Recent zbMATH articles in MSC 32Ahttps://zbmath.org/atom/cc/32A2022-07-25T18:03:43.254055ZWerkzeugApproximation of functions of several variables by multidimensional \(S\)-fractions with independent variableshttps://zbmath.org/1487.320062022-07-25T18:03:43.254055Z"Dmytryshyn, R. I."https://zbmath.org/authors/?q=ai:dmytryshyn.r-i"Sharyn, S. V."https://zbmath.org/authors/?q=ai:sharyn.sergiiSummary: The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called ``multidimensional \(S\)-fraction with independent variables''. As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional \(S\)-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.Real analyticity of a \(C^{\infty}\)-germ at the originhttps://zbmath.org/1487.320072022-07-25T18:03:43.254055Z"Sadullaev, Azimbay"https://zbmath.org/authors/?q=ai:sadullaev.azimbai-sadullaevichSummary: The work is devoted to real analyticity in the ball \(B(0,1)\) of a function \(f\) that is infinitely smooth at \(0\in\mathbb{R}^n\) and whose restrictions to lines \(l\ni 0\) are real analytic in the interval \(l\cap B(0,1)\approx (-1,1)\).Slice holomorphic functions in the unit ball: boundedness of \(L\)-index in a direction and related propertieshttps://zbmath.org/1487.320082022-07-25T18:03:43.254055Z"Bandura, A. I."https://zbmath.org/authors/?q=ai:bandura.a-i"Salo, T. M."https://zbmath.org/authors/?q=ai:salo.tetyana-mykhailivna"Skaskiv, O. B."https://zbmath.org/authors/?q=ai:skaskiv.oleg-bSummary: Let \(\mathbf{b}\in\mathbb{C}^n\setminus\{\boldsymbol{0}\}\) be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice \(\{z^0+t\mathbf{b}: t\in\mathbb{C}\}\) with the unit ball \(\mathbb{B}^n=\{z\in\mathbb{C}^: |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}<1\}\) for any \(z^0\in\mathbb{B}^n\). For this class of functions we consider the concept of boundedness of \(L\)-index in the direction \(\mathbf{b}\), where \(\mathbf{L}: \mathbb{B}^n\to\mathbb{R}_+\) is a positive continuous function such that \(L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}\) and \(\beta>1\) is some constant. For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications to differential equations. We introduce a concept of function having bounded value \(L\)-distribution in direction for the slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value \(L\)-distribution in a direction if and only if its directional derivative has bounded \(L\)-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function \(F\) with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function \(L\) that the function \(F\) has bounded \(L\)-index in the direction.On space of holomorphic functions with boundary smoothness and its dualhttps://zbmath.org/1487.320092022-07-25T18:03:43.254055Z"Lutsenko, Anastasiya Vladimirovna"https://zbmath.org/authors/?q=ai:lutsenko.anastasiya-vladimirovna"Musin, Il'dar Khamitovich"https://zbmath.org/authors/?q=ai:musin.ildar-khamitovichSummary: We consider a Fréchet-Schwartz space \(A_{\mathcal{H}}(\Omega)\) of functions holomorphic in a bounded convex domain \(\Omega\) in a multidimensional complex space and smooth up to the boundary with the topology defined by means of a countable family of norms. These norms are constructed via some family \(\mathcal{H}\) of convex separately radial weight functions in \(\mathbb{R}^n\). We study the problem on describing a strong dual space for this space in terms of the Laplace transforms of functionals. An interest to such problem is motivated by the researches by B. A. Derzhavets devoted to classical problems of theory of linear differential operators with constant coefficients and the researches by A. V. Abanin, S. V. Petrov and K. P. Isaev of modern problems of the theory of absolutely representing systems in various spaces of holomorphic functions with given boundary smoothness in convex domains in complex space; these problems were solved by Paley-Wiener-Schwartz type theorems. Our main result states that the Laplace transform is an isomorphism between the strong dual of our functional space and some space of entire functions of exponential type in \(\mathbb{C}^n\), which is an inductive limit of weighted Banach spaces of entire functions. This result generalizes the corresponding result of the second author in [Vladikavkaz. Mat. Zh. 22, No. 3, 100--111 (2020; Zbl 1474.32003)]. To prove this theorem, we apply the scheme proposed by M. Neymark and B. A. Taylor. On the base of results from monograph by \textit{L. Hörmander} [An introduction to complex analysis in several variables. 3rd revised ed. Amsterdam etc.: North-Holland (1990; Zbl 0685.32001)], a problem of solvability of systems of partial differential equations in \(A_{\mathcal{H}}^m (\Omega)\) is considered. An analogue of a similar result from monograph by L. Hörmander is obtained. In this case we employ essentially the properties of the Young-Fenchel transform of functions in the family \(\mathcal{H} \).Sampling in spaces of entire functions of exponential type in \(\mathbb{C}^{n + 1} \)https://zbmath.org/1487.320102022-07-25T18:03:43.254055Z"Monguzzi, Alessandro"https://zbmath.org/authors/?q=ai:monguzzi.alessandro"Peloso, Marco M."https://zbmath.org/authors/?q=ai:peloso.marco-maria"Salvatori, Maura"https://zbmath.org/authors/?q=ai:salvatori.mauraSummary: In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose on the entire functions, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary \(b \mathcal{U}\) of the Siegel upper half-space \(\mathcal{U}\) and it is fundamental that \(b \mathcal{U}\) can be identified with the Heisenberg group \(\mathbb{H}_n\). We consider entire functions in \(\mathbb{C}^{n + 1}\) of exponential type with respect to the hypersurface \(b \mathcal{U}\) whose restriction to \(b \mathcal{U}\) are square integrable with respect to the Haar measure on \(\mathbb{H}_n\). For these functions we prove a version of the Whittaker-Kotelnikov-Shannon Theorem. Instrumental in