauss-Lucas theorem states that if \(P_n\) has all zeros in \(K\) then the same is true for the zeros of the derivative \(P_n'\). In this paper, the following quantitative version of this classical result is proven: For any \(\varepsilon>0\), there is an \(\alpha_{\varepsilon}<1\) such that if \(P_n\) has \(k\geq\alpha_{\varepsilon}n\) zeros in \(K\) then \(P_n'\) has at least \(k-1\) zeros in \(K_{\varepsilon}\). The second main result consist in quantitative bounds for \(\alpha_{\varepsilon}\) in terms of \(\varepsilon\).
Reviewer: Klaus Schiefermayr (Wels)Classes of meromorphic harmonic functions defined by Sălăgean operatorhttps://zbmath.org/1496.300032022-11-17T18:59:28.764376Z"Dziok, J."https://zbmath.org/authors/?q=ai:dziok.jacekSummary: In this paper, we study classes of meromorphic harmonic functions in the punctured unit disk defined by Sălăgean operator. By using the extreme points theory we solve some classical problems in the defined classes of functions.Coefficient bounds for subclasses of bi-univalent functions defined by fractional derivative operatorhttps://zbmath.org/1496.300042022-11-17T18:59:28.764376Z"Eker, Sevtap Sümer"https://zbmath.org/authors/?q=ai:eker.sevtap-sumer"Şeker, Bilal"https://zbmath.org/authors/?q=ai:seker.bilal"Ece, Sadettin"https://zbmath.org/authors/?q=ai:ece.sadettin(no abstract)Normalized symmetric differential operators in the open unit diskhttps://zbmath.org/1496.300052022-11-17T18:59:28.764376Z"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waellSummary: The symmetric differential operator SDO is a simplification functioning of the recognized ordinary derivative. The purpose of this effort is to provide a study of SDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to deliver two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function, bounded turning function subclass and convolution structures. Consequently, we define a linear combination differential operator involving the Sàlàgean differential operator and the Ruscheweyh derivative. The new operator is a generalization of the Lupus differential operator. Moreover, we aim to solve some special complex boundary problems for differential equations, spatially the class of Briot-Bouquet differential equations. All solutions are symmetric under the suggested SDOs. Additionally, by using the SDOs, we introduce a generalized class of Briot-Bouquet differential equations to deliver, what is called the symmetric Briot-Bouquet differential equations. We shall show that the upper solution is symmetric in the open unit disk by considering a set of examples of univalent functions.
For the entire collection see [Zbl 1485.65002].A class of close-to-convex functions satisfying a differential inequalityhttps://zbmath.org/1496.300062022-11-17T18:59:28.764376Z"Kaur, Pardeep"https://zbmath.org/authors/?q=ai:kaur.pardeep"Billing, Sukhwinder Singh"https://zbmath.org/authors/?q=ai:billing.sukhwinder-singhSummary: Let \(\mathcal{H}^{\phi}_{\alpha}(\beta)\) denote the class of functions \(f,\) analytic in the open unit disk \(\mathbb{E}\), which satisfy the condition
\[
\Re\left[(1-\alpha)\frac{zf^\prime(z)}{\phi(z)}+\alpha\left(2+\frac{zf^{\prime\prime}(z)}{f^\prime(z)}-\frac{z\phi^\prime(z)}{\phi(z)}\right)\right]>\beta, \quad z\in\mathbb{E},
\] where \(\alpha,~\beta\) are pre-assigned real numbers and \(\phi\) is a starlike function in \(\mathbb{E}\). In the present paper, we prove that members of the class \(\mathcal{H}^{\phi}_{\alpha}(\beta)\) are close-to-convex and hence univalent for real numbers \(\alpha,~ \beta,~\alpha\leq\beta<1\) and for a starlike function \(\phi\).Coefficient functionals for starlike functions of reciprocal orderhttps://zbmath.org/1496.300072022-11-17T18:59:28.764376Z"Kumar, Virendra"https://zbmath.org/authors/?q=ai:kumar.virendra"Kumar, Sushil"https://zbmath.org/authors/?q=ai:kumar.sushil"Cho, Nak Eun"https://zbmath.org/authors/?q=ai:cho.nak-eun|cho.nakeunSummary: Several properties of the class \(\mathcal{S}^*_r(\alpha)\) of starlike functions of reciprocal order \(\alpha\) (\(0\leq \alpha <1\)) defined on the open unit disk have been studied in this paper. The paper begins with a sufficient condition for analytic functions to be in the class \(\mathcal{S}^*_r(\alpha)\). Further, the sharp bounds on third order Hermitian-Toeplitz determinant, initial inverse coefficients and initial logarithmic coefficients for functions in the class \(\mathcal{S}^*_r(\alpha)\) are derived.Third order differential subordination and superordination results for analytic functions involving the Hohlov operatorhttps://zbmath.org/1496.300082022-11-17T18:59:28.764376Z"Mishra, A. K."https://zbmath.org/authors/?q=ai:mishra.akshaya-kumar"Prajapati, A."https://zbmath.org/authors/?q=ai:prajapati.anuja"Gochhayat, P."https://zbmath.org/authors/?q=ai:gochhayat.priyabrat(no abstract)Second Hankel determinant for a subclass of analytic functions defined by Sălăgean-difference operatorhttps://zbmath.org/1496.300092022-11-17T18:59:28.764376Z"Panigrahi, T."https://zbmath.org/authors/?q=ai:panigrahi.trailokya"Murugusundaramoorthy, G."https://zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharanSummary: In the present investigation, inspired by the work on Yamaguchi type class of analytic functions satisfying the analytic criteria \(\mathfrak{Re}\{\frac{f (z)}{z}\} > 0,\) in the open unit disk \(\Delta=\{z \in \mathbb{C}\colon |z|<1\}\) and making use of Sǎlǎgean-difference operator, which is a special type of Dunkl operator with Dunkl constant \(\vartheta\) in \(\Delta \), we designate definite new classes of analytic functions \(\mathcal{R}_{\lambda}^{\beta}(\psi)\) in \(\Delta \). For functions in this new class, significant coefficient estimates \(|a_2|\) and \(a_3|\) are obtained. Moreover, Fekete-Szegő inequalities and second Hankel determinant for the function belonging to this class are derived. By fixing the parameters a number of special cases are developed are new (or generalization) of the results of earlier researchers in this direction.Coefficient bounds for a subclass of bi-univalent functions associated with Dziok-Srivastava operatorhttps://zbmath.org/1496.300102022-11-17T18:59:28.764376Z"Shabani, Mohammad Mehdi"https://zbmath.org/authors/?q=ai:shabani.mohammad-mehdi"Sababe, Saeed Hashemi"https://zbmath.org/authors/?q=ai:hashemi-sababe.saeedSummary: In this article, we represent and examine a new subclass of holomorphic and bi-univalent functions defined in the open unit disk \(\mathfrak{U} \), which is associated with the Dziok-Srivastava operator. Additionally, we get upper bound estimates on the Taylor-Maclaurin coefficients \(|a_2|\) and \(|a_3|\) of functions in the new class and improve some recent studies.Hankel determinant problems for certain subclasses of Sakaguchi type functions defined with subordinationhttps://zbmath.org/1496.300112022-11-17T18:59:28.764376Z"Singh, Gagandeep"https://zbmath.org/authors/?q=ai:singh.gagandeep"Singh, Gurcharanjit"https://zbmath.org/authors/?q=ai:singh.gurcharanjitSummary: The present investigation is concerned with the estimation of initial coefficients, Fekete-Szegö inequality, second Hankel determinants, Zalcman functionals and third Hankel determinants for certain subclasses of Sakaguchi type functions defined with subordination in the open unit disc \(E=\{z\in\mathbb{C}: |z|<1\}\). The results derived in this paper will pave the way for the further study in this direction.The second Hankel determinant for subclasses of bi-univalent functions associated with a nephroid domainhttps://zbmath.org/1496.300122022-11-17T18:59:28.764376Z"Srivastava, Hari Mohan"https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Murugusundaramoorthy, Gangadharan"https://zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharan"Bulboacă, Teodor"https://zbmath.org/authors/?q=ai:bulboaca.teodorSummary: In the present paper, we obtain the estimates for the first two initial Taylor-Maclaurin coefficients and for the upper bounds of the Fekete-Szegö functional for new subclasses of the class \(\Sigma\) of normalized analytic and bi-univalent functions, which are defined here with the aid of the Nephroid function. We also determine upper bounds of the functional \(|a_2a_4-a_3^2|\) for the functions that belong to these classes. A related open problem as well some potential directions for further researches are posed in the concluding section.Integral means and Yamashita's conjecture associated with the Janowski type \((j, k)\)-symmetric starlike functionshttps://zbmath.org/1496.300132022-11-17T18:59:28.764376Z"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Prajapati, A."https://zbmath.org/authors/?q=ai:prajapati.anuja"Gochhayat, P."https://zbmath.org/authors/?q=ai:gochhayat.priyabratSummary: Let \(L_1(r, f)\) and \(\Delta(r, f)\) denote, respectively, the integral means and the area of the image of the subdisk
\[
\mathbb{D}_r:= \{z: z \in\mathbb{C}\text{ and }|z|<r; 0\leqq r<1\}
\]
of a function \(f\), which is analytic in \(\mathbb{D}\). For \(j=0, 1, 2, \dots, k-1\) (\(k = 1, 2, 3, \dots\)), \(A\in\mathbb{C}\); \(-1\leqq B \leqq 0\) with \(A\neq B\), we introduce the family of the Janowski type \((j, k)\)-symmetric starlike functions, which is denoted by \(\mathcal{ST}_{[j, k]}(A, B)\). Here, in this article, we first derive the bounds on \(L_1(r, f_{j, k})\) for every \(f_{j, k}\in\mathcal{ST}_{[j, k]}(A, B)\). The necessary coefficient condition for functions in the class \(\mathcal{ST}_{[j, k]}(A, B)\) is then presented. Our investigation leads us to get the sharp bounds on Yamashita's functional of the form \(\Delta \left(r, \frac{z}{f_{j, k}}\right)\). Finally, we provide the sharp estimate of the \(n\)th logarithmic coefficient.Conformally natural extensions of vector fields and applicationshttps://zbmath.org/1496.300142022-11-17T18:59:28.764376Z"Fan, Jinhua"https://zbmath.org/authors/?q=ai:fan.jinhua"Hu, Jun"https://zbmath.org/authors/?q=ai:hu.jun.2The authors study an integral operator extending tangent vector fields along the unit circle \(\mathbb S^1\). For an orientation-preserving homeomorphism \(h\) of \(\mathbb S^1\), denote \(\|h\|_{\text{cr}}\) the cross-ratio distortion norm of \(h\). The authors investigate a conformally natural extension of a continuous tangent vector field along \(\mathbb S^1\). Let \(C^0(\mathbb S^1,\mathbb C)\) be the collection of continuous functions from \(\mathbb S^1\) to \(\mathbb C\). Given an element \(V\in C^0(\mathbb S^1,\mathbb C)\) and \(z\in \mathbb D=\{z\in\mathbb C:|z|<1\}\), define
\[
L_0(V)(z)=\frac{(1-|z|^2)^3}{2\pi i}\int_{\mathbb S^1}\frac{V(\zeta)d\zeta}{(1-\overline z\zeta)^3(\zeta-z)}.
