Recent zbMATH articles in MSC 32https://zbmath.org/atom/cc/322021-01-08T12:24:00+00:00WerkzeugACC conjecture for weighted log canonical thresholds.https://zbmath.org/1449.140022021-01-08T12:24:00+00:00"Hong, N. X."https://zbmath.org/authors/?q=ai:hong.nguyen-xuan.1"Long, T. V."https://zbmath.org/authors/?q=ai:long.tang-van"Trang, P. N. T."https://zbmath.org/authors/?q=ai:trang.pham-nguyen-thuFor \(n\geq 1\) and \(\mu\) a Borel measure in \(\mathbb{C}^n\), the weighted log canonical threshold of a holomorphic function \(f\) defined in a neighborhood of the origin of \(\mathbb{C}^n\) is defined as \(c_{\mu}(f):=\)sup\(\{ c\geq 0\) : \(|f|^{-2c}\) is \(L^1(\mu)\) in a neighbourhood of \(0~\}\). Set
\(\mathcal{C}(\mu):=\{ c_{\mu}(f)\) : \(f\) is holomorphic in a neighbourhood of \(0~\}\).
The ACC conjecture for weight \(\mu\) reads:
\(\mathcal{C}(\mu)\) satisfies the ascending chain condition every convergent increasing sequence in \(\mathcal{C}(\mu)\) to be stationary.
The authors show that the conjecture holds true for \(n=2\) and \(\mu =\| z\| ^{2t}dV_4\), \(t\geq 0\).
Reviewer: Vladimir P. Kostov (Nice)Hypercyclic multiplication composition operators on weighted Banach space.https://zbmath.org/1449.470562021-01-08T12:24:00+00:00"Wang, Cui"https://zbmath.org/authors/?q=ai:wang.cui"Lu, Huiqiang"https://zbmath.org/authors/?q=ai:lu.huiqiangSummary: This paper characterizes some sufficient and necessary conditions for the hypercyclicity of multiple composition operators on \(H_{\log, 0}^\infty\).The Hardy-Dirichlet space \(\mathcal{H}^p\) and its composition operators.https://zbmath.org/1449.320012021-01-08T12:24:00+00:00"Queffélec, Hervé"https://zbmath.org/authors/?q=ai:queffelec.herveSummary: We present some recent results on composition operators acting on a Hardy space \(\mathcal{H}^p\) of a new type, formed by Dirichlet series. This study was initiated by Hedenmalm-Lindqvist-Seip for \(p = 2\) to answer a question of Beurling, and then it was continued for \(p\neq 2\) by Bayart. We get new results on the spectrum and the approximation numbers of such operators, especially when \(p = 1\). The proofs use interpolation sequences, Carleson measures and extensions of the Weyl inequalities to the Banach space setting, as well as the prime number theorem. Many interesting problems remain open.
For the entire collection see [Zbl 1404.42002].A generalization of the theorem of Von Staudt-Hua-Buekenhout-Cojan in the real \(\overset{=}{\partial}-\mathcal{F}\mathbb{R}^k_{td}\), \(1\le k\le 2n+1\), space on real geometric projective \(P_k\), \(1\le k\le 2n+1\), finite dimensional space. II.https://zbmath.org/1449.510012021-01-08T12:24:00+00:00"Cojan, Stelian Paul"https://zbmath.org/authors/?q=ai:cojan.stelian-paulSummary: Quite often it is possible to discover an alternative way to define a geometric locus which is totally different from the original one. When this is possible we obtain new interesting insight on the geometric object analogous at the improvement achieved when different ways to prove a given theorem are discovered. The purpose of our article is to describe some well-known loci using an alternative approach.
For Part I see [the author, Int. J. Geom. 7, No. 2, 50--58 (2018; Zbl 1412.58002)].Gromov hyperbolicity of the Kobayashi metric.https://zbmath.org/1449.320092021-01-08T12:24:00+00:00"Andreev, Lyubomir"https://zbmath.org/authors/?q=ai:andreev.lyubomir"Nikolov, Nikolai"https://zbmath.org/authors/?q=ai:nikolov.nikolai-marinov"Trybula, Maria"https://zbmath.org/authors/?q=ai:trybula.mariaThis paper deals with the Kobayashi distance \( k_{\Omega} \) on domains \( \Omega \) in \( \mathbb{C}^{n} \). The authors propose a collection of many known results on the subject. More specially, suppose that \( \Omega \)is bounded convex domain with \( C^{\infty} \) boundary. Then \( (\Omega, k_{\Omega}) \) is Gromov hyperbolic if and only if \( \partial \Omega \) has finite type in the sense of D'Angelo. A special attention is paid on the cases when Gromov hyperbolicity is violated. For example, let \( \Omega \subset \mathbb{C}^{n} \) be a bounded \( \mathbb{C} \) convex domain and \( S \) is a complex affine hyperplane such that \( \Omega \cap S \neq \emptyset \). Then \( (\Omega \setminus S, k_{\Omega \setminus S})\) is not Gromov hyperbolic.
