Recent zbMATH articles in MSC 46Bhttps://zbmath.org/atom/cc/46B2024-04-15T15:10:58.286558ZWerkzeugExistence in the nonlinear Schrödinger equation with bounded magnetic fieldhttps://zbmath.org/1530.352442024-04-15T15:10:58.286558Z"Schindler, Ian"https://zbmath.org/authors/?q=ai:schindler.ian"Tintarev, Cyril"https://zbmath.org/authors/?q=ai:tintarev.kyrilThis paper investigates the existence of ground states for the nonlinear Schrödinger equation with a bounded external magnetic field. The equation includes an external magnetic field represented by a real-valued covector field \(A\), and an external potential \(V\). The study focuses on the existence of solutions without requiring lattice periodicity or symmetry of the magnetic field, or the presence of an external electric field. The paper builds on previous research on the existence of solutions for the nonlinear magnetic Schrödinger equation, extending the analysis to more general cases of bounded external magnetic fields. The authors provide new results and theorems related to the existence of ground states for this equation, considering critical exponents and concentration-compactness principles. The paper is structured into sections covering preliminary concepts, profile decomposition, and critical exponent problems. The results presented in the paper contribute to the understanding of the behavior of solutions to the nonlinear Schrödinger equation in the presence of bounded external magnetic fields.
\begin{itemize}
\item [1)] The existence of ground states for the nonlinear Schrödinger equation with a general external magnetic field, without requiring lattice periodicity, symmetry of the magnetic field, or the presence of an external electric field.
\item [2)] The existence conditions for ground states are refined, using a concentration-compactness argument that overcomes the lack of compactness of Sobolev embeddings in the whole space.
\item [3)] The study introduces the concept of energy at infinity, which is evaluated via lattice shifts, and provides a refined existence condition based on comparing the magnetic field and electric potential with their respective limits at infinity.
\item [4)] The paper builds upon previous research on the existence of solutions to the nonlinear magnetic Schrödinger equation, extending the analysis to cases with bounded magnetic fields and without relying on strong electric fields dominating the entire space.
\end{itemize}
The theorems and lemmas presented in the paper.
\begin{itemize}
\item [Theorem 4.5: ] Addresses the existence of minimizers in the constraint problem under a penalty condition, providing insights into the minimum in the problem and the convergence of minimizing sequences.
\item [Theorem 4.2: ] Focuses on the existence of minimizers in a model minimization problem involving the Aharonov-Bohm magnetic potential, a singular electric potential, and critical Sobolev nonlinearity.
\item [Theorem 5.3: ] Explores the critical exponent problem, specifically addressing the minimum in the problem and the convergence of minimizing sequences under certain conditions.
\item [Lemma 4.1: ] Provides a proof of the existence of minimizers in the constraint problem, demonstrating the convergence of minimizing sequences to a minimizer
\item [Lemma 4.3: ] Offers insights into the relaxation of conditions, allowing for a broader range of physical scenarios in the analysis of the nonlinear Schrödinger equation.
\item [Lemma 5.1: ] Addresses the minimum in the critical exponent problem, providing a proof of the attainment of the minimum under specific conditions.
\end{itemize}
These theorems and lemmas collectively contribute to the understanding of the behavior of solutions to the nonlinear Schrödinger equation in the presence of bounded external magnetic fields, offering valuable insights and implications for further research in this area.
Previous research mentioned in references.
The paper mentions previous research on the existence of solutions for the nonlinear magnetic Schrödinger equation
\begin{itemize}
\item [Existence Results:] [\textit{P.-L. Lions}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109--145 (1984; Zbl 0541.49009)]: This paper is one of the earliest existing results for nonlinear magnetic Schrödinger equations. It considers the case where the magnetic field is assumed to be constant.
\item [Generalization:] [\textit{G. Arioli} and \textit{A. Szulkin}, Arch. Ration. Mech. Anal. 170, No. 4, 277--295 (2003; Zbl 1051.35082)]: This paper generalizes the existence result of Esteban and Lions to the case of periodic magnetic fields. It introduces the concept of energy-preserving operators called ''magnetic shifts'' to control the loss of compactness in problems with periodic magnetic fields.
\item [Quasiclassical Asymptotics:] The paper also mentions the study of quasiclassical asymptotics in the context of the magnetic Schrödinger equation. This line of research explores the behavior of solutions in the limit of large quantum numbers, providing insights into the semiclassical behavior of the system.
\item [Properties of Solutions:] The paper refers to studies of the properties of solutions to the magnetic Schrödinger equation.
\end{itemize}
Conclusion.