\]
The authors prove the following theorems.
Theorem 1. The operator \(L_0\) is conformally natural in the following sense:
1. If \(V\in C^0(\mathbb S^1,\mathbb C)\) has a continuous extension \(H\) to the closure \(\overline{\mathbb D}\) of \(\mathbb D\) that is holomorphic in \(\mathbb D\), then \(L_0(V)=H\).
2. For any element \(g\) in \(\text{Möb}(\mathbb S^1)\) and \(V\in C^0(\mathbb S^1,\mathbb C)\),
\[
L_0(g^*V)=g^*(L_0(V)),
\]
where \(g^*(V)=(V\circ g^{-1})/(g^{-1})'.\)
Let \(\Lambda(\mathbb S^1)\) be the collection of all Zygmund bounded tangent vector fields along \(\mathbb S^1\).
Theorem 2. There is a constant \(C>0\) such that for any \(V\in\Lambda(\mathbb S^1)\),
\[
\frac{1}{C}\|V\|_{\text{cr}}\leq\|\overline{\partial} L_0(V)\|_{\infty}\leq C\|V\|_{\text{cr}}.
\]
There are more characteristics of \(L_0\) and applications to the tangent spaces of the Teichmüller space.
Reviewer: Dmitri V. Prokhorov (Saratov)Conformally formal manifolds and the uniformly quasiregular non-ellipticity of \((\mathbb{S}^2 \times\mathbb{S}^2)\#(\mathbb{S}^2\times\mathbb{S}^2)\)https://zbmath.org/1496.300152022-11-17T18:59:28.764376Z"Kangasniemi, Ilmari"https://zbmath.org/authors/?q=ai:kangasniemi.ilmariThe paper is devoted to quasiregular self-maps answering an open question. Given two oriented Riemann \(n\)-manifolds \(M\) and \(N\), a map \(f:M\to N\) is \(K\)-quasiregular, \(K\geq1\), if \(f\) is continuous, \(f\in W_{\text{loc}}^{1,n}(M,N)\) and \(|Df(x)|^n\leq KJ_f(x)\) for almost every \(x\in M\). Here, \(|\cdot|\) is the operator norm, and \(J_f\) is the Jacobian determinant. A \(K\)-quasiregular homeomorphism is called \(K\)-quasiconformal. A self-map \(M\to M\) is uniformly \(K\)-quasiregular if every iterate of \(f\) is \(K\)-quasiregular. A closed, connected, oriented Riemann \(n\)-manifold \(M\) is quasiregularly elliptic if there exists a non-constant quasiregular map \(f:\mathbb R^n\to M\). Similarly, \(M\) is uniformly quasiregularly elliptic if there exists a non-constant non-injective uniformly quasiregular self-map \(f:M\to M\). The author resolves negatively an open question in the following theorem.
Theorem 1.1. The manifold \((\mathbb S^2\times\mathbb S^2)\#(\mathbb S^2\times\mathbb S^2)\) is not uniformly quasiregularly elliptic.
The author gives also the concrete topological obstruction for uniformly quasiregular ellipticity.
Reviewer: Dmitri V. Prokhorov (Saratov)Gromov hyperbolicity, John spaces, and quasihyperbolic geodesicshttps://zbmath.org/1496.300162022-11-17T18:59:28.764376Z"Zhou, Qingshan"https://zbmath.org/authors/?q=ai:zhou.qingshan"Li, Yaxiang"https://zbmath.org/authors/?q=ai:li.yaxiang"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.anttiAssume that \((D, d)\) is a locally compact, rectifiably connected and noncomplete metric space. For a constant \(a\geq 1\), the space \(D\) is called an \(a\)-John domain if each pair of points \(x, y\in D\) can be joined by a rectifiable arc \(\alpha\) in \(D\) such that for all \(z\in \alpha\),
\[
\min\big\{\text{length }(\alpha[x,z]),\text{length }(\alpha[z,y])\leq ad(z)\big\}.
\]
The arc \(\alpha\) is called a double \(a\)-cone arc. In this paper, the authors focus on the geometric properties of quasihyperbolic geodesics in John metric spaces. There is a question whether or not each quasihyperbolic geodesic in a John domain is a double cone arc. \textit{F. W. Gehring} et al. [Math. Scand. 65, No. 1, 75--92 (1989; Zbl 0702.30007)] had already considered this question in Euclidean John domains. The answer to this question is negative in general. There is a positive answer in the case of simply connected plane John domains. This raises the natural problem of determining necessary and sufficient conditions for quasihyperbolic geodesics in a John space to be double cone arcs. Various authors have contributed to this line of work. For example,
\textit{J. Heinonen} [Rev. Mat. Iberoam. 5, No. 3-4, 97--123 (1989; Zbl 0712.30017)] posed the question of whether or not an \(a\)-John domain in \(\mathbb{R}^{n}\), which is quasiconformally equivalent to the unit ball, has the property that every quasihyperbolic geodesic is a double cone arc. Using the conformal modulus of path families and Ahlfors \(n\)-regularity of \(n\)-dimensional Hausdorff measure of \(\mathbb{R}^{n}\) , \textit{M. Bonk} et al. [Uniformizing Gromov hyperbolic spaces. Paris: Société Mathématique de France (2001; Zbl 0970.30010)] gave an affirmative answer for bounded domains with a constant depending on the space dimension \(n\). In particular, they proved that every quasihyperbolic geodesic in a bounded Gromov hyperbolic John domain in \(\mathbb{R}^{n}\) is a double cone arc. In this paper, the authors show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a double cone arc. This result provides a new approach to the elementary metric geometry question, formulated by Heinonen [loc. cit] and which has been studied by Gehring et al. [loc. cit.]. Also the authors obtain a simple geometric condition connecting uniformity of a metric space with the existence of a Gromov hyperbolic quasihyperbolization.
Reviewer: Shengjin Huo (Tianjin)Unified value sharing of meromorphic functionshttps://zbmath.org/1496.300172022-11-17T18:59:28.764376Z"Charak, Kuldeep Singh"https://zbmath.org/authors/?q=ai:charak.kuldeep-singh"Korhonen, Risto"https://zbmath.org/authors/?q=ai:korhonen.risto"Kumar, Gaurav"https://zbmath.org/authors/?q=ai:kumar.gauravIt is known as the Rubel and Yang theorem that if a non-constant entire function \(f\) and its derivative \(f'\) share two distinct values CM, then \(f'\equiv f\). This result was extended for the case when \(f\) is a meromorphic function in the complex plane. Further, by means of Nevanlinna theory, this type uniqueness problems have been studied with higher differential operator, difference operator, \(q\)-difference operator, in place of the differential operator \(d/dz\), e.g., [\textit{J. Heittokangas} et al., Complex Var. Elliptic Equ. 56, No. 1--4, 81--92 (2011; Zbl 1217.30029)]. The authors introduce a general class
\[
\mathcal{B}_f=\Big\{g: g\text{ is meromorphic in }\mathbb{C}\text{ and }m\Big(r,\frac{g}{f}\Big)=S(r,f)\Big\},
\]
which includes as a special case the derivative functions \(f^{(n)}\), \(n\in\mathbb{N}\), the differences \(\Delta^nf(z)\) and shifts \(f(z+c)\), \(c\in \mathbb{C}\setminus \{0\}$ of \(f$ if the hyper-order of \(f$ is less than one, and the \(q\)-shifts and \(q\)-differences of \(f\) in the case \(f\) is of order zero. As the authors point out, using the class \(\mathcal{B}_f\), uniqueness problems concerning \(f\) and its derivatives, shifts, differences and their combinations, can be considered in a unified way. One of the main results is the following: Let \(f\) be a meromorphic function and \(g\in \mathcal{B}_f\) be such that \(f\) and \(g\) share \(0\), \(1\), \(\infty\) CM with \(\overline{N}_{1)}(r,f)=S(r,f)\). Then \(f\equiv g\). For the treatments of difference operators, the proofs require the recent developments of Nevanlinna theory, e.g., [\textit{R. Halburd} and the third author, Proc. Edinb. Math. Soc., II. Ser. 57, No. 2, 493--504 (2014; Zbl 1343.30020)].
For the entire collection see [Zbl 1484.34002].