Reviewer: Petar Popivanov (Sofia)Sharp distortion theorems for quasi-convex mapping of order \(\alpha\) on the unit ball.https://zbmath.org/1449.300202021-01-08T12:24:00+00:00"Guo, Lijuan"https://zbmath.org/authors/?q=ai:guo.lijuan"Zhang, Xiaofei"https://zbmath.org/authors/?q=ai:zhang.xiaofei"Zhang, Xinhong"https://zbmath.org/authors/?q=ai:zhang.xinhongSummary: In this article, we obtained the sharp distortion theorems of determinant and sharp distortion theorems of matrix at the extreme points for quasi-convex mapping of order \(\alpha\) using the Schwarz lemma at the boundary of unit ball in Euclidean space.Distributional boundary values of generalized Hardy functions in Beurling's tempered distributions.https://zbmath.org/1449.320072021-01-08T12:24:00+00:00"Sohn, Byung Keun"https://zbmath.org/authors/?q=ai:sohn.byung-keunLet \(C\) be an open convex cone in \(\mathbb{R}^N\) and let \(T^C=\mathbb{R}^N+iC\) in \(\mathbb{C}^N\). In this paper the author defines a generalization of Hardy functions (\(1 \leq p < \infty\)) on \(T^C\) and extended tempered distribution space \(S_{w}'\) of Beurling's tempered distribution space \(S_{(w)}'\) for a weight function \(w\). The author obtains the analytical and topological properties of \(S_{w}'\) and shows that the generalized Hardy functions (\(1< p \leq 2\)), have distributional boundary values in the weak topology of \(S_{(w)}'\) using the analytical properties of \(S_{w}'\) .
Reviewer: Koichi Saka (Akita)Hankel operators on generalized Fock spaces.https://zbmath.org/1449.470652021-01-08T12:24:00+00:00"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.1"Xia, Jin"https://zbmath.org/authors/?q=ai:xia.jin"Chen, Jianjun"https://zbmath.org/authors/?q=ai:chen.jianjunSummary: We characterize boundedness and compactness of Hankel operators on a very general class of weighted Fock spaces over \({\mathbb{C}^n}\) in terms of a certain notion of bounded and vanishing mean oscillation. The analogous description holds for the commutators \([{M_f}, P]\) where \({{M_f}}\) denotes the multiplication operator with symbol \(f\) and \(P\) is the Toeplitz projection. We also give geometric descriptions for the spaces BMO and VMO which are defined in terms of the Berezin transform.A subclass of boundary measures and the convex combination problem for Herglotz-Nevanlinna functions in several variables.https://zbmath.org/1449.320052021-01-08T12:24:00+00:00"Nedic, Mitja"https://zbmath.org/authors/?q=ai:nedic.mitjaSummary: In this paper, we begin by investigating a particular subclass of boundary measures of Herglotz-Nevanlinna functions in two variables. Based on this, we then proceed to solve the convex combination problem for Herglotz-Nevanlinna functions in several variables.On existence of main polynomial for analytic vector-valued functions of bounded \(L\)-index in the unit ball.https://zbmath.org/1449.320022021-01-08T12:24:00+00:00"Baksa, V. P."https://zbmath.org/authors/?q=ai:baksa.v-p"Bandura, A. I."https://zbmath.org/authors/?q=ai:bandura.a-i"Skaskiv, O. B."https://zbmath.org/authors/?q=ai:skaskiv.oleh-bohdanovych|skaskiv.oleg-bSummary: In this paper, we present necessary and sufficient conditions of boundedness of \(L\)-index in joint variables for vector-functions analytic in the unit ball \(\mathbb{B}^2\), where \(L = (l_1,l_2) : \mathbb{B}^2\to\mathbb{R}^2_+\) is a positive continuous vector-function. These conditions describe local behavior of homogeneous polynomials (so-called main polynomial) with power series expansion for analytic vector-valued functions in the unit ball. These results use a bidisc exhaustion of a unit ball.An integral-type operators from weighted Bergman space to Zygmund type spaces on the unit