our work are spaces of entire functions in \(\mathbb{C}^{n + 1}\) of exponential type with respect to the hypersurface \(b \mathcal{U}\) whose restrictions to \(b \mathcal{U}\) belong to some homogeneous Sobolev space on \(\mathbb{H}_n\). For these spaces, using the group Fourier transform on \(\mathbb{H}_n\), we prove a Paley-Wiener type theorem and a Plancherel-Pólya type inequality.Factorization of linearly independent operators and prime solutions of partial differential equationshttps://zbmath.org/1487.320112022-07-25T18:03:43.254055Z"Wang, Qiong"https://zbmath.org/authors/?q=ai:wang.qiong"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.4"Zhan, Guoping"https://zbmath.org/authors/?q=ai:zhan.guopingSummary: In this paper, we obtain the precise form of meromorphic functions \(u(z_1, z_2)\) in \(\mathbb{C}^2\) when the linearly independent operators \(au_{z_1} + bu_{z_2}\) and \(cu_{z_1} + du_{z_2}\) have a common right factor with constants \(a, b, c, d\). We also describe entire solutions of the partial differential equation
\[
F(u_{z_1}, u_{z_2}, \ldots, u_{z_n})=1
\]
in \(\mathbb{C}^n\) under a more general definition in the sense of the prime functions. In addition, our results generalize the recent results in
[\textit{B. Q. Li}, Trans. Am. Math. Soc. 357, No. 8, 3169--3177 (2005; Zbl 1073.35057); Int. J. Math. 15, No. 5, 473--485 (2004; Zbl 1053.35042)].Transcendental entire solutions for several quadratic binomial and trinomial PDEs with constant coefficientshttps://zbmath.org/1487.320122022-07-25T18:03:43.254055Z"Xu, Hong Yan"https://zbmath.org/authors/?q=ai:xu.hongyan"Xu, Ling"https://zbmath.org/authors/?q=ai:xu.lingSummary: This article is concerned with the description of the existence of entire solutions of several quadratic binomial and trinomial partial differential equations (PDEs) with constant coefficients. We established a series of theorems on the forms of finite order transcendental entire solutions for such PDEs, which are some generalization and improvement of the previous theorems given by
\textit{B. Cao} [J. Fuzzy Math. 14, No. 1, 1--14 (2006; Zbl 1108.90052)] and \textit{J. Cao} and \textit{C. Xu} [in: Recent advances in scientific computing and applications. Eighth international conference on scientific computing and applications, University of Nevada, Las Vegas, NV, USA, April 1--4, 2012. Proceedings. Providence, RI: AMS. 93--103 (2013; Zbl 1280.65066)]. Moreover, a series of examples are given to show that the existence conditions and the forms of transcendental entire solutions with finite order of such equations are precise.On normal families in \(C^n\)https://zbmath.org/1487.320132022-07-25T18:03:43.254055Z"Dovbush, P. V."https://zbmath.org/authors/?q=ai:dovbush.peter-vSummary: We show that a family \(\mathcal{F} = \{f\}\) of functions holomorphic in a domain \(\Omega \subset C^n\) is normal if all eigenvalues of the complex Hessian matrix of \(\log (1+|f|^2)\) are uniformly bounded away from zero on compact subsets of \(\Omega \).A geometric approach to the theory of normal familieshttps://zbmath.org/1487.320142022-07-25T18:03:43.254055Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-georgeSummary: We treat normal families of holomorphic mappings from the point of view of invariant geometry. We relate these ideas to the concept of normal function.A property of the spherical derivative of an entire curve in complex projective spacehttps://zbmath.org/1487.320152022-07-25T18:03:43.254055Z"Nguyen Thanh Son"https://zbmath.org/authors/?q=ai:nguyen-thanh-son."Tran Van Tan"https://zbmath.org/authors/?q=ai:tran-van-tan.Summary: We establish a type of the Picard's theorem for entire curves in \(P^n(\mathbb{C})\) whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union \(D\) of finite number of hypersurfaces in the complex projective space \(P^n(\mathbb{C})\) such that for every entire curve \(f\) in \(P^n(\mathbb{C})\), if the spherical derivative \(f^{\#}\) of \(f\) is bounded on \(f^{-1}(D)\), then \(f^{\#}\) is bounded on the entire complex plane, and hence, \(f\) is a Brody curve.On generalized Fermat Diophantine functional and partial differential equations in \(\mathbf{C}^2\)https://zbmath.org/1487.320162022-07-25T18:03:43.254055Z"Wang, Qiong"https://zbmath.org/authors/?q=ai:wang.qiong"Han, Qi"https://zbmath.org/authors/?q=ai:han.qi.1"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.4Summary: In this paper, we characterize meromorphic solutions \(f(z_1, z_2)\), \(g(z_1, z_2)\) to the generalized Fermat Diophantine functional equations \(h(z_1, z_2)f^m + k(z_1, z_2)g^n=1\) in \(\mathbf{C}^2\) for integers \(m,n\ge 2\) and nonzero meromorphic functions \(h(z_1, z_2)\), \(k(z_1, z_2)\) in \(\mathbf{C}^2\). Meromorphic solutions to associated partial differential equations are also studied.Asymptotic expansion of the Bergman kernel via semi-classical symbolic calculushttps://zbmath.org/1487.320172022-07-25T18:03:43.254055Z"Hou, Yu-Chi"https://zbmath.org/authors/?q=ai:hou.yu-chiSummary: We give a new proof on the pointwise asymptotic expansion for Bergman kernel associated to \(k\)-th tensor power of a hermitian holomorphic line bundle on the points where the curvature of the line bundle is positive and satisfies local spectral gap condition. The main point is to introduce a suitable semi-classical symbol space and related symbolic calculus inspired from recent work of Hsiao and Savale. Particularly, we establish the existence of pointwise asymptotic expansion on the positive part for certain semi-positive line bundles.Entropy of Bergman measures of a toric Kaehler manifoldhttps://zbmath.org/1487.320182022-07-25T18:03:43.254055Z"Zelditch, Steve"https://zbmath.org/authors/?q=ai:zelditch.steve"Flurin, Pierre"https://zbmath.org/authors/?q=ai:flurin.pierreSummary: Associated to the Bergman kernels of a polarized toric Kähler manifold \((M,\omega, L, h)\) are sequences of measures \(\{\mu^z_k\}^{\infty}_{k=1}\) parametrized by the points \(z\in M\). We determine the asymptotics of the entropies \(H(\mu^z_k)\) of these measures. The sequence \(\mu^z_k\) in some ways resembles a sequence of convolution powers; we determine precisely when it actually is such a sequence. When \((M,\omega)\) is a Fano toric manifold with positive Ricci curvature, we show that there exists a unique point \(z_0\) (up to the real torus action) for which \(\mu^z_k\) has asymptotically maximal entropy. If the Kähler metric is Kähler-Einstein, we show that the image of \(z_0\) under the moment map is the center of mass of the polytope. We also show that the Gaussian measure on the space \(H^0 (M, L^k)\) induced by the Kähler metric has maximal entropy at the balanced metric.Koppelman formulas on smooth compact toric varietieshttps://zbmath.org/1487.320192022-07-25T18:03:43.254055Z"Tryfonos, C."https://zbmath.org/authors/?q=ai:tryfonos.c"Vidras, A."https://zbmath.org/authors/?q=ai:vidras.alekosSummary: In this paper we derive an explicit Koppelman integral representation formula in terms of the combinatorial data on smooth compact toric varieties for \((0, q)\) smooth forms taking values in specific line bundles. The n-dimensional toric varieties are such that their Newton polyhedron contains the origin and the standard base \(\{e_1,\dots, e_n\}\) of \(\mathbb{R}^n \). Applying the above formula one obtains an alternative proof about vanishing of the Dolbeault cohomology groups of \((0, q)\) forms over such smooth compact toric varieties with values in various lines bundles.Distortion results for a certain subclass of biholomorphic mappings in \(\mathbb{C}^n\)https://zbmath.org/1487.320202022-07-25T18:03:43.254055Z"Xiong, Liangpeng"https://zbmath.org/authors/?q=ai:xiong.liangpengSummary: Let \(\mathbb{C}^n\) be the space of \(n\)-dimensional complex variables and \(\mathbb{D}^n\) be the unit polydisc in \(\mathbb{C}^n\). We obtain the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for a certain subclass of normalized biholomorphic mappings defined on \(\mathbb{D}^n\). Also, the distortion theorem of Jacobi-determinant type for the corresponding subclass defined on the unit ball in \(\mathbb{C}^n\) with arbitrary norm is established. Our results allow each component of complex vectors to have different dimensions, which extends severl previous works being closely related to some subclasses of starlike mappings.Area operators on Hardy spaces in the unit ball of \(\mathbb{C}^n\)https://zbmath.org/1487.320212022-07-25T18:03:43.254055Z"Liu, Xiaosong"https://zbmath.org/authors/?q=ai:liu.xiaosong"Lou, Zengjian"https://zbmath.org/authors/?q=ai:lou.zengjian"Zhao, Ruhan"https://zbmath.org/authors/?q=ai:zhao.ruhanSummary: We characterize boundedness and compactness of area operators from \(H^p\) into \(L^q( \mathbb{S}_n)\) in terms of Carleson measures. Some of the tools used in the proof of the one dimensional case are not available in the higher dimension case, such as the strong factorization of Hardy spaces and the Calderón-Zygmund decomposition. Our approach involves a duality argument and maximal functions of \(L^p( \mathbb{S}_n)\) functions in the unit ball of \(\mathbb{C}^n\).On the rigidity of proper holomorphic mappings for the Bergman-Hartogs domainshttps://zbmath.org/1487.320222022-07-25T18:03:43.254055Z"Bi, Enchao"https://zbmath.org/authors/?q=ai:bi.enchaoSummary: In this paper, we mainly discuss the proper holomorphic mappings of the Bergman-Hartogs domains raised by Roos, which can also be regarded as a natural generalization of Cartan-Hartogs domains. We show that any proper holomorphic mapping between two equidimensional Bergman-Hartogs domains over bounded circular homogeneous domains is a biholomorphism. As an application of our result, we can firstly obtain the rigidity of the proper holomorphic mappings for the Cartan-Hartogs domains. Secondly, we are able to describe the biholomorphism between two Bergman-Hartogs domains over any bounded circular homogeneous domains, and thus we can completely determine its automorphism group without fibre's restriction.Some remarks on the Kobayashi-Fuks metric on strongly pseudoconvex domainshttps://zbmath.org/1487.320232022-07-25T18:03:43.254055Z"Borah, Diganta"https://zbmath.org/authors/?q=ai:borah.diganta"Kar, Debaprasanna"https://zbmath.org/authors/?q=ai:kar.debaprasannaSummary: The Ricci curvature of the Bergman metric on a bounded domain \(D \subset \mathbb{C}^n\) is strictly bounded above by \(n + 1\) and consequently \(\log( K_D^{n + 1} g_{B , D})\), where \(K_D\) is the Bergman kernel for \(D\) on the diagonal and \(g_{B , D}\) is the Riemannian volume element of the Bergman metric on \(D\), is the potential for a Kähler metric on \(D\) known as the Kobayashi-Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.Hankel operators between Bergman spaces with variable exponents on the unit ball of \(\mathbb{C}^n\)https://zbmath.org/1487.320242022-07-25T18:03:43.254055Z"Dieudonne, Agbor"https://zbmath.org/authors/?q=ai:agbor.dieudonneSummary: We characterize boundedness and compactness of Hankel operators between Bergman spaces of variable exponent and the Lebesque spaces of variable exponents. We also give some characterizations of the symbol class which is some \textit{BMO}-type spaces with variable exponent on the unit ball of \(\mathbb{C}^n \).Dyadic decomposition of convex domains of finite type and applicationshttps://zbmath.org/1487.320252022-07-25T18:03:43.254055Z"Gan, Chun"https://zbmath.org/authors/?q=ai:gan.chun"Hu, Bingyang"https://zbmath.org/authors/?q=ai:hu.bingyang"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyasSummary: In this paper, we introduce a dyadic structure on convex domains of finite type via the so-called dyadic flow tents. This dyadic structure allows us to establish weighted norm estimates for the Bergman projection \(P\) on such domains with respect to Muckenhoupt