Reviewer: Katsuya Ishizaki (Chiba)Uniqueness and periodicity for meromorphic functions with partial sharing valueshttps://zbmath.org/1496.300182022-11-17T18:59:28.764376Z"Lin, Weichuan"https://zbmath.org/authors/?q=ai:lin.weichuan"Chen, Shengjiang"https://zbmath.org/authors/?q=ai:chen.shengjiang"Gao, Xiaoman"https://zbmath.org/authors/?q=ai:gao.xiaomanSummary: We prove a periodic theorem of meromorphic functions of hyper-order \(\rho_2(f) < 1\). As an application, we obtain the corresponding uniqueness theorem on periodic meromorphic functions. In addition, we show the accuracy of the results by giving some examples.Approximation by polyanalytic functions in Hölder spaceshttps://zbmath.org/1496.300192022-11-17T18:59:28.764376Z"Mazalov, M. Ya."https://zbmath.org/authors/?q=ai:mazalov.maksim-yaSummary: The problem of approximation of functions on plane compact sets by polyanalytic functions of order higher than two in the Hölder spaces \(C^m\), (\(m\in(0, 1)\)), is significantly more complicated than the well-studied problem of approximation by analytic functions. In particular, the fundamental solutions of the corresponding operators belong to all the indicated Hölder spaces, but this does not lead to the triviality of the approximation conditions. In the model case of polyanalytic functions of order 3, approximation conditions and a constructive approximation method generalizing the Vitushkin localization method are studied. Some unsolved problems are formulated.A Neumann problem for the polyanalytic operator in planar domains with harmonic Green functionhttps://zbmath.org/1496.300202022-11-17T18:59:28.764376Z"Akel, Mohamed"https://zbmath.org/authors/?q=ai:akel.mohamed-s|akel.mohamed-s-m"Begehr, Heinrich"https://zbmath.org/authors/?q=ai:begehr.heinrich"Mohammed, Alip"https://zbmath.org/authors/?q=ai:mohammed.alipAuthors' abstract: The higher order Pompeiu integral operator providing a particular solution to the inhomogeneous polyanalytic equation provides via the higher order Cauchy-Pompeiu representation an attempt for solving the respective higher order Dirichlet problem under certain solvability conditions. An alteration of this Pompeiu operator, the Schwarz-Pompeiu integral serves to solve the well-posed Schwarz problem for the polyanalytic operator via the Cauchy-Schwarz-Pompeiu representation. This alteration, explicitly known for the unit disc, is available for domains with the harmonic Green function. Another alteration of the Pompeiu operator with regard to the Neumann problem is introduced also for domains with the harmonic Green function. In fact, the novelty of the new integral operator is not the operator itself. For the unit disc, it is just the known higher order polyanalytic Pompeiu operator. For other domains however, an explicit evaluation of the integrals involved is not possible. The novelty is the combination with the effect of the harmonic Green function, which enables to handle the Neumann problem. As the higher order Neumann problem, however, is also over-determined for the polyanalytic operator, again solvability conditions on the data are required. They are worked out here in combination with the representation formula for the solution.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Trudinger-Moser type inequalities with a symmetric conical metric and a symmetric potentialhttps://zbmath.org/1496.300212022-11-17T18:59:28.764376Z"de Souza, Manassés X."https://zbmath.org/authors/?q=ai:de-souza.manasses-xSummary: Let \((M, g)\) be a closed Riemann surface, where the metric \(g\) has certain conical singularities at finite points. Suppose \(\boldsymbol{\Gamma}\) is a group with elements of isometries acting on \((M, g)\). In this paper, Trudinger-Moser inequalities involving \(\boldsymbol{\Gamma}\) and the operador \(\Delta_g + V\) are established, where \(\Delta_g\) denotes the Laplace-Beltrami operator associated to \(g\) and the potential \(V : M \to (0, \infty)\) belongs to a class of symmetric and continuous functions. Moreover, via the method of blow-up analysis, the corresponding extremals are also obtained.The Poincaré exponent and the dimensions of Kleinian limit setshttps://zbmath.org/1496.300222022-11-17T18:59:28.764376Z"Fraser, Jonathan M."https://zbmath.org/authors/?q=ai:fraser.jonathan-mA Kleinian group \(\Gamma\) is a discrete group of orientation-preserving isometries of the hyperbolic space \( \mathbb H^n\). In the Poincaré disk model, the boundary at infinity of \( \mathbb H^n\) can be identified with the unit sphere \( S^{n-1}\) in \( \mathbb R^n\), and hyperbolic isometries act by conformal diffeomorphisms of the unit disk \( \mathbb D^n\).
An interesting phenomenon is that the orbit of a point under the action of a Kleinian group in \( \Gamma \) can accumulate to the sphere at infinity. The rate at which this happens is measured by the so-called Poincaré exponent \( \delta (\Gamma) \) of \( \Gamma \). The subspace of the sphere at infinity consisting of all the accumulation points of an orbit is called the limit set \( L (\Gamma) \) of \( \Gamma \), and often displays intricate fractal geometry. A celebrated result in this area is that for a geometrically finite nonelementary Kleinian group \( \Gamma \) the Poincaré exponent is equal to the Hausdorff dimension and the upper box dimension of the limit set.
In this paper the author proposes an elementary proof of the fact that the Poincaré exponent \( \delta (\Gamma) \) of a nonelementary Kleinian group is a lower bound for the upper box dimension of the limit set \( L (\Gamma) \). The proof is based on simple estimates for the Euclidean volume of hyperbolic balls, and only involves elementary methods from hyperbolic and fractal geometry.
Reviewer: Lorenzo Ruffoni (Medford)Thurston's boundary for Teichmüller spaces of infinite surfaces: the length spectrumhttps://zbmath.org/1496.300232022-11-17T18:59:28.764376Z"Šarić, Dragomir"https://zbmath.org/authors/?q=ai:saric.dragomirLet \(X_0\) be an infinite area geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. The Teichmüller space \(T(X_0)\) of \(X_0\) is the space of all quasiconformal deformations of \(X_0\) modulo post-compositions by conformal maps and homotopies. Unlike surfaces of finite type, \(T(X_0)\) is an infinite-dimensional manifold. In analogy with the finite-type surfaces, the Teichmüller space \(T(X_0)\) is determined by its marked length spectrum. By a result of Shiga, the topology induced by the length spectrum distance on \(T(X_0)\) is equal to the Teichmüller topology when the surface \(X_0\) has a geodesic pants decomposition with lengths of cuffs pinched between two positive constants.
In this paper the author introduces the Thurston's boundary to \(T(X_0)\) using a construction which is analogue to Thurston's one for finite-type surfaces. He defines an appropriate space of functionals \(l^\infty_{X_0}\) endowed with a normalized supremum norm. He proves that the length spectrum Thurston's boundary of \(T(X_0)\) is the closure of the space of projective bounded measured laminations \(PML_{bdd}(X_0)\) in \(Pl^\infty_{X_0}\), where \(Pl^\infty_{X_0}\) has the quotient topology induced by the topology on \(l^\infty_{X_0}\) coming from the normalized supremum norm. Thuston's boundary coincides with \(PML_{bdd}(X_0)\) when \(X_0\) can be decomposed into a countable union of geodesic pairs of pants whose boundary geodesics \(\{a_n\}_{n\in \mathbb{N}}\) have lengths pinched between two positive constants. When a subsequence of the lengths of the boundary curves of the geodesic pairs of pants \(\{a_n\}_{n \in \mathbb N}\) converges to zero, Thurston's boundary using the length spectrum is strictly larger than the space of projective bounded measured laminations.
The author also proves that the natural action of the quasi-conformal mapping class group on \(T(X_0)\) extends to a continuous action on the length spectrum Thurston's closure of \(T(X_0)\).
Reviewer: Valentina Disarlo (Heidelberg)Asymptotic first boundary value problem for elliptic operatorshttps://zbmath.org/1496.300242022-11-17T18:59:28.764376Z"Falcó, Javier"https://zbmath.org/authors/?q=ai:falco.javier"Gauthier, Paul M."https://zbmath.org/authors/?q=ai:gauthier.paul-mSummary: In 1955, Lehto showed that, for every measurable function \(\psi\) on the unit circle \(\mathbb{T},\) there is a function \(f\) holomorphic in the unit disc, having \(\psi\) as radial limit a.e. on \(\mathbb{T}\). We consider an analogous problem for solutions \(f\) of homogenous elliptic equations \(Pf=0\) and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.Some remarks on the growth of composite \(p\)-adic entire functionhttps://zbmath.org/1496.300252022-11-17T18:59:28.764376Z"Biswas, Tanmay"https://zbmath.org/authors/?q=ai:biswas.tanmay"Biswas, Chinmay"https://zbmath.org/authors/?q=ai:biswas.chinmaySummary: In this paper we wish to introduce the concept of generalized relative index-pair \((\alpha ,\beta)\) of a \(p\)-adic entire function with respect to another \(p\)-adic entire function and then prove some results relating to the growth rates of composition of two \(p\)-adic entire functions with their corresponding left and right factors.On the Dirichlet problem for not strongly elliptic second-order equationshttps://zbmath.org/1496.300262022-11-17T18:59:28.764376Z"Bagapsh, Astamur O."https://zbmath.org/authors/?q=ai:bagapsh.astamur-o"Mazalov, Maksim Ya."https://zbmath.org/authors/?q=ai:mazalov.maksim-ya"Fedorovskiy, Konstantin Yu."https://zbmath.org/authors/?q=ai:fedorovskiy.konstantin-yuThe authors consider the problem on the regularity of domains in \(\mathbb C\), with respect to second-order operators, which are not strongly elliptic.Approximation theorems for Pascali systemshttps://zbmath.org/1496.300272022-11-17T18:59:28.764376Z"Drnovšek, Barbara Drinovec"https://zbmath.org/authors/?q=ai:drnovsek.barbara-drinovec"Kuzman, Uroš"https://zbmath.org/authors/?q=ai:kuzman.urosLet \(\Omega\subset\mathbb{C}\) a domain in the complex plane. In the paper, the Pascali system on \(\Omega\) is considered. The Pascali system is an elliptic system which can be written in the following normalized form (as the Bers-Vekua equation):
\[
w_{\overline{\zeta}}+B_1w + B_2\overline{w}=0,
\]
where \(w:\Omega\rightarrow\mathbb{C}^n\) is a complex vector function, while \(B_1\) and \(B_2\) are \(n\times n\) matrix functions. In the paper, \(B_1\) and \(B_2\) are smooth matrix functions. The solutions of this system form a subclass \(\mathcal{O}_B(\Omega)\) of generalized analytic vectors, which correspond to elliptic systems with vanishing Beltrami coefficient. Since \(B_1\) and \(B_2\) are smooth matrix functions then the elements of the set \(\mathcal{O}_B(\Omega)\) are smooth vector functions. Approximation theorems are considered. The proofs are relied on a version of the Runge approximation theorem for generalized analytic vectors, which was proved by \textit{B. Goldschmidt} [Math. Nachr. 90, 57--90 (1979; Zbl 0432.30037)], and the work on Pascali systems by \textit{A. Sukhov} and \textit{A. Tumanov} [J. Anal. Math. 116, 1--16 (2012; Zbl 1273.32032)]. The following result is a Mergelyan-type theorem.
Theorem 1.1. Let \(\Omega\subset\mathbb{C}\) be an open subset, let \(S\subset\mathbb{C}\) be an admissible compact set, and let \(B_1\) and \(B_2\) be \(n\times n\) matrix functions with coefficients in \(\mathcal{C}^\infty(\Omega)\). Given \(\epsilon>0\) and a smooth map \(f:S\rightarrow\mathbb{C}^n\) whose restriction to the interior \(\stackrel{\circ}S\) lies in \(\mathcal{O}_B(\stackrel{\circ}S)\), there exists \(w\in\mathcal{O}_B(\Omega)\) such that \(|f(\zeta)-w(\zeta)|<\epsilon\) for all \(\zeta\in S\).
A version of Carleman's approximation theorem for solutions of Pascali systems is provided. All results are valid for \(n=1\).
Reviewer: Konstantin Malyutin (Kursk)Slice regular functions as covering maps and global \(\star\)-rootshttps://zbmath.org/1496.300282022-11-17T18:59:28.764376Z"Altavilla, Amedeo"https://zbmath.org/authors/?q=ai:altavilla.amedeo"Mongodi, Samuele"https://zbmath.org/authors/?q=ai:mongodi.samueleThe present work aims at studying slice regular functions of a quaternionic variable as covering maps and the existence and nature of global \(k\)-th \(\star\)-roots of slice functions. There are several steps done to reach the result. The authors introduce the formalism of stem functions and of slice preserving functions and their relations with the \(\star\)-product. They further show that, under suitable natural hypotheses, a slice regular function that is also a finite map is a covering map. Section 5 contains the main outcomes of this paper. On the basis of the results of the previous section, they are able to prove that, under suitable natural hypotheses, any slice regular function defined on a domain without real points admits \(k^2\) \(k\)-th \(\star\)-roots.