In this paper, the authors contribute to the existing literature on the nonlinear Schrödinger equation with a bounded external magnetic field by providing new results and theorems related to the existence of ground states. Specifically, the paper extends the analysis to more general cases of bounded external magnetic fields, without requiring lattice periodicity or symmetry of the magnetic field, or the presence of an external electric field. The new contributions in this paper include the development of theorems and results that address critical exponents and concentration-compactness principles in the context of the nonlinear Schrödinger equation with a bounded external magnetic field. These contributions expand the understanding of the behavior of solutions to the nonlinear Schrödinger equation in the presence of bounded external magnetic fields, providing valuable insights into this area of study.
Reviewer: Mustafa Moumni (Batna)On the convexity coefficient of Musielak-Orlicz function spaces equipped with the Orlicz normhttps://zbmath.org/1530.390222024-04-15T15:10:58.286558Z"Guo, Tianbao"https://zbmath.org/authors/?q=ai:guo.tianbao"Cui, Yunan"https://zbmath.org/authors/?q=ai:cui.yunanAuthors' abstract: In [\textit{H. Hudzik} and \textit{T. Landes}, Commentat. Math. Univ. Carol. 33, No. 4, 615--621 (1992; Zbl 0779.46029)], the convexity coefficient of Musielak-Orlicz function spaces over a non-atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak-Orlicz functions are strictly convex see [loc. cit]. In this paper, we extend this result to the case of Musielak-Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak-Orlicz function spaces as well as \(k\)-uniformly convex Musielak-Orlicz spaces equipped with the Orlicz norm is given.
Reviewer: Hark-Mahn Kim (Daejeon)On matrix-valued Gabor frames over locally compact abelian groupshttps://zbmath.org/1530.420552024-04-15T15:10:58.286558Z"Sinha, Uttam Kumar"https://zbmath.org/authors/?q=ai:sinha.uttam-kumar"Vashisht, Lalit Kumar"https://zbmath.org/authors/?q=ai:vashisht.lalit-kumar"Das, Pankaj Kumar"https://zbmath.org/authors/?q=ai:das.pankaj-kumarSummary: In this paper, we study Gabor frames in the matrix-valued signal space \(L^2 (G, \mathbb{C}^{n\times n})\), where \(G\) is a locally compact abelian group which is metrizable and \(\sigma\)-compact, and \(n\) is a positive integer. First, we give sufficient conditions on scalars in an infinite combination of vectors (from a given matrix-valued Gabor frame) to constitute a new frame for the space \(L^2 (G, \mathbb{C}^{n\times n})\). This generalizes a result due to \textit{A. Aldroubi} [Proc. Am. Math. Soc. 123, No. 6, 1661--1668 (1995; Zbl 0851.42030)]. Second, we discuss frame conditions for finite sums of matrix-valued Gabor frames. Sufficient conditions for finite sums of matrix-valued Gabor frames in terms of frame bounds are established. It is shown that the sum of images of matrix-valued Gabor frames under bounded linear operators acting on \(L^2 (G, \mathbb{C}^{n\times n})\) constitute a frame for the space \(L^2 (G, \mathbb{C}^{n\times n})\) provided operators are adjointable with respect to the matrix-valued inner product and satisfy a majorization. Finally, we show that matrix-valued Gabor frames are stable under small perturbations.Characterization of \(p\)-Banach spaces based on a reverse triangle inequalityhttps://zbmath.org/1530.460022024-04-15T15:10:58.286558Z"Dadipour, Farzad"https://zbmath.org/authors/?q=ai:dadipour.farzad"Rezaei, Asiyeh"https://zbmath.org/authors/?q=ai:rezaei.asiyehSummary: In this paper, we deal with the reverse of the generalized triangle inequality of the second type in quasi-Banach spaces. More exactly, by using the concept of equivalent \(p\)-norms, we provide some necessary and sufficient conditions for \(n\)-tuples to satisfy the mentioned inequality. As applications, we improve some already known results and present some characterizations of \(p\)-Banach spaces among quasi-Banach spaces. In particular, we prove that a quasi-Banach space such as \(X\) is a \(p\)-Banach space if and only if for all \((\mu_1, \dots, \mu_n) \in \mathbb{R}^n\) satisfying \(\mu_j > 0\) for some \(j\) and \(\mu_i < 0\) for all \(i \neq j\), the generalized triangle inequality of the second type \(\sum_{i=1}^n \frac{\| x_i \|^p}{\mu_i} \leq \| \sum_{i=1}^n x_i \|^p\) \((x_i \in X)\) holds only with the assumption \(\mu_j \geq \max_{i \in \{1, \dots, n\} {\setminus} \{j\}} \{1, | \mu_i | \}\).A topological generalization of orthogonality in Banach spaces and some