weights. In particular, this result gives an alternative proof of the \(L^p\) boundedness of \(P\). Moreover, using extrapolation, we are also able to derive weighted vector-valued estimates and weighted modular inequalities for the Bergman projection.Volume integral means over spherical shellhttps://zbmath.org/1487.320262022-07-25T18:03:43.254055Z"Karapetrović, Boban"https://zbmath.org/authors/?q=ai:karapetrovic.bobanSummary: We investigate integral means over spherical shell of holomorphic functions in the unit ball \(\mathbb{B}_n\) of \(\mathbb{C}^n\) with respect to the weighted volume measures and their relation with the weighted Hadamard product. The main result of this paper has many consequences which improve some well-known estimates related to the Hadamard product in Hardy spaces and weighted Bergman spaces.On an invariant distance induced by the Szegő kernelhttps://zbmath.org/1487.320272022-07-25T18:03:43.254055Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-george"Wójcicki, Paweł M."https://zbmath.org/authors/?q=ai:wojcicki.pawel-mSummary: In this paper we introduce a new distance by means of the so-called Szegő kernel and examine some basic properties and its relationship with the so-called Skwarczyński distance. We also examine the relationship between this distance, and the so-called Bergman distance and Szegő distance.Weighted \(L^2\) approximation of analytic sectionshttps://zbmath.org/1487.320282022-07-25T18:03:43.254055Z"Li, Zhi"https://zbmath.org/authors/?q=ai:li.zhi|li.zhi.1"Zhou, Xiangyu"https://zbmath.org/authors/?q=ai:zhou.xiangyu.1|zhou.xiangyuSummary: In this paper, we obtain a global weighted \(L^2\) approximation result for holomorphic sections in weighted Bergman spaces, generalizing the approximation theorems of Taylor, Fornæss, and Wu. The main novelty here is a combination of the \(L^2\) estimates with twisted Bochner-Kodaira technique, optimal \(L^2\) extension technique and the solution of the strong openness conjecture on multiplier ideal sheaves.Carleson measures on the generalized Hartogs triangleshttps://zbmath.org/1487.320292022-07-25T18:03:43.254055Z"Zhang, Shuo"https://zbmath.org/authors/?q=ai:zhang.shuo.1The author characterizes Carleson measures and vanishing Carleson measures of weighted Bergman spaces on the generalized Hartogs triangle
\[H=H^n_{\{k_i\}, \gamma}:=\{z\in\mathbb C^n: \max\limits_{i=1,\dots,\ell}\|\widetilde z_i\|<|z_{k+1}|^\gamma<\dots<|z_n|^\gamma<1\},\]
where \(n\geq2\), \(\gamma, k_1,\dots,k_\ell\in\mathbb N\), \(k:=\sum_{i=1}^\ell k_i<n\), \(\widetilde z_i:=(z_{m_{i-1}+1},\dots,z_{m_i})\), \(m_0:=0\), \(m_i:=\sum_{j=1}^i k_j\). Define \(\varphi(z):=|z_{k+1}|^{-2k\gamma}\prod_{j=k+2}^n|z_j|^{-2}\) and
\[\psi(z):=\prod_{i=1}^\ell(1-\|\frac{\widetilde z_i}{z_{k+1}^\gamma}\|^2)^{-(k_i+1)}\cdot\prod_{j=k+1}^n(1-|\frac{z_j}{z_{j+1}}|^2)^{-2},\]
\(z\in H\).
Let \(\mu\) be a finite positive Borel measure on \(H\) and let \(1\leq p<+\infty\). We say that \(\mu\) is a \(p\)-Carleson measure (resp. vanishing \(p\)-Carleson measure) of the weighted Bergman space \(A^2_\alpha(H):=\{f\in\mathcal O(H): \int_H|f|^2\psi^\alpha dV<+\infty\}\) if the inclusion operator \(A^2_\alpha(H)\longrightarrow L^p(H,\mu)\) is bounded (resp. compact).
Let \(\mu\) be a finite positive Borel measure on \(H\), \(p, \alpha\in\mathbb R\), \(2\leq p\leq\infty\), \(-1<\alpha<\min\limits_{i=1,\dots,\ell}\frac1{k_i+1}\). Let \(\mu^p\) be the measure on \(H\) defined by \(d\mu^p=\varphi^{p/2}d\mu\). The main results of the paper are the following two theorems:
Theorem. The following conditions are equivalent:
(1) \(\mu\) is a \(p\)-Carleson measure of \(A^2_\alpha(H)\);
(2) the function \(H\ni z\longmapsto\psi(z)^{-\frac12p\alpha}B^p\mu(z)\) is bounded, where \(B^p_\mu(z):=\int_H|\frac{K_H(z,w)}{K_H(z,z)^{1/2}}|^pd\mu(w)\);
(3) the function \(H\ni z\longmapsto\psi(z)^{\frac12(1-\alpha)p-1}\widehat{\mu^p}_r(z)\) is bounded, where \(\widehat{\mu^p}_r(z):=\frac{\mu^p(B^c_H(z,r))}{V(B^c_H(z,r))}\), \(B^c_H(z,r):=\{w\in H: \tanh c_H(w,z)<r\}\), \(c_H\) stands for the Möbius distance;
(4) for any \(r\in(0,1)\) and \(r\)-lattice \(\{a_j\}_{j=1}^\infty\) the sequence \(\{\psi(a_j)^{\frac12(1-\alpha)p-1}\widehat{\mu^p}_r(a_j)\}_{j=1}^\infty\) is bounded.
Theorem. If \(\mu\) is a vanishing \(p\)-Carleson of \(A^2_\alpha(H)\), then for any \(r\in(0,1)\) we have \(\psi(z)^{\frac12(1-\alpha)p-1}\widehat{\mu^p}_r(z)\longrightarrow0\) as \(z\longrightarrow\partial H\setminus\{0\}\).
Reviewer: Marek Jarnicki (Kraków)\(L^p\) Sobolev mapping properties of the Bergman projections on \(n\)-dimensional generalized Hartogs triangleshttps://zbmath.org/1487.320302022-07-25T18:03:43.254055Z"Zhang, Shuo"https://zbmath.org/authors/?q=ai:zhang.shuo.1Summary: The \(n\)-dimensional generalized Hartogs triangles \(\mathbb{H}_{\textbf{p}}^n\) with \(n\geq 2\) and \(\textbf{p}:=(p_1,\ldots,p_n)\in(\mathbb{R}^+)^n\) are the domains defined by
\[
\mathbb{H}_{\textbf{p}}^n:=\{z=(z_1,\ldots,z_n)\in\mathbb{C}^n:|z_1|^{p_1}< \cdots < |z_n|^{p_n}< 1\}.
\]
In this paper, we study the \(L^p\) Sobolev mapping properties for the Bergman projections on the \(n\)-dimensional generalized Hartogs triangles \(\mathbb{H}^n_\mathbf{p}\) which can be viewed as a continuation of the work by \textit{S. Zhang} in [Complex Var. Elliptic Equ. 66, No. 9, 1591--1608 (2021; Zbl 1478.32019)] and a higher-dimensional generalization of the work by \textit{L. D. Edholm} and \textit{J. D. McNeal} in [J. Geom. Anal. 30, No. 2, 1293--1311 (2020; Zbl 1446.32005)].Multidimensional singular integrals and integral equations in fractional spaces. IIhttps://zbmath.org/1487.320312022-07-25T18:03:43.254055Z"Bliev, N. K."https://zbmath.org/authors/?q=ai:bliev.nazarbai-kadyrovichSummary: In this paper, boundedness, noetherity and smoothness properties of multidimensional singular integral operators and solvability of the corresponding singular integral equations in Besov spaces are studied.