Reviewer: Swanhild Bernstein (Freiberg)Ahlfors regular conformal dimension of metrics on infinite graphs and spectral dimension of the associated random walkshttps://zbmath.org/1496.300292022-11-17T18:59:28.764376Z"Sasaya, Kôhei"https://zbmath.org/authors/?q=ai:sasaya.koheiSummary: Quasisymmetry is a well-studied property of homeomorphisms between metric spaces, and the Ahlfors regular conformal dimension is a quasisymmetric invariant. In the present paper, we consider the Ahlfors regular conformal dimension of metrics on infinite graphs, and show that this notion coincides with the critical exponent of \(p\)-energies. Moreover, we give a relation between the Ahlfors regular conformal dimension and the spectral dimension of a graph.Large scale conformal mapshttps://zbmath.org/1496.300302022-11-17T18:59:28.764376Z"Pansu, Pierre"https://zbmath.org/authors/?q=ai:pansu.pierreThe author introduces rich definitions and ideas in the paper, such as, the definitions of an \((l,R,S)\)-packing of a metric space, large-scale conformal maps between metric spaces and the \(L^{p,q}\)-cohomology on metric spaces. There are two main applications of the theory in the paper.
The first one shows that if there exists a large-scale conformal map between nilpotent or hyperbolic groups \(G\) and \(G'\), then \(d_1(G)\leq d_2(G')\). Here \(d_1(G):=\) the infimal \(p\) such that the \(L^p\)-cohomology of \(G\) does not vanish and \(d_2(G):=\) the Ahlfors-regular conformal dimension of the ideal boundary of \(G\). The second one shows that if the homeomorphisms \(f\) and \(f^{-1}\) between bounded geometry manifolds or polyhedras with isoperimetric dimension \(>1\) are large-scale conformal, then \(f\) is a quasi-isometry.
The starting point of the paper is the definition of an \((l,R,S)\)-packing, which is a collection of balls \(\{B_j\}\), each with radius between \(R\) and \(S\), such that the concentric balls \(lB_j\) are pairwise disjoint. An \((N,l,R,S)\)-packing is the union of at most \(N\) \(N(l,R,S)\)-packings. Using \((N,l,R,S)\)-packings on metric spaces, the author also defines coarsely conformal, uniformly conformal and roughly conformal of maps between metric spaces. The relations between the new classes of maps and quasi-symmetric/quasi-Möbius maps are discussed.
The \(p\)-energy with parameters \(l\geq 1\), \(R\) and \(S\geq R\) of a map between metric spaces are introduced so that the \((p, l, R, S)\)-modulus of maps and \((p, l, R, S)\)-parabolicity of a locally compact metric space can be defined. Then the author defines \(L^{p,q}\)-cohomology and uses it to prove the main results above.
Reviewer: Jialong Deng (Beijing)Moduli of doubly connected domains under univalent harmonic mapshttps://zbmath.org/1496.310012022-11-17T18:59:28.764376Z"Bshouty, Daoud"https://zbmath.org/authors/?q=ai:bshouty.daoud-h"Lyzzaik, Abdallah"https://zbmath.org/authors/?q=ai:lyzzaik.abdallah"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.antti"Vasudevarao, Allu"https://zbmath.org/authors/?q=ai:vasudevarao.alluLet \(\mathcal{T}(t)=\mathbb{C}\setminus([-1,1]\cup[t,\infty))\), \(t>1\), be a Teichmüller doubly connected domain. The Teichmüller-Nitsche problem is formulated as follows:
For which values \(s,t\), \(1<s,t<\infty\), does a harmonic homeomorphism \(f:\mathcal{T}(s)\rightarrow\mathcal{T}(t)\) exist?
In the paper, the Teichmüller-Nitsche problem is solved for symmetric harmonic homeomorphisms between \(\mathcal{T}(s)\) and \(\mathcal{T}(t)\). This problem is solved by using the method of extremal length.
The following question suggested by \textit{T. Iwaniec} et al. [Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 5, 1017--1030 (2011; Zbl 1267.30059)] is also considered:
Characterize pairs \((\Omega,\Omega^*)\) of doubly connected domains that admit a univalent harmonic mapping from \(\Omega\) onto \(\Omega^*\).
This question is tested regarding the moduli of the doubly connected domains related by harmonic homeomorphisms. The paper concludes with relevant questions.
Reviewer: Konstantin Malyutin (Kursk)Short closed geodesics on cusped hyperbolic surfaceshttps://zbmath.org/1496.320142022-11-17T18:59:28.764376Z"Vo, Hanh"https://zbmath.org/authors/?q=ai:vo.hanh-nguyenSummary: This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any nonnegative integer \(k\), we consider the set of closed geodesics that self-intersect at least \(k\) times and investigate those of minimal length. The main result is that, if the surface has at least one cusp, their self-intersection numbers are exactly \(k\) for large enough \(k\).Quotient of Bergman kernels on punctured Riemann surfaceshttps://zbmath.org/1496.320232022-11-17T18:59:28.764376Z"Auvray, Hugues"https://zbmath.org/authors/?q=ai:auvray.hugues"Ma, Xiaonan"https://zbmath.org/authors/?q=ai:ma.xiaonan"Marinescu, George"https://zbmath.org/authors/?q=ai:marinescu.georgeLet \(\overline\Sigma\) be a compact Riemann surface, \(D=\{a_1,\ldots,a_N\}\subset\overline\Sigma\) a finite set, and \(\Sigma=\overline\Sigma\setminus D\). Assume that \(\omega_\Sigma\) is a Hermitian metric on \(\Sigma\) and that \(L\) is a holomorphic line bundle on \(\overline\Sigma\) endowed with a metric \(h\) which is smooth on \(\Sigma\) and such that every \(a_j\in D\) has a neighborhood \(\overline V_j\) in \(\overline\Sigma\) with coordinate \(z_j\) on which there is a trivialization of \(L\) with \(|1|^2_h(z_j)=\big|\log(|z_j|^2)\big|\). Moreover, assume that the curvature \(R^L\) of \(h\) satisfies \(iR^L=\omega_\Sigma\) on \(V_j=\overline V_j\setminus\{a_j\}\) and \(iR^L\geq\varepsilon\omega_\Sigma\) on \(\Sigma\), for some \(\varepsilon>0\). Hence \((\Sigma,\omega_\Sigma,L,h)|_{{\mathbb D}_r^\star}=({\mathbb D}^\star,\omega_{{\mathbb D}^\star},\mathbb C,h_{{\mathbb D}^\star})|_{{\mathbb D}_r^\star}\) in the local coordinate \(z_j\) near \(a_j\) for some \(r>0\), where \({\mathbb D}_r^\star\) is the punctured disc of radius \(r\) centered at \(a_j\). Here \(\omega_{{\mathbb D}^\star}\) is the standard Poincaré metric of the punctured unit disc \({\mathbb D}^\star\) and \(h_{{\mathbb D}^\star}=\big|\log(|z|^2)\big|h_0\), where \(h_0\) is the flat Hermitian metric on the trivial bundle on \(\mathbb C\).
Let \(H^0_{(2)}(\Sigma,L^p)\) be the space of \(L^2\)-holomorphic sections of \(L^p:=L^{\otimes p}\) with respect to the metrics \(h^p:=h^{\otimes p}\) and \(\omega_\Sigma\), and let \(B_p\) be the corresponding Bergman kernel function. Let \(B^{{\mathbb D}^\star}_p\) be the Bergman kernel function of \(({\mathbb D}^\star,\omega_{{\mathbb D}^\star},\mathbb C,h_{{\mathbb D}^\star}^p)\). In an earlier paper [Math. Ann. 379, No. 3--4, 951--1002 (2021; Zbl 1480.30034)], the authors showed that \(B^{{\mathbb D}^\star}_p\) is the local model for \(B_p\) near \(a_j\): if \(r>0\) is sufficiently small then for any \(k\in\mathbb N\), \(\ell>0\), \(\delta\geq0\) there exists a constant \(C=C_{k,\ell,\alpha}\) such that, for all \(p\geq1\),
\[
\Big|B^{{\mathbb D}^\star}_p(x)-B_p(x)\Big|_{C^k}\leq Cp^{-\ell}\big|\log(|x|^2)\big|^{-\delta}
\]
holds on \({\mathbb D}_r^\star\).
The main result of the present paper is a uniform estimate on the ratio of the two Bergman kernels near the punctures. Namely, in the above setup, for any \(\ell>0\) there exists a constant \(C\) such that for any \(p\geq1\) one has
\[
\sup_{x\in V_1\cup\ldots\cup V_N}\Big|\frac{B_p}{B^{{\mathbb D}^\star}_p}(x)-1\Big|\leq Cp^{-\ell}.