applicationshttps://zbmath.org/1530.460032024-04-15T15:10:58.286558Z"Sain, Debmalya"https://zbmath.org/authors/?q=ai:sain.debmalya"Roy, Saikat"https://zbmath.org/authors/?q=ai:roy.saikat"Paul, Kallol"https://zbmath.org/authors/?q=ai:paul.kallolSummary: We introduce a topological notion of orthogonality in a vector space. We show that for a suitable choice of orthogonality space, Birkhoff-James orthogonality in a Banach space is a particular case of the orthogonality introduced by us. We characterize the right additivity of orthogonality in our setting and obtain a necessary and sufficient condition for a Banach space to be smooth, as a corollary to our characterization. Finally, using our notion of orthogonality, we obtain a topological generalization of the Bhatia-Šemrl Theorem.Duals of Cesàro sequence vector lattices, Cesàro sums of Banach lattices, and their finite elementshttps://zbmath.org/1530.460062024-04-15T15:10:58.286558Z"Gönüllü, Uğur"https://zbmath.org/authors/?q=ai:gonullu.ugur"Polat, Faruk"https://zbmath.org/authors/?q=ai:polat.faruk"Weber, Martin R."https://zbmath.org/authors/?q=ai:weber.martin-rThe authors study the ideals of finite elements in special vector lattices of real sequences. In particular, in the first part of the paper, they study the ideals of finite elements in the dual spaces \(d_p\) of the Cesàro sequence spaces \(\mathrm{ces}_p\), for \(p\in \{0\}\cup (1,\infty)\). In the second part of the paper, the authors introduce the so-called Cesàro sum for a sequence of Banach spaces, study the dual space, and characterize the ideals of finite elements if the summed up spaces are Banach lattices.
Reviewer: Angela Albanese (Lecce)Renormings preserving local geometry at countably many points in spheres of Banach spaces and applicationshttps://zbmath.org/1530.460082024-04-15T15:10:58.286558Z"Quilis, Andrés"https://zbmath.org/authors/?q=ai:quilis.andresRenorming theory investigates whether equivalent norms with certain properties of convexity or smoothness (or both) can be constructed on a given Banach space \(X\). It is well known that the existence of specific norms on \(X\) are tighly related with the isomorphic properties of \(X\), and in fact several natural isomorphism classes can be characterized by the existence of specific norms.
The present article answers two renorming problems which were stated in the 2022 monograph by \textit{A. J. Guirao} et al. [Renormings in Banach spaces. A toolbox. Cham: Birkhäuser (2022; Zbl 1508.46001)]. More precisely, the author shows that if \(X\) is a separable Banach space with an equivalent \(\mathcal{C}^k\)-smooth norm, then \(X\) has a \(\mathcal{C}^k\)-smooth norm which is not uniformly Gateaux smooth in any direction, with a dentable unit ball. The proof of this result uses in particular a theorem by
\textit{P.~Hájek} and \textit{J.~Talponen} [Q. J. Math. 65, No.~3, 957--969 (2014; Zbl 1315.46009)]
which states that if a separable space has a \(\mathcal{C}^k\)-smooth norm, any equivalent norm can be uniformly approximated on bounded sets by \(\mathcal{C}^k\)-smooth norms. It is also shown that for any set \(\Gamma\), the space \(c_0(\Gamma)\) has an equivalent \(\mathcal{C}^\infty\)-smooth norm which locally depends on finitely many coordinates, with a dentable unit ball, which is not uniformly Gateaux smooth in any direction.
Reviewer: Gilles Godefroy (Paris)Projections in Lipschitz-free spaces induced by group actionshttps://zbmath.org/1530.460092024-04-15T15:10:58.286558Z"Cúth, Marek"https://zbmath.org/authors/?q=ai:cuth.marek"Doucha, Michal"https://zbmath.org/authors/?q=ai:doucha.michalGiven a metric space \(\mathcal M\) and a group \(G\) acting by isometries on \(\mathcal M\), one can consider the space \(\mathcal M/G\) of the closures of the orbits endowed with the Hausdorff metric. The authors show that the space of Lipschitz functions \(\operatorname{Lip}_0(\mathcal M/G)\) is isometric to the subspace of \(\operatorname{Lip}_0(\mathcal M)\) made up of the \(G\)-invariant functions, and that the Lipschitz-free space \(\mathcal F(\mathcal M/G)\) is isometric to a quotient of \(\mathcal F(\mathcal M)\). The main results of the paper provide conditions on \(\mathcal M\) and \(G\) ensuring that \(\operatorname{Lip}_0(\mathcal M/G)\) is complemented in \(\operatorname{Lip}_0(\mathcal M)\) or that \(\mathcal F(\mathcal M/G)\) is complemented in \(\mathcal F(\mathcal M)\). For instance,
\begin{itemize}
\item if \(G\) is compact, then \(\mathcal F(\mathcal M/G)\) is complemented in \(\mathcal F(\mathcal M)\);
\item if \(G\) is abelian or locally compact and amenable, and the orbits are bounded, then \(\operatorname{Lip}_0(\mathcal M/G)\) is complemented in \(\operatorname{Lip}_0(\mathcal M)\).