ball.https://zbmath.org/1449.320042021-01-08T12:24:00+00:00"Zhao, Yanhui"https://zbmath.org/authors/?q=ai:zhao.yanhui"Liao, Chunyan"https://zbmath.org/authors/?q=ai:liao.chunyan"Deng, Chunhong"https://zbmath.org/authors/?q=ai:deng.chunhongSummary: Some questions of integral-type operator were studied on Zygmund type spaces on the unit ball. By the methods of functional analysis and several complex variables, the necessary and sufficient conditions are given for integral-type operators to be bounded and compact on Zygmund type spaces on the unit ball. At the same time, the corresponding conclusions are obtained on the disk \(D\) and \(\varphi (z) = z\), respectively.Essential norm estimates for little Hankel operators with anti holomorphic symbols on weighted Bergman spaces of the unit ball.https://zbmath.org/1449.470632021-01-08T12:24:00+00:00"Tanaka, K."https://zbmath.org/authors/?q=ai:tanaka.keigo|tanaka.kohei|tanaka.kazuyuki|tanaka.keisuke|tanaka.kazuhito|tanaka.kanji|tanaka.ken-ichi|tanaka.kazunori|tanaka.katsunori|tanaka.koichiro|tanaka.kazunaga|tanaka.katsuhiko|tanaka.kaori|tanaka.katsuaki|tanaka.kazuyo|tanaka.kazuhiko|tanaka.kokoro|tanaka.katsuto|tanaka.keiji|tanaka.kenichiro|tanaka.kazuaki|tanaka.koumei|tanaka.keiichi|tanaka.kazuhide|tanaka.katsuhiro|tanaka.kiyoki|tanaka.kiyoshi|tanaka.kengo|tanaka.katsuki|tanaka.katsuya|tanaka.kotaro|tanaka.koichi|tanaka.kikuaki|tanaka.koji|tanaka.kazuo|tanaka.kakuji|tanaka.kenji|tanaka.kentaro|tanaka.k.4|tanaka.kanya|tanaka.katsuyuki|tanaka.kimiyuki|tanaka.kensuke|tanaka.katsumi|tanaka.k.3|tanaka.kazuhiro|tanaka.kiyoaki"Yamaji, S."https://zbmath.org/authors/?q=ai:yamaji.satoshiSummary: We give estimates for the essential norm of a little Hankel operator with anti holomorphic symbol on weighted Bergman spaces of the unit ball in terms of the Bloch semi-norm of its symbol function.The singular integrals on the closed piecewise smooth manifolds of octonions.https://zbmath.org/1449.320082021-01-08T12:24:00+00:00"Gong, Dingdong"https://zbmath.org/authors/?q=ai:gong.dingdongSummary: The solid-angle coefficient method is used to study the principal value on the closed piecewise smooth manifolds in octonions, and a corresponding Sokhotski-Plemel formula is obtained. These results are proved to be useful in the further study of the singular integral theory in octonions.Several properties on the normal weight Zygmund space in several complex variables.https://zbmath.org/1449.320032021-01-08T12:24:00+00:00"Li, Shenlian"https://zbmath.org/authors/?q=ai:li.shenlian"Zhang, Xuejun"https://zbmath.org/authors/?q=ai:zhang.xuejunSummary: In this paper, the authors investigate some properties of the normal weight Zygmund space \({Z_\mu} (B)\) in several complex variables. Firstly, the authors establish an integral representation of function in \({Z_\mu} (B)\). Secondly, the authors show that \({Z_\mu} (B)\) can be identified with the dual space of the normal weight Bergman space \(A_v^1 (B)\) under the integral pairing \[\langle {f, g} \rangle = \lim\limits_{\rho\to{1^-}} \int_B {f (\rho z)} \overline{ ({R^{\beta-2,1}}g) (\rho z)} d{v_{\beta-1}} (z)\;\;\; (f \in A_v^1 (B), g \in {Z_\mu} (B)), \] where \(v (r) = (1-r^2)^{\beta+1}{\mu^{-1}} (r) (0 \le r < 1)\) and \(\beta > \max \{0, b -1\}\). Finally, as an application of the integral representation and the dual, the authors give an atomic decomposition for every function in \({Z_\mu} (B)\).Uniqueness theorem on meromorphic mappings with few moving targets.https://zbmath.org/1449.320102021-01-08T12:24:00+00:00"Liu, Zhixue"https://zbmath.org/authors/?q=ai:liu.zhixue"Zhang, Qingcai"https://zbmath.org/authors/?q=ai:zhang.qingcaiSummary: In this paper, concerning some truncated counting functions with different weights, we prove a new second main theorem for meromorphic mappings from \({\mathbb{C}^n}\) into \({\mathbb{P}^N} (\mathbb{C})\). By