\]
Reviewer: Dan Coman (Syracuse)Fekete-Szegö problem for Bavrin's functions and close-to-starlike mappings in \(\mathbb{C}^n\)https://zbmath.org/1496.320312022-11-17T18:59:28.764376Z"Długosz, Renata"https://zbmath.org/authors/?q=ai:dlugosz.renata"Liczberski, Piotr"https://zbmath.org/authors/?q=ai:liczberski.piotrSummary: The paper is devoted to the study of a family of complex-valued holomorphic functions and a family of holomorphic mappings in \(\mathbb{C}^n\). More precisely, the article concerns a Bavrin's family of functions defined on a bounded complete \(n\)-circular domain \(\mathcal{G}\) of \(\mathbb{C}^n\) and a family of biholomorphic mappings on the Euclidean open unit ball in \(\mathbb{C}^n\). The presented results include some estimates of a combination of the Fréchet differentials at the point \(z=0\), of the first and second order for Bavrin's functions, also of the second and third order for biholomorphic close-to-starlike mappings in \(\mathbb{C}^n\), respectively. These bounds give a generalization of the Fekete-Szegö coefficients problem for holomorphic functions of a complex variable on the case of holomorphic functions and mappings of several variables.On weighted solutions of \(\overline{\partial} \)-equation in the unit dischttps://zbmath.org/1496.320562022-11-17T18:59:28.764376Z"Hayrapetyan, F. V."https://zbmath.org/authors/?q=ai:hayrapetyan.f-vSummary: In the paper an equation \(\partial g(z)/\partial \overline{z} = v(z)\) is considered in the unit disc \(\mathbb{D} \). For \(C^k\)-functions \(v\) (\(k = 1,2,3,\dots, \infty)\) from weighted \(L^p\)-classes (\(1 \leq p < \infty\)) with weight functions of the type \(|z|^{2\gamma} (1-|z|^{2\rho})^{\alpha}, z \in \mathbb{D} \), a family \(g_{\beta}\) of solutions is constructed (\(\beta\) is a complex parameter).Cauchy-Riemann \(\bar{\partial}\)-equations with some applicationshttps://zbmath.org/1496.320582022-11-17T18:59:28.764376Z"Xiao, Jie"https://zbmath.org/authors/?q=ai:xiao.jie.1"Yuan, Cheng"https://zbmath.org/authors/?q=ai:yuan.chengSummary: This paper shows that given \(0 < p < 3\) and a complex Borel measure \(\mu\) on the unit disk \(\mathbb{D}\) the inhomogeneous Cauchy-Riemann \(\bar{\partial}\)-equation
\[
\partial_{\bar{z}}u(z) = \frac{d\mu(z)}{(2\pi i)^{-1}d\bar{z}\wedge dz} \text{ -- a complex Gauss curvature of the weighted disk }(\mathbb{D}, \mu)\;\forall z\in\mathbb{D},
\]
has a distributional solution (initially defined on \(\overline{\mathbb{D}} = \mathbb{D} \cup \mathbb{T}\)) \(u\in \mathcal{L}^{2, p}(\mathbb{T})\) (formed of: (i) Morrey's space \(M^{2, 0< p <1}(\mathbb{T})\); (ii) John-Nirenberg's space \(BMO(\mathbb{T}) = \mathcal{L}^{2,1}(\mathbb{T})\); (iii) Hölder-Lipschitz's space \(C^{0 < \frac{p - 1}{2} < 1}(\mathbb{T})\)), if and only if
\[
\overline{\mathbb{D}}\ni z \mapsto \int\limits_{\mathbb{D}}(1 - z\bar{w})^{-1}d\bar{\mu}(w)\text{ belongs to the analytic Campanato space }\mathcal{CA}_p (\mathbb{D}),
\]
thereby not only extending Carleson's corona \& Wolff's ideal theorems to the algebra \(M(\mathcal{CA}_p(\mathbb{D}))\) of all analytic pointwise multiplications of \(\mathcal{CA}_p(\mathbb{D})\), but quadratically generalizing Brownawell's result on Hilbert's Nullstellensatz for the analytic polynomial class \(\mathcal{P}(\mathbb{C})\).When does a hypergeometric function \(_pF_q\) belong to the Laguerre-Pólya class \(LP^+\)?https://zbmath.org/1496.330062022-11-17T18:59:28.764376Z"Sokal, Alan D."https://zbmath.org/authors/?q=ai:sokal.alan-dSummary: I show that a hypergeometric function \(_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; \cdot)\) with \(p \leq q\) belongs to the Laguerre-Pólya class \(L P^+\) for arbitrarily large \(b_{p + 1}, \ldots, b_q > 0\) if and only if, after a possible reordering, the differences \(a_i - b_i\) are nonnegative integers. This result arises as an easy corollary of the case \(p = q\) proven two decades ago by Ki and Kim. I also give explicit examples for the case \(_1F_2\).Differential polynomials generated by solutions of second order non-homogeneous linear differential equationshttps://zbmath.org/1496.341292022-11-17T18:59:28.764376Z"Belaïdi, Benharrat"https://zbmath.org/authors/?q=ai:belaidi.benharratSummary: This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation
\[
f^{\prime\prime}+Ae^{a_1 z} f^\prime+B(z) e^{a_2 z}f = F(z) e^{a_1 z},
\]
where \(A\), \(a_1\), \(a_2\) are complex numbers, \(B (z)\) (\(\not\equiv 0\)) and \(F (z)\) (\(\not\equiv 0\)) are entire functions with order less than one. Moreover, we investigate the growth and the oscillation of some differential polynomials generated by solutions of the above equation.Integrability by compensation for Dirac equationhttps://zbmath.org/1496.351952022-11-17T18:59:28.764376Z"Da Lio, Francesca"https://zbmath.org/authors/?q=ai:da-lio.francesca"Rivière, Tristan"https://zbmath.org/authors/?q=ai:riviere.tristan"Wettstein, Jerome"https://zbmath.org/authors/?q=ai:wettstein.jeromeThe authors consider the Dirac operator acting on the Clifford algebra. Under critical assumptions on the potential and the spinor field, they prove that the equation is subject to an integrability by compensation phenomenon and has a sub-critical behaviour below some positive energy threshold.
Reviewer: Zhipeng Yang (Göttingen)On ergodicity of foliations on \({\mathbb{Z}^d}\)-covers of half-translation surfaces and some applications to periodic systems of Eaton lenseshttps://zbmath.org/1496.370322022-11-17T18:59:28.764376Z"Frączek, Krzysztof"https://zbmath.org/authors/?q=ai:fraczek.krzysztof-m"Schmoll, Martin"https://zbmath.org/authors/?q=ai:schmoll.martin-johannesSummary: We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for \({\mathbb{Z}^d}\)-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.McMullen's and geometric pressures and approximating the Hausdorff dimension of Julia sets from belowhttps://zbmath.org/1496.370432022-11-17T18:59:28.764376Z"Przytycki, Feliks"https://zbmath.org/authors/?q=ai:przytycki.feliksSummary: We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope that some of them will turn out useful to achieve fast approximation from below of the hyperbolic Hausdorff dimension of Julia sets.Factorizations of bivariate Taylor series via inverse power productshttps://zbmath.org/1496.410132022-11-17T18:59:28.764376Z"Elewoday, Mohamed"https://zbmath.org/authors/?q=ai:elewoday.mohamed"Gingold, Harry"https://zbmath.org/authors/?q=ai:gingold.harry"Quaintance, Jocelyn"https://zbmath.org/authors/?q=ai:quaintance.jocelynIn this paper, the authors convert \(f(x, y)\) into the formal product \(\Pi^{\infty} _{\substack{p=1 \\ m+n=p}}(1-h_{m,n}x^{m}y^{n})^{ -1}\), namely the inverse power product expansion in two independent variables and also discuss various combinatorial interpretations provided by these inverse power product expansions.
English is clear in style and consistent with the standards of usage.
Reviewer: Aida Tagiyeva (Baku)Completeness conditions of systems of Bessel functions in weighted \(L^2\)-spaces in terms of entire functionshttps://zbmath.org/1496.420082022-11-17T18:59:28.764376Z"Khats', Ruslan"https://zbmath.org/authors/?q=ai:khats.r-vSummary: Let \(J_\nu\) be the Bessel function of the first kind of index \(\nu\geq 1/2\), \(p\in\mathbb{R}\) and \((\rho_k)_{k\in\mathbb{N}}\) be a sequence of distinct nonzero complex numbers. Sufficient conditions for the completeness of the system \(\left\{x^{-p-1}\sqrt{x\rho_k}J_\nu(x\rho_k): k\in\mathbb{N}\right\}\) in the weighted space \(L^2((0;1); x^{2p} dx)\) are found in terms of an entire function with the set of zeros coinciding with the sequence \((\rho_k)_{k\in\mathbb{N}}\).Convergence and almost sure properties in Hardy spaces of Dirichlet serieshttps://zbmath.org/1496.430022022-11-17T18:59:28.764376Z"Bayart, Frédéric"https://zbmath.org/authors/?q=ai:bayart.fredericThis is a remarkable continuation of a series of recent articles trying to establish a modern theory of general Dirichlet series \(D = \sum_n a_n e^{-\lambda_n s}\); here \(s \in \mathbb{C}\) is a complex variable, the \(a_n\)'s form the coefficients, and \(\lambda = (\lambda_n)\) is the frequency (i.e., a sequence of non-negative, strictly increasing real numbers tending to \(\infty\)).
The article is divided into three sections. The first section isolates a new condition on \(\lambda\) (denoted by ($NC$)), which ensures that a somewhere convergent \(\lambda\)-Dirichlet series defining a bounded function on the right half-plane, converges uniformly on every smaller half-plane. Conditions of this type are fundamental for the understanding of general Dirichlet series, and \((NC)\) indeed extends famous work of H.~Bohr (condition \((BC)\)) and E.~Landau (condition \((LC)\)). Using alternative techniques (as e.g. Saksman's vertical convolution formula), Bayart's new condition \((NC)\) is applied to improve recent maximal inequalities from [\textit{I. Schoolmann}, Math. Nachr. 293, No. 8, 1591--1612 (2020; Zbl 07261807)].
For \(1 \leq p \leq \infty \) the Hardy space \(\mathcal{H}_p(\lambda)\) of \(\lambda\)-Dirichlet series is defined as the completion of all finite \(\lambda\)-Dirichlet polynomials \(D = \sum_{n=1}^N a_n e^{-\lambda_n s}\) under the \(\mathcal{H}_p\)-norm \[\|D\|_p = \lim_{T \to \infty} \frac{1}{2T} \Big(\int_{-T}^{T} |D(it)|^p dt \Big)^{1/p}\,.\] However, this internal description is often not sufficient to understand the structure of these Banach spaces. Following earlier work of Bayart for ordinary Dirichlet series (\(\lambda = (\log n)\)), a~group approach is suggested in [\textit{A. Defant} and \textit{I. Schoolmann}, J. Fourier Anal. Appl. 25, No.~6, 3220--3258 (2019; Zbl 1429.43004)], and among others carefully studied in [\textit{A. Defant} and \textit{I. Schoolmann}, Math. Ann. 378, No.~1--2, 57--96 (2020; Zbl 1483.43006)] and [\textit{A. Defant} and \textit{I. Schoolmann}, J. Funct. Anal. 279, No.~5, Article ID 108569, 36~p. (2020; Zbl 1470.43006)].
The second section of this article answers non-trivial questions arising from these works -- mainly extending a famous theorem of Helson to Dirichlet series in \(\mathcal{H}_1(\lambda).\) It shows that if \(\lambda\) satisfies the new condition \((NC)\) and \(D = \sum_n a_n e^{-\lambda_n s} \in \mathcal{H}_1(\lambda),\) then for almost all homomorphisms \(\omega: \mathbb{R} \to \mathbb{T}\) the Dirichlet series \(D = \sum_n \omega(\lambda_n)a_n e^{-\lambda_n s}\) converges pointwise on \([\Re s>0]\). Moreover a maximal inequality is added, and a non-trivial counterexample showing that such a result is false for arbitrary \(\mathcal{H}_1(\lambda)\)-Dirichlet series (we note that for \(1 < p < \infty\) Helson's theorem in the above sense holds for any \(\mathcal{H}_p(\lambda)\)-Dirichlet series without any further assumption on~\(\lambda\)). These results indicate that the Hardy spaces \(\mathcal{H}_p(\lambda)\) seem to behave well whenever we consider their `almost everywhere properties'.