For Part I, see [ibid. 66, No. 5, 819--825 (2021; Zbl 1470.32017)].Estimates for the volume of the zeros of a holomorphic function depending on a complex parameterhttps://zbmath.org/1487.320322022-07-25T18:03:43.254055Z"Kytmanov, Aleksandr M."https://zbmath.org/authors/?q=ai:kytmanov.alexander-mechislavovich"Sadullaev, Azimbay"https://zbmath.org/authors/?q=ai:sadullaev.azimbai-sadullaevichExtensions of abstract Loewner chains and spirallikenesshttps://zbmath.org/1487.320792022-07-25T18:03:43.254055Z"Muir, Jerry R. jun."https://zbmath.org/authors/?q=ai:muir.jerry-r-junSummary: We generate a variety of Loewner chains on the Euclidean unit ball in \(\mathbb{C}^n\) by extending chains from lower-dimensional disks or balls. Using these extended Loewner chains, we produce an assortment of spirallike mappings. Because of the Loewner chains used, these spirallike mappings are extensions, via either a modified Roper-Suffridge extension operator introduced by the author or a perturbation of the Pfaltzgraff-Suffridge extension operator, of lower-dimensional spirallike mappings. The Loewner chains under consideration are not normalized, but are of order \(p\), meaning only a locally uniform local \(L^p\)-continuity condition is imposed on the real parameter of the family. Therefore, the resulting spirallike mappings are not normalized and may be spirallike with respect to a boundary point. Furthermore, mappings are produced that satisfy a generalized form of spirallikeness with respect to a locally integrable operator-valued function \(A\) on \([0,\infty)\) rather than a fixed linear operator. There is a natural link between the function \(\Vert A(\cdot)\Vert\) being locally \(L^p\) and the \(L^p\)-continuity condition on the corresponding Loewner chain. Despite the abstract nature of these results, they remain novel even in the case where \(A\) is constant and the mappings are normalized; that is, we obtain new normalized biholomorphic mappings that are spirallike with respect to a linear operator.On the Weyl-Ahlfors theory of derived curveshttps://zbmath.org/1487.320842022-07-25T18:03:43.254055Z"Huynh, Dinh Tuan"https://zbmath.org/authors/?q=ai:huynh.dinh-tuan"Xie, Song-Yan"https://zbmath.org/authors/?q=ai:xie.song-yanSummary: For derived curves intersecting a family of decomposable hyperplanes in \textit{subgeneral position}, we obtain an analog of the Cartan-Nochka Second Main Theorem, generalizing a classical result of Fujimoto about decomposable hyperplanes in \textit{general position}.Nevanlinna theory for holomophic curves from annuli into semi-abelian varietieshttps://zbmath.org/1487.320852022-07-25T18:03:43.254055Z"Quang, Si Duc"https://zbmath.org/authors/?q=ai:si-duc-quang.Summary: In this paper, we prove a lemma on logarithmic derivative for holomorphic curves from annuli into Kähler compact manifolds. As its application, a second main theorem for holomophic curves from annuli into semi-abelian varieties intersecting with only one divisor is given.Two meromorphic mappings having the same inverse images of some moving hyperplanes with truncated multiplicityhttps://zbmath.org/1487.320862022-07-25T18:03:43.254055Z"Quang, Si Duc"https://zbmath.org/authors/?q=ai:si-duc-quang.Summary: Let \(f\) and \(g\) be two meromorphic mappings of \(\mathbb{C}^m\) into \(\mathbb{P}^n(\mathbb{C})\) and let \(a_1, \ldots, a_{2n + 2}\) be \(2n + 2\) moving hyperplanes which are slow with respect to \(f\) and \(g\). We will show that if \(f\) and \(g\) have the same inverse images for all \(a_i\) (\(1 \leq i \leq 2n + 2\)) with multiplicities counted to level \(l_i\) such that \[\mathop{\sum}_{i = 1}^{2n + 2} \frac{1}{l_i} < \frac{2}{3n^{2}q(q-2)},\] where \(q = \binom{2n + 2}{n + 1}\), then the map \(f \times g\) into \(\mathbb{P}^n (\mathbb{C}) \times \mathbb{P}^n (\mathbb{C})\) must be algebraically degenerate over the field \(\mathscr{R}_{\{a_{i}\}_{i = 1}^{2n + 2}}\). Our result extends and improves the previous result in this topic.A second main theorem for holomorphic maps into the projective space with hypersurfaceshttps://zbmath.org/1487.320872022-07-25T18:03:43.254055Z"Shi, Lei"https://zbmath.org/authors/?q=ai:shi.lei.1|shi.lei.4|shi.lei.3|shi.lei.2|shi.lei"Yan, Qiming"https://zbmath.org/authors/?q=ai:yan.qimingSummary: In this paper, the study will focus on the hypersurfaces in the projective space located in subgeneral position. By considering the number of irreducible components of these hypersurfaces, a new second main theorem is established for algebraically non-degenerate holomorphic maps from \(\mathbb{C}\) into the projective space with truncated counting functions. Moreover, as the counterpart of this second main theorem, a Schmidt's subspace type theorem in Diophantine approximation is also given for this case.Meromorphic mappings into projective varieties with arbitrary families of moving hypersurfaceshttps://zbmath.org/1487.320882022-07-25T18:03:43.254055Z"Si, Duc Quang"https://zbmath.org/authors/?q=ai:si-duc-quang.Summary: In this paper, we prove a general second main theorem for meromorphic mappings into a subvariety \(V\) of \(\mathbb{P}^N (\mathbb{C})\) with an arbitrary family of moving hypersurfaces. Our second main theorem generalizes and improves all previous results for meromorphic mappings with