\end{itemize}
The case in which \(\mathcal F(\mathcal M)\) is a dual space is also analysed.
The results are interesting for the study of properties preserved by complemented subspaces, like the bounded approximation property. The authors address these applications in the last section, posing also several open questions.
It is remarkable that the results are proven from an abstract study of the projection associated with the action of an amenable group on a Banach space that might be useful for other settings.
Reviewer: Luis C. García Lirola (Zaragoza)Phase-isometries on the unit sphere of CL-spaceshttps://zbmath.org/1530.460102024-04-15T15:10:58.286558Z"Tan, Dongni"https://zbmath.org/authors/?q=ai:tan.dongni"Zhang, Fan"https://zbmath.org/authors/?q=ai:zhang.fan.12|zhang.fan.1|zhang.fan"Huang, Xujian"https://zbmath.org/authors/?q=ai:huang.xujianSummary: A mapping \(f:S_X\to S_Y\) between the unit spheres of two real Banach spaces \(X\) and \(Y\) is said to be a phase-isometry if it satisfies \(\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}\) for all \(x,y\in S_X\). When \(f\) is surjective, \(X\) is a real CL-space and \(Y\) is an arbitrary real Banach space, we establish in this paper that there exists a phase-function \(\varepsilon:S_X\to\{-1,1\}\) such that \(\varepsilon\cdot f\) is an isometry which is the restriction of a linear isometry from \(X\) to \(Y\).On some properties of weighted Hilbert spaceshttps://zbmath.org/1530.460112024-04-15T15:10:58.286558Z"Branishti, Vladislav V."https://zbmath.org/authors/?q=ai:branishti.vladislav-vSummary: We describe the weighted Hilbert spaces \(L_{2,w}(\Omega)\) with positive weight functions \(w(x)\) which are summable on every bounded interval. We give sufficient condition for \(L_{2,w_1}(\Omega)\) space to be extension of \(L_{2,w_2}(\Omega)\) space. We also describe how to use given result in statistical probability density estimation.Isolated vertices and diameter of the \textit{BJ}-orthograph in \(C^\ast\)-algebrashttps://zbmath.org/1530.460122024-04-15T15:10:58.286558Z"Kečkić, Dragoljub J."https://zbmath.org/authors/?q=ai:keckic.dragoljub-j"Stefanović, Srdjan"https://zbmath.org/authors/?q=ai:stefanovic.srdjanSummary: We give necessary and sufficient condition that an element of an arbitrary \(C^\ast\)-algebra is an isolated vertex of the orthograph related to the mutual strong Birkhoff-James orthogonality. Also, we prove that for all \(C^\ast\)-algebras except \(\mathbb{C}\), \(\mathbb{C} \oplus \mathbb{C}\) and \(M_2 (\mathbb{C})\) all non isolated points make a single connected component of the orthograph which diameter is less than or equal to 4, i.e. any two non isolated points can be connected by a path with at most 4 edges. Some related results are given.A lower bound for the constant \(A_1(X)\) in normed linear spaceshttps://zbmath.org/1530.460132024-04-15T15:10:58.286558Z"Mizuguchi, Hiroyasu"https://zbmath.org/authors/?q=ai:mizuguchi.hiroyasuSummary: To describe the geometry of normed spaces, many geometric constants have been investigated. Among them, there are two geometric constants related to Minkowski ellipses. These constants measure how large the sum of the distances from a point of the unit sphere to two antipodal points can be. In this paper we investigate the lower bound for one of them. We also treat isosceles and Birkhoff orthogonalities. The usual orthogonality in inner product spaces and isosceles orthogonality in normed spaces are symmetric. However, Birkhoff orthogonality is not symmetric in general normed spaces. A two-dimensional normed plane in which Birkhoff orthogonality is symmetric is called Radon plane. The upper and lower bound for that two constants have been studied in general spaces, and in Radon planes. An inequality had shown in Radon planes is proved in general normed spaces.How the distance between subspaces in the metric of a spherical opening affects the geometric structure of a symmetric spacehttps://zbmath.org/1530.460142024-04-15T15:10:58.286558Z"Strakhov, Stepan Igorevich"https://zbmath.org/authors/?q=ai:strakhov.stepan-igorevichSummary: A relationship is found between the metric of a spherical opening on the space of all subspaces of a symmetric space and some numerical characteristic of the subspace. It is known that, for example, in \(L_1\) this characteristic takes only two values (i.e. this is a binary space), while in \(L_2\) there are infinitely many values. Using the connection