A further non-trivial problem is to determine the optimal half-plane \([\Re s > \sigma_{\mathcal{H}_p(\lambda)} ]\), where \(\sigma_{\mathcal{H}_p(\lambda)}\) is defined to be the best \(\sigma \in \mathbb{R}\) such that the convergence abscissa \(\sigma_{c}(D)\) of all \(D \in \mathcal{H}_p(\lambda)\) is \(\leq \sigma\). Previous work of Bayart shows that, based on the multiplicativity of \(\lambda = (\log n)\) and the hypercontractivity of the Poisson kernel acting on the Hardy space \(H_1(\mathbb{T})\), we have \(\sigma_{\mathcal{H}_p((\log n))} = \frac{1}{2}\) for all \(1 \leq p < \infty\). A natural guess would be that \(\sigma_{\mathcal{H}_p(\lambda)} = \frac{L(\lambda)}{2} \) for all \(1 \leq p < \infty\), where \(L(\lambda) = \limsup_n \frac{\log n}{\lambda_n} \) (following Bohr, this number equals the width of the largest possible strip on which a \(\lambda\)-Dirichlet series converges but does not converge absolutely). For \(p=2\) this is indeed true, but in contrast to the ordinary case, there is no hope to get a similar result for \(p\neq 2\). Among others, it is proved that \(\sigma_{\mathcal{H}_1(\lambda)} \leq 2 \sigma_{\mathcal{H}_2(\lambda)}\) for all frequencies, and that there exists a frequency with Bohr's condition \((BC)\) for which we here even have equality.
Reviewer: Andreas Defant (Oldenburg)Unconditional bases in radial Hilbert spaceshttps://zbmath.org/1496.460092022-11-17T18:59:28.764376Z"Isaev, Konstantin P."https://zbmath.org/authors/?q=ai:isaev.konstantin-petrovich"Yulmukhametov, Rinad S."https://zbmath.org/authors/?q=ai:yulmukhametov.rinad-salavatovichRigidity of the Pu inequality and quadratic isoperimetric constants of normed spaceshttps://zbmath.org/1496.460112022-11-17T18:59:28.764376Z"Creutz, Paul"https://zbmath.org/authors/?q=ai:creutz.paulThe author furnishes an enhanced bound on the filling areas curves (not closed geodesics) in Banach spaces. He shows rigidity of \textit{P. M. Pu}'s classical systolic inequality [Pac. J. Math. 2, 55--71 (1952; Zbl 0046.39902)] and examines the isoperimetric constants of normed spaces.
Reviewer: Mohammed El Aïdi (Bogotá)Two classes of de Branges spaces that are really onehttps://zbmath.org/1496.460192022-11-17T18:59:28.764376Z"Arov, Damir Z."https://zbmath.org/authors/?q=ai:arov.damir-zyamovich"Dym, Harry"https://zbmath.org/authors/?q=ai:dym.harrySummary: It is well known that if \(J\) is an \(m\times m\) signature matrix and \(U\) is \(J\)-inner with respect to the open upper half-plane \(\mathbb{C}_+\), then the kernel
\[
K_\omega^U(\lambda)=\frac{J-U(\lambda)JU(\omega)^\ast}{-2\pi i(\lambda-\overline{\omega})}
\]
is positive and hence is the reproducing kernel of a reproducing kernel Hilbert space \(\mathcal{H}(U)\) of a space of \(m\times 1\) vector valued functions that are holomorphic in the domain of holomorphy of \(U\).
It seems, however, to be not so well known that this reproducing kernel Hilbert space coincides with the de Branges space \(\mathcal{B}(\mathfrak{E})\) based on an appropriately defined de Branges matrix \(\mathfrak{E}=[E_-\;\; E_+]\) with \(m\times m\) components and reproducing kernel
\[
K_\omega^{\mathfrak{E}}(\lambda)=\frac{E_+(\lambda)E_+(\omega)^\ast-E_-(\lambda)E_-(\omega)^\ast}{-2\pi i(\lambda-\overline{\omega})}.
\]
This connection is significant, because it yields a recipe for the inner product in \(\mathcal{H}(U)\) that is not available from Aronszjan's theorem.
Enroute, a pleasing characterization of a class of finite dimensional de Branges spaces \(\mathcal{B}(\mathfrak{E})\) is developed.Unconditional bases of reproducing kernels for Fock spaces with nonradial weightshttps://zbmath.org/1496.460202022-11-17T18:59:28.764376Z"Isaev, K. P."https://zbmath.org/authors/?q=ai:isaev.konstantin-petrovich"Lutsenko, A. V."https://zbmath.org/authors/?q=ai:lutsenko.anastasiya-vladimirovna"Yulmukhametov, R. S."https://zbmath.org/authors/?q=ai:yulmukhametov.rinad-salavatovichSummary: We prove that the Fock space \(\mathcal{F}_\varphi\) with a nonradial weight \(\varphi\) has an unconditional basis of reproducing kernels if and only if such a basis exists for the Fock space \(\mathcal{F}_\varphi\) with a certain radial weight \(\upsilon\) determined by \(\varphi \).Stable Gabor phase retrieval for multivariate functionshttps://zbmath.org/1496.460232022-11-17T18:59:28.764376Z"Grohs, Philipp"https://zbmath.org/authors/?q=ai:grohs.philipp"Rathmair, Martin"https://zbmath.org/authors/?q=ai:rathmair.martinSummary: In recent work [\textit{P.~Grohs} and \textit{M.~Rathmair}, Commun. Pure Appl. Math. 72, No.~5, 981--1043 (2019; Zbl 1460.94022)]
the instabilities of the Gabor phase retrieval problem, i.e., the problem of reconstructing a function \(f\) from its spectrogram \(|\mathcal{G}f|\), where
\[
\mathcal{G}f(x,y)=\int_{\mathbb{R}^d} f(t) e^{-\pi|t-x|^2} e^{-2\pi i t\cdot y} \, dt, \quad x,y\in \mathbb{R}^d,
\] have been completely classified in terms of the disconnectedness of the spectrogram. These findings, however, were crucially restricted to the one-dimensional case (\(d=1\)) and therefore not relevant for many practical applications.
In the present paper we not only generalize the aforementioned results to the multivariate case but also significantly improve on them. Our new results have comprehensive implications in various applications such as ptychography, a highly popular method in coherent diffraction imaging.Pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness and their applicationshttps://zbmath.org/1496.460332022-11-17T18:59:28.764376Z"Li, Zi Wei"https://zbmath.org/authors/?q=ai:li.ziwei"Yang, Da Chun"https://zbmath.org/authors/?q=ai:yang.dachun.1|yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wen|yuan.wen.1The nowadays well-known homogeneous spaces \(\dot{A}^s_{p,q} (\mathbb R^n)\) with \(A \in \{B,F \}\), \(s\in \mathbb R\) and \(0<p,q \le \infty\) have been modified in several ways. The smoothness \(s\), characterized by \(\{ 2^{js} \}^\infty_{j=0}\), is generalized by suitable sequences \(\{\sigma_j \}^\infty_{j=0}\) of positive numbers. Furthermore, \(\mathbb R^n\) is replaced by metric spaces \((X, d, \mu)\) with the metric \(d\) and the Borel measure \(\mu\) on the set \(X\), where the smoothness is expressed by so-called Hajłasz gradients. More recently, there is some type of discretization, called hyperbolic filling. The paper deals with spaces based on these ingredients and their relations, especially to \(\dot{A}^\sigma_{p,q} (\mathbb R^n)\).
Reviewer: Hans Triebel (Jena)Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacianshttps://zbmath.org/1496.470112022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arneSummary: We consider the discrete Laplacian on \(\mathbb{Z}^d\), and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if \(d\) is odd, and a logarithm branching if \(d\) is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensionshttps://zbmath.org/1496.470122022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.2|ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arne-m|jensen.arne-skov|jensen.arneThe authors have obtained some closed formulae for lattice Green functions of the form \[ G(z,n)=(2\pi)^{-d}\int_{\mathbb{T}^d}\dfrac{e^{in\theta}}{2d-2\cos(\theta_1)-\dots-2\cos(\theta_d)-z}d\theta.\]
Such investigation was mainly restricted to dimensions \(d=1,2\). In the Introduction, they start to depict that \(2dG(0,n)\) (\(z=0\)) shall be represented as the expectation value \( \mathbb{E}[n]=\sum_{k=0}^\infty P(X_k=n)\) that counts the number of times that a walker visits \(n\in \mathbb{Z}^d\). To get rid of the fact that \(\mathbb{E}[n]\) is divergent for dimensions \(d=1,2\) (see also Appendix B), they propose a renormalization technique to approximate \(\mathbb{E}[n]\) by \(\mathbb{E}[\epsilon,n]=\frac{2d}{1-\epsilon}G(\frac{-2d\epsilon}{1-\epsilon},n)\), for values of \(\epsilon\in (0,1]\).
In this way, they succeed in representing \(G(z,n)\) as a convergent series (see, e.g., Theorem 2.2. and Theorem 2.3.). Such analysis goes far beyond the asymptotic analysis, in the limit \(z\rightarrow 0\), considered by so many authors in the past.
As a whole, this paper is complementary to the thors' previous paper [J. Funct. Anal. 277, No. 4, 965--993 (2019; Zbl 1496.47011)] in which the authors have shown that \(G(z,n)\) admits, for each threshold \(4q\), \(q=0,\dots,n\), the splitting formula \[ G(z,n)=\mathcal{E}_q(z,n)+f_q(z)\mathcal{F}_q(z,n), \] whereby \(\mathcal{F}_q(z,n)\) -- the singular part of \(G(z,n)\) -- was represented in terms of the so-called Appell-Lauricella hypergeometric function of type \(B\), \(F_B^{(d)}\). Further comparisons between both approaches may be found in Appendix~A.