moving hypersurfaces, in particular for meromorphic mappings and families of moving hypersurfaces in subgeneral position. The method of our proof is different from that of previous authors used for the case of moving hypersurfaces.Non-integrated defect relation and uniqueness theorem for meromorphic mappings on Kähler manifoldshttps://zbmath.org/1487.320892022-07-25T18:03:43.254055Z"Si Duc Quang"https://zbmath.org/authors/?q=ai:si-duc-quang."Tran Duc Ngoc"https://zbmath.org/authors/?q=ai:tran-duc-ngoc.Improvement of the uniqueness theorems of meromorphic maps of \(\mathbb{C}^m\) into \(\mathbb{P}^n(\mathbb{C})\)https://zbmath.org/1487.320902022-07-25T18:03:43.254055Z"Zhou, Kai"https://zbmath.org/authors/?q=ai:zhou.kai"Jin, Lu"https://zbmath.org/authors/?q=ai:jin.luSummary: There have been many extensions of the Nevanlinna's five-values theorem for meromorphic functions to the case of meromorphic maps into \(\mathbb{P}^n(\mathbb{C})\). We improve these results by considering the degenerate case and using the weaker condition `\(f^{-1}(H) \subseteq g^{-1}(H)\)' instead of the usual one `\(f^{-1}(H)=g^{-1}(H)\)' for some hyperplanes \(H\) among the given hyperplanes. Our main theorems contain not only many well-known results but also more new conclusions. The sharpness of our results in some aspects is also explained.Multiplication operators on the Bergman space by proper holomorphic mappingshttps://zbmath.org/1487.320922022-07-25T18:03:43.254055Z"Ghosh, Gargi"https://zbmath.org/authors/?q=ai:ghosh.gargiSummary: Suppose that \(\boldsymbol{f}:=(f_1,\dots,f_d):\Omega_1\to\Omega_2\) is a proper holomorphic map between two bounded domains in \(\mathbb{C}^d\). We show that the multiplication operator (tuple) \(\mathbf{M}_{\boldsymbol{f}}=(M_{f_1}, \dots,M_{f_d})\) on the Bergman space \(\mathbb{A}^2(\Omega_1)\) admits a non-trivial minimal joint reducing subspace, say \(\mathcal{M}\) and the restriction of \(\mathbf{M}_{\boldsymbol{f}}\) to \(\mathcal{M}\) is unitarily equivalent to the Bergman operator on \(\mathbb{A}^2(\Omega_2)\). A number of interesting consequences of this result have been observed.Estimates of the Bergman kernel for É. Cartan's classical domainshttps://zbmath.org/1487.321172022-07-25T18:03:43.254055Z"Abdullaev, Zhonibek Shokirovich"https://zbmath.org/authors/?q=ai:abdullayev.jonibek-shokirovichSummary: The aim of this work is to find optimal estimates for the Bergman kernels for the classical domains \(\Re_I(m,k)\), \(\Re_{II} (m)\), \(\Re_{III} (m)\) and \(\Re_{IV}(n)\) through the Bergman kernels of balls in the spaces \(\mathbb{C}^{mk}\), \(\mathbb{C}^{\frac{m(m+1)}{2}}\), \(\mathbb{C}^{\frac{m(m-1)}{2}}\) and \(\mathbb{C}^n\), respectively. For this, we use the statements of the Summer-Mehring theorem on the extension of the Bergman kernel and some properties of the Bergman kernel.Schatten class Bergman-type and Szegö-type operators on bounded symmetric domainshttps://zbmath.org/1487.321182022-07-25T18:03:43.254055Z"Ding, Lijia"https://zbmath.org/authors/?q=ai:ding.lijiaSummary: In this paper, we investigate singular integral operators induced by the Bergman kernel and Szegö kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szegö-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symmetric domain. These results not only generalize a recent result on the Hilbert unit ball due to the author and his coauthor but also cover all irreducible bounded symmetric domains. Moreover, we obtain two trace formulae and a new integral estimate related to the Forelli-Rudin estimate. The key ingredient of the proofs involves the function theory on the bounded symmetric domain and the spectrum estimate of Bergman-type and Szegö-type operators.Non-Archimedean normal familieshttps://zbmath.org/1487.321252022-07-25T18:03:43.254055Z"Rodríguez Vázquez, Rita"https://zbmath.org/authors/?q=ai:rodriguez-vazquez.ritaSummary: We present several results on the compactness of the space of morphisms between analytic spaces in the sense of Berkovich. We show that under certain conditions on the source, every sequence of analytic maps having an affinoid target has a subsequence that converges pointwise to a continuous map. We also study the class of continuous maps that arise in this way. Locally, they turn to be analytic after a certain base change. Our results naturally lead to a definition of normal families. We give some applications to the dynamics of an endomorphism of the projective space. We introduce two natural notions of Fatou set and generalize to the non-Archimedan setting a theorem of Ueda stating that every Fatou component is hyperbolically imbedded in the projective space.\(L^p\) regularity of Toeplitz operators on generalized Hartogs triangleshttps://zbmath.org/1487.321262022-07-25T18:03:43.254055Z"Balay, Meijke"https://zbmath.org/authors/?q=ai:balay.meijke"Neutgens, Trent"https://zbmath.org/authors/?q=ai:neutgens.trent"Rosen, Nick"https://zbmath.org/authors/?q=ai:rosen.nick"Wagner, Nathan"https://zbmath.org/authors/?q=ai:wagner.nathan-a"Zeytuncu, Yunus E."https://zbmath.org/authors/?q=ai:zeytuncu.yunus-ergynSummary: We obtain \(L^p\) estimates for Toeplitz operators on the generalized Hartogs triangles \(\mathbb{H}_\gamma = \{(z_1,z_2) \in \mathbb{C}^2\,{:}\, |z_1|^\gamma \!< |z_2|<1\}\) for two classes of positive radial symbols, one a power of the distance to the origin, and the other a power of the distance to the boundary.Bergman type operators on some generalized Cartan-Hartogs domainshttps://zbmath.org/1487.321282022-07-25T18:03:43.254055Z"He, Le"https://zbmath.org/authors/?q=ai:he.le"Tang, Yanyan"https://zbmath.org/authors/?q=ai:tang.yanyan"Tu, Zhenhan"https://zbmath.org/authors/?q=ai:tu.zhenhanSummary: For \(\mu = (\mu_1,\dots, \mu_t)\) (\(\mu_j > 0\)), \(\xi = (z_1,\dots, z_t, w) \in\mathbb{C}^{n_1} \times\dots\times\mathbb{C}^{n_t}\times\mathbb{C}^{m}\), define