found, the necessary conditions for the binarity of a symmetric space were generalized.Revisiting Mazur separable quotient problem (1932)https://zbmath.org/1530.460152024-04-15T15:10:58.286558Z"López-Pellicer, Manuel"https://zbmath.org/authors/?q=ai:lopez-pellicer.manuel"López-Alfonso, Salvador"https://zbmath.org/authors/?q=ai:lopez-alfonso.salvador"Moll-López, Santiago"https://zbmath.org/authors/?q=ai:moll-lopez.santiagoSummary: The centenary of the still open problem of the existence of a separable quotient of infinite dimension in a Banach space \(X\) will be in 2032. This note aims to give a quick overview of this problem, where it is known that it has a positive solution for Banach spaces with, rough speaking, ``large density character'' and also with ``small density character''. Proofs of several results have been reproduced or simplified to motivate interest in solving this so-called ``Mazur separable quotient problem''.Approximation properties in Lipschitz-free spaces over groupshttps://zbmath.org/1530.460162024-04-15T15:10:58.286558Z"Doucha, Michal"https://zbmath.org/authors/?q=ai:doucha.michal"Kaufmann, Pedro L."https://zbmath.org/authors/?q=ai:kaufmann.pedro-levitThis interesting article investigates the validity of the metric approximation property (in the sense of Grothendieck) in Lipschitz-free spaces on metric groups equipped with an invariant metric. We recall that the Lipschitz-free space \(\mathcal{F}(M)\) over a pointed metric space \(M\) is the isometric predual of the space of Lipschitz functions from \(M\) to the real line. The main result of this work is Theorem~3.5, which asserts that if a metric compact group \(G\) is equipped with an arbitrary left-invariant metric, then the corresponding free space \(\mathcal{F}(G)\) has the M.A.P. The proof relies on a penetrating analysis of the special case of compact Lie groups, where harmonic analysis is of course employed, and then on classical unitary representations of compact groups.
A similar result is shown for \(G\)-homogeneous metric spaces equipped with a quotient invariant metric. Moreover, the case of discrete finitely generated groups is investigated, and it is shown that the corresponding free space has a Schauder basis when the group is ``shortlex combable''. This applies in particular to the free groups \(F_n\) and to hyperbolic groups. If \(\mathcal{N}\) is a net in the real hyperbolic $n$-space \(\mathbb H_n\), then \(\mathcal{F}(\mathcal{N})\) has a Schauder basis as well. A list of problems concludes the article, which should motivate researchers in this very active and rapidly expanding field.
Reviewer: Gilles Godefroy (Paris)The generalized Banach sequence spaces and their dual spaces based on permutation symmetric gauge normshttps://zbmath.org/1530.460172024-04-15T15:10:58.286558Z"Chen, Yanni"https://zbmath.org/authors/?q=ai:chen.yanni"Fu, Tiantian"https://zbmath.org/authors/?q=ai:fu.tiantian"Zhang, Ye"https://zbmath.org/authors/?q=ai:zhang.yeSummary: In the present paper, we introduce and study a class \(\mathcal{N}\) of norms \(\alpha\) on the sequence space \(\mathbb{C}_{00}\), called permutation symmetric gauge norms, which properly contains the classical class of \(\{\| \cdot \|_p : 1 \leq p \leq \infty\}\). For each \(\alpha \in \mathcal{N}\), we define the generalized sequence space \(\ell^\alpha\), which is proved to be a Banach space, and then we obtain a characterization of \(\ell^\alpha\) in terms of the projection onto \(\mathbb{C}_{00}\). For the duality, the expected results in the classical sequence spaces \(\ell^p\) \((1 \leq p \leq \infty)\) are still valid in these new settings, including the characterization of dual space and the existence of Hölder's inequality. It is worthy pointing out that the generalized sequence space \(\ell^\alpha\) is not a reflexive space, which is different from the usual sequence spaces.Compact and weakly compact Lipschitz operatorshttps://zbmath.org/1530.460182024-04-15T15:10:58.286558Z"Abbar, Arafat"https://zbmath.org/authors/?q=ai:abbar.arafat"Coine, Clément"https://zbmath.org/authors/?q=ai:coine.clement"Petitjean, Colin"https://zbmath.org/authors/?q=ai:petitjean.colinGiven a Lipschitz map \(f:M\to N\) between metric spaces, there is a unique operator \(\hat{f}:\mathcal{F}(M)\to \mathcal{F}(N)\) between the corresponding Lipschitz-free spaces extending \(f\). Its adjoint is the operator \(C_f:g\in \operatorname{Lip}_0(N)\mapsto g\circ f\in \operatorname{Lip}_0(M)\).