Reviewer: Nelson Faustino (Alfeizerão)Almost invariant subspaces of the shift operator on vector-valued Hardy spaceshttps://zbmath.org/1496.470192022-11-17T18:59:28.764376Z"Chattopadhyay, Arup"https://zbmath.org/authors/?q=ai:chattopadhyay.arup"Das, Soma"https://zbmath.org/authors/?q=ai:das.soma"Pradhan, Chandan"https://zbmath.org/authors/?q=ai:pradhan.chandanThe authors characterize nearly invariant subspaces of finite defect for the backward shift operator acting on vector-valued Hardy spaces \(H^2_{\mathbb C^m}(\mathbb D)\), generalizing the scalar-valued result by \textit{I. Chalendar} et al. [J. Oper. Theory 83, No. 2, 321--331 (2020; Zbl 1463.47096)]. Given a bounded analytic function \(\Theta\) with values in the space of linear operators \(\mathcal L(\mathbb C^r,\mathbb C^m)\), we can induce the multiplier \(T_\theta F(z)=\Theta(z)F(z)\) from \(H^2_{\mathbb C^r}(\mathbb D)\) into \(H^2_{\mathbb C^m}(\mathbb D)\). They are determined by the condition \(ST_\Theta=T_\Theta S\) where \(S\) denotes the forward shift operator \(SF(z)=zF(z)\) which acts on the corresponding space in each case. We write, as usual, the backward shift \(S^* F(z)=\frac{F(z)-F(0)}{z}\). A closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called almost-invariant for \(S\) if there exists a finite-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that \(S(\mathcal M)\subset \mathcal M\oplus\mathcal F\). Similarly, a closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called nearly invariant for \(S^*\) if any \(F\in \mathcal M\) with \(F(0)=0\) satisfies that \(S^*F\in \mathcal M\) and it is called nearly \(S^*\)-invariant with defect \(p\) if there exists a \(p\)-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that, if \(F\in \mathcal M\) with \(F(0)=0\), then \(S^*F\in \mathcal M \oplus \mathcal F\). Also, \(\mathcal M\) is called \(S^*\)-almost invariant with defect \(p\) if \(S^*\mathcal M\subset\mathcal M \oplus \mathcal F\) and \(\dim \mathcal F=p\).
In the paper under review, the authors present a characterization of nearly invariant subspaces for \(S^*\) with finite defect in the vector-valued Hardy spaces. Using such a result, they also manage to obtain the description of almost invariant subspaces for the shift and its adjoint acting on vector-valued Hardy spaces.
Reviewer: Oscar Blasco (València)Interpolating matriceshttps://zbmath.org/1496.470262022-11-17T18:59:28.764376Z"Dayan, Alberto"https://zbmath.org/authors/?q=ai:dayan.albertoThe main result is a matrix version of the Carleson theorem [\textit{L. Carleson}, Am. J. Math. 80, 921--930 (1958; Zbl 0085.06504)] which says that a sequence of points \(\Lambda=(\lambda_k)_{n\in\mathbb{N}}\) in the open unit disk \(\mathbb{D}\) is an interpolation sequence in \(H^\infty\) if and only if it is strongly separated. Here, \(\Lambda\) is replaced by a matrix sequence \(A=(A_n)_{n\in\mathbb{N}}\) (possibly with varying dimensions) with spectra in \(\mathbb{D}\) and interpolating means that for any bounded sequence \((\phi_n)_{n\in\mathbb{N}}\) in \(H^\infty\) there is a \(\varphi\in H^\infty\) such that \(\varphi(A_n)=\phi_n(A_n)\), \(n\in\mathbb{N}\). Strong separated means \(\inf_{z\in\mathbb{D}}\sup_{n\in\mathbb{N}}\prod_{k\ne n}|B_{A_k}(z)|>0\) in \(\mathbb{D}\) where \(B_M(z)\) is a Blaschke product whose zeros are eigenvalues of \(M\) (including multiplicity). An equivalent formulation is that the model spaces \(H_n=H^2\ominus B_{A_n}H^2\) are strongly separated. If the dimensions of the matrices \(A_n\) are uniformly bounded, then \((H_n)_{n\in\mathbb{N}}\) being a weakly separated Bessel system is an alternative equivalent condition for \(A\) to be interpolating.
Reviewer: Adhemar Bultheel (Leuven)On a class of operator equations in locally convex spaceshttps://zbmath.org/1496.470282022-11-17T18:59:28.764376Z"Mishin, Sergeĭ N."https://zbmath.org/authors/?q=ai:mishin.sergey-nSummary: We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. The research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.Liouville operators over the Hardy spacehttps://zbmath.org/1496.470442022-11-17T18:59:28.764376Z"Russo, Benjamin P."https://zbmath.org/authors/?q=ai:russo.benjamin-p"Rosenfeld, Joel A."https://zbmath.org/authors/?q=ai:rosenfeld.joel-aSummary: The role of Liouville operators in the study of dynamical systems through the use of occupation measures has been an active area of research in control theory over the past decade. This manuscript investigates Liouville operators over the Hardy space, which encode complex ordinary differential equations in an operator over a reproducing kernel Hilbert space.Generalized weighted composition operators from the Bloch-type spaces to the weighted Zygmund spaceshttps://zbmath.org/1496.470452022-11-17T18:59:28.764376Z"Abbasi, Ebrahim"https://zbmath.org/authors/?q=ai:abbasi.ebrahim"Vaezi, Hamid"https://zbmath.org/authors/?q=ai:vaezi.hamidSummary: The boundedness and compactness of generalized weighted composition operators from Bloch-type spaces and little Bloch-type spaces into weighted Zygmund spaces on the unit disc are characterized in this paper.A class of \(C\)-normal weighted composition operators on Fock space \(\mathcal{F}^2 (\mathbb{C})\)https://zbmath.org/1496.470462022-11-17T18:59:28.764376Z"Bhuia, Sudip Ranjan"https://zbmath.org/authors/?q=ai:bhuia.sudip-ranjanSummary: In this paper we study a class of \(C\)-normal weighted composition operators \(W_{\psi, \varphi}\) on the Fock space \(\mathcal{F}^2 (\mathbb{C})\). We provide some properties of \(\psi\) and \(\varphi\) when a weighted composition operator \(W_{\psi, \varphi}\) is \(C\)-normal with a conjugation \(C\) defined on \(\mathcal{F}^2 (\mathbb{C})\). We also show that hyponormal \(C\)-normal operators are normal. We investigate \(C\)-normality of \(W_{\psi, \varphi}\) with weighted composition conjugation and the weight function as a kernel function. Alongside we give eigenvalues and eigenvectors of \(C\)-normal \(W_{\psi, \varphi}\). We establish a condition on the symbols for a class of \(C\)-normal weighted composition operators to be normal.Differences of weighted differentiation composition operators from \(\alpha\)-Bloch space to \(H^\infty\) spacehttps://zbmath.org/1496.470492022-11-17T18:59:28.764376Z"Wang, Cui"https://zbmath.org/authors/?q=ai:wang.cui"Zhou, Ze-Hua"https://zbmath.org/authors/?q=ai:zhou.zehuaSummary: This paper characterizes the boundedness and compactness of the differences of weighted differentiation composition operators acting from the \(\alpha\)-Bloch space \(B^\alpha\) to the space \(H^\infty\) of bounded holomorphic functions on the unit disk \(\mathbb D\).Conjugations in \(L^2(\mathcal{H})\)https://zbmath.org/1496.470502022-11-17T18:59:28.764376Z"Câmara, M. Cristina"https://zbmath.org/authors/?q=ai:camara.m-cristina"Kliś-Garlicka, Kamila"https://zbmath.org/authors/?q=ai:klis-garlicka.kamila"Łanucha, Bartosz"https://zbmath.org/authors/?q=ai:lanucha.bartosz"Ptak, Marek"https://zbmath.org/authors/?q=ai:ptak.marekLet \(L_2=L_2(\mathbb T,\mathfrak{m})\) (resp., \(L_\infty=L_\infty(\mathbb T,\mathfrak{m})\)) be the space of square-integrable (resp., bounded) functions on \(\mathbb T\) (the unit circle) w.r.t the normalized Lebesgue measure \(\mathfrak{m}\). Let \(\mathbf{M}_z=\begin{bmatrix} M_z & 0 \\
0 & M_z \end{bmatrix}\) where \(M_z\) is the operator defined on \(L_2\) of multiplication by \(z\in L_\infty\). Let \((\mathcal{H},\langle\cdot\rangle_{\mathcal{H}})\) be a complex Hilbert space. We say that \(C:\mathcal{H}\to\mathcal{H}\) is a conjugation in \(\mathcal{H}\) if it is an antilinear isometric involution and \(\langle Cf,Cg\rangle_{\mathcal{H}}=\langle g,f\rangle_{\mathcal{H}}\) for all \((f,g)\in\mathcal{H}^2\) and we denote \(\mathbf{C}=\begin{bmatrix} D_1& D_2 \\
D_3 & D_4 \end{bmatrix}\) where \((D_j)_{1\le j\le 4}\) are antilinear operators on \(\mathcal{H}.\) The purpose of the authors is to characterize all \(\mathbf{M}_z\)-conjugations \(\mathbf{C}\) in \(L_2(\mathcal{H})\) and all \(\mathbf{M}_z\)-commuting conjugations in \(L_2(\mathcal{H})\). Also, the authors study these conjugations for which vector valued model spaces are invariant.