\[
\Omega(\mu,t)=\Big\{\xi\in\mathbb{B}_{n_1}\times\dots\times\mathbb{B}_{n_t}\times\mathbb{C}^m:\| w\|^2< C(\chi, \mu)\prod\nolimits^t_{j=1}(1-\|z_j\|^2)^{\mu_j}\Big\},
\]
where \(\mathbb{B}_{n_j}\) is the unit ball in \(\mathbb{C}^{n_j}\) (\(1 \leq j \leq t\)), \(C(\chi, \mu)\) is a constant only depending on \(\chi = (n_1,\dots, n_t)\) and \(\mu = (\mu_1,\dots, \mu_t)\), which is a special type of generalized Cartan-Hartogs domain. We will give some sufficient and necessary conditions for the boundedness of some type of operators on \(L^p(\Omega(\mu, t), \omega)\) (the weighted \(L^p\) space of \(\Omega(\mu, t)\) with weight \(\omega\), \(1 < p < \infty\)). This result generalizes the works from certain classes of generalized complex ellipsoids to the generalized Cartan-Hartogs domain \(\Omega(\mu, t)\).Special Toeplitz operators on elementary Reinhardt domainshttps://zbmath.org/1487.321292022-07-25T18:03:43.254055Z"Tang, Yanyan"https://zbmath.org/authors/?q=ai:tang.yanyan"Zhang, Shuo"https://zbmath.org/authors/?q=ai:zhang.shuo.1Summary: The elementary Reinhardt domain, a generalization of the classical Hartogs triangle, is defined by
\[
\mathcal{H}(\mathbf{k}) := \{z\in\mathbb{D}^n: z^k \text{ is defined and } |z^k|<1\},
\]
where \(\mathbf{k} =(k_1, \dots, k_n)\in\mathbb{Z}^n\). In this paper, a sharp criteria for the \(L^p\)-\(L^q\) boundedness of the Toeplitz operator with a class of positive radial symbol is obtained on \(\mathcal{H}(\mathbf{k})\), which generalizes the previous results of \textit{T. V. Khanh} et al. [Proc. Am. Math. Soc. 147, No. 1, 327--338 (2019; Zbl 1404.32003)] and \textit{M. Balay} et al. [Eur. J. Math. 8, No. 1, 403--416 (2022; Zbl 1487.32126)] to a more general setting.Zero problems of the Bergman kernel function on the first type of Cartan-Hartogs domainhttps://zbmath.org/1487.321302022-07-25T18:03:43.254055Z"Zhao, Xin"https://zbmath.org/authors/?q=ai:zhao.xin"Wang, An"https://zbmath.org/authors/?q=ai:wang.anSummary: The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros. If the Bergman kernel function of this type of domain has zeros, the zero set is composed of several path-connected branches, and there exists a continuous curve to connect any two points in the non-zero set.A \(L^2\) estimate for the minimal solution of \(\overline{\partial}\) on the unit ballhttps://zbmath.org/1487.322002022-07-25T18:03:43.254055Z"Thuc, Phung Trong"https://zbmath.org/authors/?q=ai:thuc.phung-trongSummary: We extend the result of \textit{A. P. Schuster} and \textit{D.Varolin} [J. Reine Angew. Math. 691, 173--201 (2014; Zbl 1309.32002)] on \(L^2\)-estimates for the minimal solution of \(\overline{\partial}\) on the unit ball \(\mathbb{B}\) in \(\mathbb{C}^n\). Estimates of weighted Bergman kernels are also considered.The fundamental solution to \(\Box_b\) on quadric manifolds. I: General formulashttps://zbmath.org/1487.322012022-07-25T18:03:43.254055Z"Boggess, Albert"https://zbmath.org/authors/?q=ai:boggess.albert"Raich, Andrew"https://zbmath.org/authors/?q=ai:raich.andrew-sSummary: This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of \(\mathbb{C}^n\times\mathbb{C}^m\). In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator \(N\) and the projection onto the nullspace of \(\Box_b\). The main application of our formulas is the critical case of codimension two quadrics in \(\mathbb{C}^4\) where we discuss the known solvability and hypoellipticity criteria of \textit{M. M. Peloso} and \textit{F. Ricci} [Funct. Anal. 203, No. 2, 321--355 (2003; Zbl 1043.32021)]. We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.Beurling quotient modules on the polydischttps://zbmath.org/1487.460562022-07-25T18:03:43.254055Z"Bhattacharjee, Monojit"https://zbmath.org/authors/?q=ai:bhattacharjee.monojit"Krishna Das, B."https://zbmath.org/authors/?q=ai:krishna-das.b"Debnath, Ramlal"https://zbmath.org/authors/?q=ai:debnath.ramlal"Sarkar, Jaydeb"https://zbmath.org/authors/?q=ai:sarkar.jaydebSummary: Let \(H^2(\mathbb{D}^n)\) denote the Hardy space over the polydisc \(\mathbb{D}^n\), \(n\geq 2\). A closed subspace \(\mathcal{Q}\subseteq H^2(\mathbb{D}^n)\) is called Beurling quotient module if there exists an inner function \(\theta\in H^\infty(\mathbb{D}^n)\) such that \(\mathcal{Q}=H^2(\mathbb{D}^n)/\theta H^2( \mathbb{D}^n)\). We present a complete characterization of Beurling quotient modules of \(H^2(\mathbb{D}^n)\): Let \(\mathcal{Q}\subseteq H^2(\mathbb{D}^n)\) be a closed subspace, and let \(C_{z_i}=P_{\mathcal{Q}}M_{z_i}|_{\mathcal{Q}}\), \(i=1,\ldots,n\). Then \(\mathcal{Q}\) is a Beurling quotient module if and only if
\[
(I_{\mathcal{Q}}-C_{z_i}^\ast C_{z_i})(I_{\mathcal{Q}}-C_{z_j}^\ast C_{z_j})=0\quad(i \neq j).