using the new second main theorem, we consider the uniqueness problem for the case of degenerate meromorphic mappings sharing moving hyperplanes located in general position, and a uniqueness result is obtained under some weak conditions, which can be seen as an improvement of previous well-known results.Algebraic properties of Toeplitz operators on cutoff harmonic Bergman space.https://zbmath.org/1449.470662021-01-08T12:24:00+00:00"Yang, Jingyu"https://zbmath.org/authors/?q=ai:yang.jingyu"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufeng"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huoSummary: In this paper, we first investigate the finite-rank product problems of several Toeplitz operators with quasihomogeneous symbols on the cutoff harmonic Bergman space \(b_n^2\). Next, we characterize the finite rank commutators and semi-commutators of two Toeplitz operators with quasihomogeneous symbols on \(b_n^2\).Testing Lipschitz non-normally embedded complex spaces.https://zbmath.org/1449.140012021-01-08T12:24:00+00:00"Denkowski, Maciej"https://zbmath.org/authors/?q=ai:denkowski.maciej-p"Tibăr, Mihai"https://zbmath.org/authors/?q=ai:tibar.mihai-mariusOne can define two metrics on a locally closed path-connected set \(X\subset \mathbb{R}^n\): the Euclidean metric \(d(x,y)\) and the inner metric \(d_X(x,y)\), where \(d_x,y)\) is the length of the shortest path belonging to \(X\) and joining the points \(x,y\in X\). The set \(X\) is said to be (Lipschitz) normally embedded if there exists \(C>0\) such that \(d_X(x,y)\leq Cd(x,y)\) for all \(x,y\in X\). The authors introduce a sectional criterion for testing if complex analytic germs \((X,0)\subset (\mathbb{C}^n,0)\) are Lipschitz non-normally embedded.
Reviewer: Vladimir P. Kostov (Nice)On the non-existence of negative weight derivations of the new moduli algebras of singularities.https://zbmath.org/1449.140042021-01-08T12:24:00+00:00"Ma, Guorui"https://zbmath.org/authors/?q=ai:ma.guorui"Yau, Stephen S.-T."https://zbmath.org/authors/?q=ai:yau.stephen-shing-toung"Zuo, Huaiqing"https://zbmath.org/authors/?q=ai:zuo.huaiqingTo any isolated hypersurface singularity \((V,0)\subset (\mathbb{C}^{n+1},0),\) defined by a holomorphic function \(f:(\mathbb{C}^{n+1},0)\rightarrow (\mathbb{C},0),\) there are associated many algebraic objects, for instance the Tjurina algebra \(A(V):=\mathcal{O}_{n+1}/(f,\frac{\partial f}{\partial z_{0}},\ldots ,\frac{\partial f}{\partial z_{n+1}})\) which completely determines the analytic structure of \((V,0).\) A weaker invariant is the Lie algebra of derivations of \(A(V)\) i.e. \(L(V):=\operatorname{Der}(A(V),A(V)).\) The authors consider more subtle invariants: the local algebra
\[
A^{\ast }(V):=\mathcal{O}_{n+1}/(f,\frac{\partial f}{\partial z_{0}},\ldots ,\frac{\partial f}{\partial z_{n+1}},\det \left( \frac{\partial ^{2}f}{\partial z_{i}\partial z_{j}}\right) _{0\leq i,j\leq n+1})
\]
and the Lie algebra \(L^{\ast }(V):=\operatorname{Der}(A^{\ast }(V),A^{\ast }(V)).\) They prove the following property of \(L^{\ast }(V)\) for \(1\leq n\leq 3:\) Let \((V,0)\) be an isolated weighted homogeneous hypersurface singularity (then \(f
\) is a polynomial) of a weight type \((\alpha _{0},\ldots ,\alpha _{n+1};d)\) where \(d\geq 2\alpha _{0}\geq \cdots \geq 2\alpha _{n+1}>0.\) (In this case \(A^{\ast }(V)\) and \(L^{\ast }(V)\) are graded algebra). Then in \(L^{\ast }(V)\)
there are no derivations of negative weight i.e. derivations \(D:A^{\ast}(V)\rightarrow A^{\ast }(V)\) which sends elements of degree \(i\) into elements of degree \(i-k_{0},\) where \(k_{0}\in \mathbb{N}.\)
Reviewer's remark: In the paper there is a strange statement ``\dots the second author discovered independently the following conjecture\ldots'' on line~1 on page~202.