Reviewer: Mohammed El Aïdi (Bogotá)Commuting Toeplitz operators on the Fock-Sobolev spacehttps://zbmath.org/1496.470532022-11-17T18:59:28.764376Z"Fan, Junmei"https://zbmath.org/authors/?q=ai:fan.junmei"Liu, Liu"https://zbmath.org/authors/?q=ai:liu.liu"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufengSummary: The purpose of this paper is to study the commutator and the semi-commutator of Toeplitz operators on the Fock-Sobolev space \(F^{2,m}({\mathbb{C}})\) in a different function theoretic way instead of Berezin transform. We determine conditions for \((T_f, T_{\overline{g}}]=0\) and \([T_f, T_{\overline{g}}]=0\) in the cases when the symbol functions \(f\) and \(g\) are both polynomials or when \(f\) is a finite linear combination of reproducing kernels and \(g\) is a polynomial. We also determine the boundedness of Hankel products on the Fock-Sobolev space for some symbol classes.A generalized Hilbert operator acting on conformally invariant spaceshttps://zbmath.org/1496.470542022-11-17T18:59:28.764376Z"Girela, Daniel"https://zbmath.org/authors/?q=ai:girela.daniel"Merchán, Noel"https://zbmath.org/authors/?q=ai:merchan.noelSummary: If \(\mu\) is a positive Borel measure on the interval \([0,1)\), we let \(\mathcal{H}_{\mu}\) be the Hankel matrix \(\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq0}\) with entries \(\mu_{n,k}=\mu_{n+k}\), where, for \(n=0,1,2,\dots\), \(\mu_{n}\) denotes the moment of order \(n\) of \(\mu\). This matrix formally induces the operator
\[
\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}(\sum_{k=0}^{\infty}\mu_{n,k}{a_{k}})z^{n}
\]
on the space of all analytic functions \(f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}\), in the unit disk \(\mathbb{D}\). This is a natural generalization of the classical Hilbert operator. The action of the operators \(H_{\mu}\) on Hardy spaces has been recently studied (cf. [\textit{C. Chatzifountas} et al., J. Math. Anal. Appl. 413, No. 1, 154--168 (2014; Zbl 1308.42021)]). This article is devoted to a study of the operators \(H_{\mu}\) acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the \(Q_{s}\)-spaces.Hardy classes and symbols of Toeplitz operatorshttps://zbmath.org/1496.470562022-11-17T18:59:28.764376Z"López-García, Marco"https://zbmath.org/authors/?q=ai:lopez-garcia.marco"Pérez-Esteva, Salvador"https://zbmath.org/authors/?q=ai:perez-esteva.salvadorSummary: The purpose of this paper is to study functions in the unit disk \(\mathbb D\) through the family of Toeplitz operators \(\{T_{\phi d\sigma_{t}}\}_{t\in[0,1)}\), where \(T_{\phi d\sigma_{t}}\) is the Toeplitz operator acting the Bergman space of \(\mathbb D\) and where \(d\sigma_t\) is the Lebesgue measure in the circle \(tS^1\). In particular for \(1\leq p < \infty\) we characterize the harmonic functions \(\phi\) in the Hardy space \(h^{p}(\mathbb D)\) by the growth in \(t\) of the \(p\)-Schatten norms of \(T_{\phi d\sigma_{t}}\). We also study the dependence in \(t\) of the norm operator of \(T_{ad\sigma_{t}}\) when \(a\in H^p_{at}\), the atomic Hardy space in the unit circle with \(1/2 < p \leq 1\).Polynomial birth-death processes and the 2nd conjecture of Valenthttps://zbmath.org/1496.470572022-11-17T18:59:28.764376Z"Bochkov, Ivan"https://zbmath.org/authors/?q=ai:bochkov.ivanSummary: The conjecture of \textit{G. Valent} [ISNM, Int. Ser. Numer. Math. 131, 227--237 (1999; Zbl 0935.30025)] about the type of Jacobi matrices with polynomially growing weights is proved.On the \(C^\ast\)-algebra generated by the Bergman operator, Carleman second-order shift, and piecewise continuous coefficientshttps://zbmath.org/1496.471302022-11-17T18:59:28.764376Z"Mozel', V. A."https://zbmath.org/authors/?q=ai:mozel.v-aSummary: We study the \(C^\ast\)-algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space \(L_2\) and extended by the Carleman rotation by an angle \(\pi\). As a result, we obtain an efficient criterion for the operators from the indicated \(C^\ast\)-algebra to be Fredholm operators.Generalized translation hypersurfaces in conformally flat spaceshttps://zbmath.org/1496.530112022-11-17T18:59:28.764376Z"Sousa, P. A."https://zbmath.org/authors/?q=ai:sousa.paulo-alexandre-araujo"Lima, B. P."https://zbmath.org/authors/?q=ai:lima.barnabe-pessoa"Vieira, B. V. M."https://zbmath.org/authors/?q=ai:vieira.b-v-mThe graph of a real function \(f\) defined in some open set of the Euclidean space of dimension \((p+q)\) is said to be a generalized translation graph (GTG) if \(f\) may be expressed as the sum of two independent functions \(\phi\) and \(\psi\) defined in open sets of the Euclidean spaces of dimension \(p\) and \(q\), respectively. In this paper, the authors study the geometry of GTG immersed in Euclidean space equipped with a metric conformal to the Euclidean metric and obtain results that characterize such hypersurfaces. Applying the characterization results, and using ODE solving techniques, they build examples of GTG satisfying geometric properties not valid in relation to the Euclidean metric.
Reviewer: Atsushi Fujioka (Osaka)Quasiconformal flows on non-conformally flat sphereshttps://zbmath.org/1496.530552022-11-17T18:59:28.764376Z"Chang, Sun-Yung Alice"https://zbmath.org/authors/?q=ai:chang.sun-yung-alice"Prywes, Eden"https://zbmath.org/authors/?q=ai:prywes.eden"Yang, Paul"https://zbmath.org/authors/?q=ai:yang.paul-c-pSummary: We study integral curvature conditions for a Riemannian metric \(g\) on \(S^4\) that quantify the best bilipschitz constant between \((S^4, g)\) and the standard metric on \(S^4\). Our results show that the best bilipschitz constant is controlled by the \(L^2\)-norm of the Weyl tensor and the \(L^1\)-norm of the \(Q\)-curvature, under the conditions that those quantities are sufficiently small, \(g\) has a positive Yamabe constant and the \(Q\)-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on \(S^4\).Curve shortening flow on Riemann surfaces with possible ambient conic singularitieshttps://zbmath.org/1496.531052022-11-17T18:59:28.764376Z"Ma, Biao"https://zbmath.org/authors/?q=ai:ma.biao.1Summary: In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles smaller than or equal to \(\pi\).Erratum to: ``A note on the McShane's identity for Hecke groups''https://zbmath.org/1496.570242022-11-17T18:59:28.764376Z"Farooq, K."https://zbmath.org/authors/?q=ai:farooq.kErratum to the author's paper [ibid. 52, No. 3, 915--931 (2021; Zbl 1489.57015)].Surfaces of Section for Seifert fibrationshttps://zbmath.org/1496.570292022-11-17T18:59:28.764376Z"Albach, Bernhard"https://zbmath.org/authors/?q=ai:albach.bernhard"Geiges, Hansjörg"https://zbmath.org/authors/?q=ai:geiges.hansjorgConsider a Seifert manifold given by a closed oriented 3-manifold \(M\) endowed with a Seifert fibration \(M\to B\) over a closed oriented surface \(B\). With these orientability assumptions, by a classical result of \textit{D. B. A. Epstein} [Ann. Math. (2) 95, 66--82 (1972; Zbl 0231.58009)], the fibration is given by an action of the circle. This structure is determined by its Seifert invariants, \(M=M(g;(\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n))\), where \(g\) is the genus of \(B\), the \(\alpha_i\in\mathbb Z^+\) are the multiplicities of the singular fibres, and every \(\beta_i\in\mathbb Z\) is coprime with \(\alpha_i\) and describes the local behaviour around the corresponding singular fibre; actually, regular fibres may be also used in this description, and they correspond to the case \(\alpha_i=1\). The Euler number of the Seifert fibration is defined as \(e=-\sum_i\beta_i/\alpha_i\). Two Seifert invariants determine the same Seifert manifold if and only if they correspond by a sequence of the following operations: permuting the pairs \((\alpha_i,\beta_i)\); adding a pair \((1,0)\); and replacing every \((\alpha_i,\beta_i)\) with \((\alpha_i,\beta_i+k_i\alpha_i)\), where the \(k_i\in\mathbb Z\) satisfy \(\sum_ik_i=0\). A global surface of section is an embedded compact surface \(\Sigma\subset M\) whose boundary \(\partial\Sigma\) is a union of fibres and whose interior intersects all other fibres transversely. The regular fibres intersecting the interior of \(\Sigma\) have the same number \(d\) of intersection points; the term a \(d\)-section is also used. Orient \(\Sigma\) so that its interior has positive intersection with the fibres. Then \(\Sigma\) is said to be positive if the induced orientation of its boundary agrees with the orientation of the fibres.
The main theorem of the paper characterizes the existence of a positive \(d\)-section \(\Sigma\) in terms of the Seifert invariants; in that case, \(\Sigma\) is unique up to isotopy. Moreover the result relates the Seifert invariants, the Euler number \(e\), the genus of \(\Sigma\), and the number of connected components of \(\Sigma\) and \(\partial\Sigma\). As a corollary, every Seifert manifold (with \(e\le0\)) admits a (positive) \(d\)-section for some \(d\in\mathbb Z^+\). The authors also describe the positive \(d\)-sections for all Seifert fibrations of \(S^3\), where the bases are weighted projective lines, and the \(d\)-sections are algebraic surfaces in weighted projective planes. These results generalize previous results for the case of Reeb dynamics.
Reviewer: Jesus A. Álvarez López (Santiago de Compostela)Roots of random functions: a framework for local universalityhttps://zbmath.org/1496.600122022-11-17T18:59:28.764376Z"Nguyen, Oanh"https://zbmath.org/authors/?q=ai:nguyen.oanh"Vu, Van"https://zbmath.org/authors/?q=ai:vu.van-hLet \(\phi_0(z),\ldots, \phi_n(z)\) be deterministic analytic functions and \(\xi_0,\ldots,\xi_n\) independent random variables. The authors study the distribution of zeroes (both, real and complex) of the random functions of the form \[ F_n(z) = \sum_{k=0}^n \xi_k \phi_k(z). \] This general setting includes many special cases such as \(\phi_k(z) = z^k\) corresponding to the so-called Kac polynomials, \(\phi_k(z) = \cos (k z)\) (corresponding to random trigonometric polynomials) or, more generally, \(\phi_k(z) = c_k z^k\) or \(\phi_k(z) = c_k \cos (k z)\) with deterministic \(c_k\)'s satisfying appropriate assumptions. The authors develop a general framework to study the local distribution of roots of such functions via ``universality theorems'' which state that the roots of \(F_n\) for general \(\xi_k\)'s can be approximated by the roots of the function \(\tilde F_n\) in which the \(\xi_k\)'s are Gaussian. More precisely, they prove explicit upper error bounds on differences of the form \[ \left|\mathbb E \sum_{i_1,\ldots, i_k} G(\zeta_{i_1},\ldots, \zeta_{i_k}) - \mathbb E \sum_{i_1,\ldots, i_k} G(\tilde \zeta_{i_1},\ldots, \tilde \zeta_{i_k})\right|, \] where \(G\) is a sufficiently smooth function. Here, the first sum runs over all \(k\)-tuples of roots of \(F_n\), while the second sum runs over all \(k\)-tuples of the roots of \(\tilde F_n\) in which the \(\xi_k\)'s are replaced by Gaussian random variables. As an application of their techniques, the authors recover, extend in a unified way and sharpen many known results on the asymptotics of the expected number of real roots of random analytic functions including such examples as Kac, Weyl and elliptic random polynomials, random trigonometric polynomials and random Taylor series with regularly varying coefficients.
Reviewer: Zakhar Kabluchko (Münster)Multiparty multicast schemes for remote state preparation of complex coefficient quantum states via partially entangled channelshttps://zbmath.org/1496.810382022-11-17T18:59:28.764376Z"Peng, Jia-yin"https://zbmath.org/authors/?q=ai:peng.jiayin"Lei, Hong-xuan"https://zbmath.org/authors/?q=ai:lei.hongxuanSummary: The goal of this paper is to further study multiparty multicast quantum communication of different quantum states via partially entangled channels. We employ the partially entangled channels to preform two multicast remote state preparation schemes for transmitting different complex coefficient states from one sender to two receivers synchronously. The first scheme is used to transmit two complex coefficient four-qubit cluster-type states to two receivers with a certain probability. In order to improve success probability of this multicast scheme, we propose another scheme, which is a synchronous transfer of a complex coefficient single-qubit state and a complex coefficient two-qubit state from one sender to two receivers. The success probability of the second scheme reaches 1, and independent of the entanglement degree of the partially entangled channel.