\]
We present two applications: first, we obtain a dilation theorem for Brehmer \(n\)-tuples of commuting contractions, and, second, we relate joint invariant subspaces with factorizations of inner functions. All results work equally well for general vector-valued Hardy spaces.Positive weighted composition operators on the Fock spacehttps://zbmath.org/1487.470422022-07-25T18:03:43.254055Z"Chaiworn, Areerak"https://zbmath.org/authors/?q=ai:chaiworn.areerak-kSummary: In this paper, we obtain a characterization for the bounded positive weighted composition operators and their spectrum on the Fock space of \(\mathbb{C}^n\).Weighted composition operators between different Hardy spaces on the unit ballhttps://zbmath.org/1487.470442022-07-25T18:03:43.254055Z"Li, Song Xiao"https://zbmath.org/authors/?q=ai:li.songxiao(no abstract)Difference of composition operators over the half-planehttps://zbmath.org/1487.470452022-07-25T18:03:43.254055Z"Pang, Changbao"https://zbmath.org/authors/?q=ai:pang.changbao"Wang, Maofa"https://zbmath.org/authors/?q=ai:wang.maofaSummary: To overcome the unboundedness of the half-plane, we use Khinchine's inequality and atom decomposition techniques to provide joint Carleson measure characterizations when the difference of composition operators is bounded or compact from standard weighted Bergman spaces to Lebesgue spaces over the half-plane for all index choices. For applications, we obtain direct analytic characterizations of the bounded and compact differences of composition operators on such spaces. This paper concludes with a joint Carleson measure characterization when the difference of composition operators is Hilbert-Schmidt.Approximation in weighted Bergman spaces and Hankel operators on strongly pseudoconvex domainshttps://zbmath.org/1487.470482022-07-25T18:03:43.254055Z"Gao, Jinshou"https://zbmath.org/authors/?q=ai:gao.jinshou"Hu, Zhangjian"https://zbmath.org/authors/?q=ai:hu.zhangjianSummary: Suppose \(D\) is a bounded strongly pseudoconvex domain in \(\mathbb{C}^n\) with smooth boundary, and let \(\rho\) be its defining function. For \(1 < p < \infty\) and \(\alpha > -1\), we show that the weighted Bergman projection \(P_\alpha\) is bounded on \(L^p (D, |\rho|^\alpha dV)\). With non-isotropic estimates for \(\overline{\partial}\) and Stein's theorem on non-tangential maximal operators, we prove that bounded holomorphic functions are dense in the weighted Bergman space \(A^p(D, |\rho|^\alpha dV)\), and hence Hankel operators, can be well defined on these spaces. For all \(1 < p, q < \infty\), we characterize bounded (resp., compact) Hankel operators from \(p\)-th weighted Bergman space to \(q\)-th weighted Lebesgue space with possibly different weights. As a consequence, we generalize the main results in [\textit{J.\,Pau} et al., Indiana Univ. Math. J. 65, No. 5, 1639--1673 (2016; Zbl 1448.47042)] and resolve a question posed in [\textit{X.-F. Lv} and \textit{K. Zhu}, Integral Equations Oper. Theory 91, No. 1, Paper No. 5, 23 p. (2019; Zbl 1486.47056)].Eigenvalues of \(K\)-invariant Toeplitz operators on bounded symmetric domainshttps://zbmath.org/1487.470512022-07-25T18:03:43.254055Z"Upmeier, Harald"https://zbmath.org/authors/?q=ai:upmeier.haraldFrom the Introduction: ``The well-known Toeplitz-Berezin calculus, acting on the Bergman space \(H^2 (D)\) of a bounded domain \(D\subset \mathbb C^d,\) is covariant under the biholomorphic group \(G\) of \(D.\) Actually, \textit{F. A. Berezin} [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 363--402 (1975; Zbl 0312.53050)] considered two kinds of symbolic calculus (contravariant and covariant symbols) which are related by the Berezin transform. For a bounded symmetric domain \(D=G/K\) of rank \(r,\) where \(G\) acts transitively on \(D\) and \(K\) is a maximal compact subgroup of \(G,\) one has a more general covariant Toeplitz-Berezin calculus acting on the weighted Bergman spaces \(H_\nu^2 (D)\) over \(D.\) Here \(\nu\) is a scalar parameter for the (scalar) holomorphic discrete series of \(G\) and its analytic continuation. Since \(G\) acts irreducibly on \(H_\nu^2 (D),\) there are no non-trivial \(G\)-invariant operators on the \(C^*\)-algebra generated by Toeplitz operators. On the other hand, there exist interesting \(K\)-invariant Toeplitz-type operators, which have been studied in relation to complex and harmonic analysis by \textit{J. Arazy} and \textit{G. Zhang} [J. Funct. Anal. 202, No. 1, 44--66 (2003; Zbl 1039.47020)] and \textit{S. Ghara} et al. [Isr. J. Math. 247, No. 1, 331--360 (2022; Zbl 07534011)]. These operators are uniquely determined by a sequence of eigenvalues indexed over all partitions of length \(r\).''
In the paper under review, the author determines the eigenvalues of certain ``fundamental'' \(K\)-invariant Toeplitz-type operators, both for the covariant and contravariant symbol. The covariant symbol is treated as a direct generalization of the work of Arazy and Zhang [loc.\,cit.]. The contravariant symbol eigenvalue formula requires more effort; a crucial ingredient there is the dimension formula for the irreducible \(K\)-types.
Reviewer: David Békollè (Yaoundé)A survey on determinantal point processeshttps://zbmath.org/1487.600942022-07-25T18:03:43.254055Z"Baverez, Guillaume"https://zbmath.org/authors/?q=ai:baverez.guillaume"Bufetov, Alexander I."https://zbmath.org/authors/?q=ai:bufetov.aleksander-igorevich"Qiu, Yanqi"https://zbmath.org/authors/?q=ai:qiu.yanqiSummary: We present a first introduction to determinantal point processes, focussing on multiplicative functionals, Palm theory, and key applications such as random matrices, Young diagrams and random holomorphic functions.
For the entire collection see [Zbl 1482.60003].Rank \(N\) Vafa-Witten invariants, modularity and blow-uphttps://zbmath.org/1487.810772022-07-25T18:03:43.254055Z"Alexandrov, Sergei"https://zbmath.org/authors/?q=ai:alexandrov.sergei-yuSummary: We derive explicit expressions for the generating functions of refined Vafa-Witten invariants \(\Omega(\gamma, y)\) of \(\mathbb{P}^2\) of arbitrary rank \(N\) and for their non- c modular completions. In the course of derivation we also provide: i) a generalization of the recently found generating functions of \(\Omega(\gamma, y)\) and their completions for Hirzebruch and del Pezzo surfaces in the canonical chamber of the moduli space to a generic chamber; ii) a version of the blow-up formula expressed directly in terms of these generating functions and its reformulation in a manifestly modular form.