Reviewer: Tadeusz Krasiński (Łódź)\(k\)-quasi-homogeneous Toeplitz operators on pluriharmonic Bergman space of the unit ball.https://zbmath.org/1449.470592021-01-08T12:24:00+00:00"Dai, Xin"https://zbmath.org/authors/?q=ai:dai.xin"Dong, Xingtang"https://zbmath.org/authors/?q=ai:dong.xingtang"Zhang, Yingying"https://zbmath.org/authors/?q=ai:zhang.yingyingSummary: In this paper, we study some basic properties of the \(k\)-quasi-homogeneous Toeplitz operators on pluriharmonic Bergman space \(b_\alpha^2\) of the unit ball, and obtain two symmetric properties of the commutator and semi-commutator consisting of two such operators on \(b_\alpha^2\). Additionally, we obtain the necessary and sufficient conditions for the finite rank of commutator and semi-commutator of two monomial-type Toeplitz operators on \(b_\alpha^2\).Gradient estimates for some evolution equations on complete smooth metric measure spaces.https://zbmath.org/1449.320112021-01-08T12:24:00+00:00"Nguyen Thac Dung"https://zbmath.org/authors/?q=ai:nguyen-thac-dung."Kieu Thi Thuy Linh"https://zbmath.org/authors/?q=ai:kieu-thi-thuy-linh."Ninh Van Thu"https://zbmath.org/authors/?q=ai:ninh-van-thu.Summary: In this paper, we consider the following general evolution equation \[u_t=\Delta_fu+au\log^{\alpha}u+bu\] on a smooth metric measure space \((M^n ,g,e^{-f} dv)\). We give a local gradient estimate of Souplet-Zhang type for positive smooth solutions of this equation provided that the Bakry-Émery curvature is bounded from below. When \(f\) is constant, we investigate the general evolution equation on compact Riemannian manifolds with nonconvex boundary satisfying an interior rolling \(R\)-ball condition. We show a gradient estimate of Hamilton type on such manifolds.Equivalent characterization of several quantities on holomorphic function spaces.https://zbmath.org/1449.320062021-01-08T12:24:00+00:00"Tang, Pengcheng"https://zbmath.org/authors/?q=ai:tang.pengcheng"Zhang, Xuejun"https://zbmath.org/authors/?q=ai:zhang.xuejun"Lv, Ruixin"https://zbmath.org/authors/?q=ai:lv.ruixinSummary: In this paper, the expression under the action of fractional derivative and fractional integral for a common function on the unit ball of several complex variables is improved. At the same time, the equivalent norms of the fractional differential on two holomorphic function spaces are improved, and the constrained conditions \(\beta = s + N\) for the fractional differential \({R^{s, t}}\) and \({R^{\beta, t}}\) in the equivalent norms are removed, where \(N\) is a positive integer.On some properties of relative capacity and thinness in weighted variable exponent Sobolev spaces.https://zbmath.org/1449.320122021-01-08T12:24:00+00:00"Unal, C."https://zbmath.org/authors/?q=ai:unal.cihan|unal.cemal"Aydin, I."https://zbmath.org/authors/?q=ai:aydin.ilknur|aydin.ismailLet \(p:\mathbb{R}^n\longrightarrow[1,+\infty)\) be a measurable function and let \(\vartheta:\mathbb{R}^n\longrightarrow(0,+\infty)\) be locally integrable. Denote by \(L^p_\vartheta(\mathbb{R}^n)\) the space of all measurable functions \(f\) such that \(\int_{\mathbb{R}^n}|f(x)|^{p(x)}\vartheta(x)dx<+\infty\) and let \(W^{1,p}_\vartheta(\mathbb{R}^n):=\{f\in L^p_\vartheta(\mathbb{R}^n): \partial{f}/\partial{x_j}\in L^p_\vartheta(\mathbb{R}^n),\;j=1,\dots,n\}\). The authors study the space \(W^{1,p}_\vartheta(\mathbb{R}^n)\) and various capacities associated with this space.
Reviewer: Marek Jarnicki (Kraków)