The most important contribution of this paper is that the authors characterize when \(\hat{f}\) is compact operator in terms of metric conditions on \(f\) (see Theorem A) and they prove that \(\hat{f}\) is compact if and only if it is weakly compact (see Theorem B).
Reviewer: Marek Cúth (Praha)Geometries of topological groupshttps://zbmath.org/1530.460192024-04-15T15:10:58.286558Z"Rosendal, Christian"https://zbmath.org/authors/?q=ai:rosendal.christianThe author gives an impressive survey on the geometry of topological groups. Many motivating and illustrating examples are given and a couple of open questions complete the presentation.
The author starts with considering separable Banach spaces in the category of metric spaces with linear isometries as homomorphisms, then he applies the forgetful functor and considers not necessarily surjective isometries. In a next step, Lipschitz functions are considered. A function is Lipschitz if and only if it is Lipschitz for large and for short distances. While Lipschitz for short distances implies uniform continuity, this yields a functor from the class of separable Banach spaces with Lipschitz for short distances mappings as homomorphisms to the same object, but with uniformly continuous mappings as homomorphisms. In the same manner, a Lipschitz for large distances mapping is bornologous which provides another forgetful functor. The author gives an overview of which of the above mentioned functors admit an inverse, respectively a reconstruction. For example, by the Corson-Klee lemma, every uniformly continuous map and every bornologous mapping is Lipschitz for large distances.
In the second part of the paper, the author turns to categories in which the objects are topological groups.
A left invariant compatible metric \(d\) on \(G\) is called minimal if for any other left invariant compatible metric \(\partial\), the identity \((G,\partial) \to (G,d)\) is Lipschitz for short distances.
He proves that if \((G,d)\) is a completely metrizable connected abelian group with ample square roots (i.e., for every neighbourhood \(W\) of \(e\) the closure of the set \(\{g^2: g\in W\}\) is a neighborhood of \(e\)) and \(d\) is a minimal metric, then the exponential function from the Banach Lie-algebra to \(G\) is a projection with discrete kernel.
Then he turns to bounded subsets in coarse structures and compares it in the case of topological vector spaces with the usual (Kolmogorov) boundedness.
A Polish group \(G\) is said to have a bounded geometry if \(G\) is coarsely equivalent to a proper metric space. A characterization applying group actions is given, when two groups of bounded geometry are coarsely equivalent.
Reviewer: Lydia Außenhofer (Passau)Spaceablity in weak Morrey spaceshttps://zbmath.org/1530.460202024-04-15T15:10:58.286558Z"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiro"Tabatabaie, Seyyed Mohammad"https://zbmath.org/authors/?q=ai:tabatabaie.seyyed-mohammadSummary: In this paper we prove that \(\mathrm{w}\mathcal{M}^p_q(\mathbb{R}^n)\setminus\mathcal{M}^p_q(\mathbb{R}^n)\) is spaceable in the weak Morrey space \(\mathrm{w}\mathcal{M}^p_q(\mathbb{R}^n)\), where \(1 \leq q \leq p< \infty\).Unconditional basis constructed from parameterised Szegö kernels in analytic \(\mathbb{H}^p(D)\)https://zbmath.org/1530.460222024-04-15T15:10:58.286558Z"Hon, Chitin"https://zbmath.org/authors/?q=ai:hon.chitin"Leong, Ieng Tak"https://zbmath.org/authors/?q=ai:leong.iengtak|leong.ieng-tak"Qian, Tao"https://zbmath.org/authors/?q=ai:qian.tao"Yang, Haibo"https://zbmath.org/authors/?q=ai:yang.haibo"Zou, Bin"https://zbmath.org/authors/?q=ai:zou.binSummary: Rational orthogonal systems in approximating analytic functions have attracted considerable interest. Among which adaptive Fourier decomposition, abbreviated as AFD, was recently established. An AFD is a sparse representation using a Takenaka-Malmquist (TM) system whose parameters are optimally selected according to the given signal. TM systems have been proved to be Schauder systems in the corresponding Banach spaces \(\mathbb{H}^p\), \(1 < p < \infty\). In the present paper, from the methodology point of view we give an alternative definition of the Hardy spaces by using the periodic Lusin area function. We extend the Botchkariev-Meyer-Wojtaszcyk Theorem to rational function systems. By using Meyer's bimodal wavelet and the Fefferman-Stein vector valued maximum operator we prove that under certain conditions the rational systems become unconditional bases in the Banach space \(\mathbb{H}^p(D)\), \(1 < p < \infty\).Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaceshttps://zbmath.org/1530.460262024-04-15T15:10:58.286558Z"Dao, Nguyen Anh"https://zbmath.org/authors/?q=ai:dao.nguyen-anhSummary: Our main purpose is to establish Gagliardo-Nirenberg-type inequalities using fractional homogeneous Sobolev spaces and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in previous studies.The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson's inequalitieshttps://zbmath.org/1530.460352024-04-15T15:10:58.286558Z"Sodini, Giacomo Enrico"https://zbmath.org/authors/?q=ai:sodini.giacomo-enricoSummary: We show that the algebra of cylinder functions in the Wasserstein Sobolev space \(H^{1, q}(\mathcal{P}_p(X,\mathsf{d}), W_{p, \mathsf{d}}, \mathfrak{m})\) generated by a finite and positive Borel measure \(\mathfrak{m}\) on the \((p, \mathsf{d})\)-Wasserstein space \((\mathcal{P}_p(X, \mathsf{d}), W_{p, \mathsf{d}})\) on a complete and separable metric space \((X, \mathsf{d})\) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space \(\mathbb{B}\), then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if \(\mathbb{B}\) is reflexive (resp. if the dual of \(\mathbb{B}\) is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space.Nuclear pseudo-differential operators in Besov spaces on compact Lie groupshttps://zbmath.org/1530.470252024-04-15T15:10:58.286558Z"Cardona, Duván"https://zbmath.org/authors/?q=ai:cardona.duvanSummary: In this work, we establish the metric approximation property for Besov spaces defined on arbitrary compact Lie groups. As a consequence of this fact, we investigate trace formulae for nuclear Fourier multipliers on Besov spaces. Finally, we study the \(r\)-nuclearity, the Grothendieck-Lidskii formula and the (nuclear) trace of pseudo-differential operators in generalized Hörmander classes acting on periodic Besov spaces. We will restrict our attention to pseudo-differential operators with symbols of limited regularity.Traces on operator ideals defined over the class of all Banach spaces and related open problemshttps://zbmath.org/1530.470262024-04-15T15:10:58.286558Z"Pietsch, Albrecht"https://zbmath.org/authors/?q=ai:pietsch.albrechtThe author is one of the main contributors to the classical theory of traces of operators on Banach spaces, and his monograph [\textit{A. Pietsch}, Eigenvalues and $s$-numbers. Cambridge University Press (1987; Zbl 0615.47019)] has become the main classical reference on the topic. This survey article addresses the extension to singular traces, giving a complete picture of the current situation of the research on this area, and a perspective of possible future developments.
The \(k\)-th ordering number of a bounded sequence \(a = (a_{h})\) is defined as \(o_{k}(a) = \sup_{h \geq k} \vert a_{h} \vert\). A~quasi-Banach shift-monotone sequence ideal is a linear subspace \(\mathfrak{z}\) of \(\ell_{\infty}\), endowed with a quari-norm \(\Vert \cdot \vert_{\mathfrak{z}} \Vert\) that is invariant under the forward shift \(S_{+}\) and satisfying that, if \(a \in \ell_{\infty}\), \(b \in \mathfrak{z}\) and \(o_{k}(a) \leq o_{k}(b)\), then \(a \in \mathfrak{z}\) (it is \(o_{k}-solid\)) and \(\Vert a \vert_{\mathfrak{z}}\Vert \leq \Vert b \vert_{\mathfrak{z}}\Vert\). It is assumed that \(\Vert e_{0} \vert_{\mathfrak{z}}\Vert =1\). For an operator \(S\) between Banach spaces, \(a_{k}(S)\) denotes the \(k\)-th approximation number. A~quasi-Banach shift-monotone sequence ideal \(\mathfrak{z}\) defines an operator ideal, denoted \(\mathfrak{L}^{\mathrm{APP}}_{\mathfrak{z}}\), consisting of those operators \(S\) for which \(\big(a_{k}(S) \big) \in \mathfrak{z}\).
The first section of the paper studies the structure of these ideals. A~\(\mathfrak{z}\)-representation of an operator \(S\) is of the form \(S=\sum_{h=0}^{\infty} S_{h}\), where each \(S_{h}\) has rank \(\leq 2^{h}\) and \(\big( S - \sum_{h=0}^{k-1} S_{h} \big) \in \mathfrak{z}\). Then the ideal \(\mathfrak{L}^{\mathrm{APP}}_{\mathfrak{z}}\) just consists of operators that are \(\mathfrak{z}\)-representable. Moreover, the quais-norm induced by \(\mathfrak{z}\) is equivalent to the norm given by \(\inf \Big\Vert \big( S - \sum_{h=0}^{k-1} S_{h} \big) \big\vert_{\mathfrak{z}} \big\Vert\). Conversely, each quasi-Banach operator ideal \(\mathfrak{A}(\ell_{2})\) defines a quasi-Banach shift-monotone sequence ideal by \(\mathfrak{A}_{\mathfrak{z}} = \{ a \in \ell_{\infty} : D_{a} \in \mathfrak{A}(\ell_{2}) \}\) (where \(D_{a}\) stands for the diagonal operator defined by \(a\)). In this way there is a one-to-one correspondence between operator ideals and shift-monotone sequence ideals. This correspondence is carefully analysed. It is shown that an operator ideal coincides with some \(\mathfrak{L}^{\mathrm{APP}}_{\mathfrak{z}}\) if and only if it is \((a_{2^{k}})\)-solid. Also, every \((a_{2^{k}})\)-solid operator ideal is uniquely determined by its Hilbert space component \(\mathfrak{A}(\ell_{2})\). Considering \(\mathfrak{A}\)-approximation numbers, \(a_{n}(S \vert \mathfrak{A})\) and a quasi-Banach shift-monotone sequence ideal \(\mathfrak{z}\), a new operator ideal (denoted \(\mathfrak{A}^{\mathrm{APP}}_{\mathfrak{z}}\)) is defined by considering those \(S \in \mathfrak{A}\) for which \(\big( a_{2^n}(S \vert \mathfrak{A}) \big) \in \mathfrak{z}\).
Traces defined on such ideals are studied. For every \(\frac{1}{2}S_{+}\)-invariant linear form \(\varphi\) on \(\mathfrak{z}\) the value \(\varphi_{\mathfrak{z}}(S) = \varphi \big(\frac{1}{2^{k}} \mathrm{trace} (S_{k}) \big)\) does not depend on the choice of the \(\mathfrak{z}\)-representation \(\sum_{h=0}^{\infty} S_{h}\), and defines a trace \(\varphi_{s}\) on \(\mathfrak{L}^{\mathrm{APP}}_{\mathfrak{z}}\). Moreover, \(\varphi_{s}\) is bounded whenever \(\varphi\) is so. The notion of exact trace type \(1/r\) operator ideal is introduced, and the previous result is extended to \(\frac{1}{2^{1/r}}S_{+}\)-invariant linear forms on exact trace \(1/r\) operator ideals and \(\mathfrak{z}\) the ideal of sequences so that \(\vert a_{h} \vert = O \big( \frac{1}{2^{h/r}} \big)\). The fist section of the paper is concluded by showing that \(\mathfrak{A}^{\mathrm{APP}}_{\mathfrak{z}}\) is a quasi-Banach operator ideal whenever \(\mathfrak{A}\) and \(\mathfrak{z}\) are quasi-Banach ideals.
The second section is dedicated to suggest possible further developments on the area, by collecting up to 27 open problems. These are grouped in different topics: spectrum of shift operators, traces and traceless operator ideals, spectral traces, the ideal of \((r,2)\)-summing operators, operator ideals generated by Weyl or entropy numbers, Banach spaces with few operators, strange traces or extending traces.
Reviewer: Pablo Sevilla Peris (València)The fixed point property for nonexpansive type mappings in nonreflexive Banach spaceshttps://zbmath.org/1530.470622024-04-15T15:10:58.286558Z"Shukla, Rahul"https://zbmath.org/authors/?q=ai:shukla.rahulSummary: In this paper, we present the fixed point property for nonexpansive type mappings in Banach spaces endowed with near-infinity concentrated norms. We also obtain a stability result. Finally, we present a nontrivial example to show the usefulness of these results.A remark on geodesics in the Banach-Mazur distancehttps://zbmath.org/1530.530512024-04-15T15:10:58.286558Z"Arias, Alvaro"https://zbmath.org/authors/?q=ai:arias.alvaro"Kovalchuk, Vladimir"https://zbmath.org/authors/?q=ai:kovalchuk.vladimirLet \((M,\rho)\) be a metric space. A \textit{geodesic} between \(x\in M\) and \(y\in M\) is a path \(\gamma\) (i.e., a continuous function \(\gamma\colon [a,b]\to (M,\rho)\)) with length equal to \(\rho(x,y),\) that starts at \(x\) and ends at \(y\). The aim of the paper under review is to show, for every \(n\geq 2\), that there are uncountably many different geodesics between any two elements of \(BM_n\), where \(BM_n\) is the set of isometry classes of \(n\)-dimensional normed spaces, endowed with the logarithm of the Banach-Mazur distance. Recall that for two \(n\)-dimensional normed spaces \(E,F\) the Banach-Mazur distance is defined by
\[
d(E,F)=\inf\{\|T\|\|T^{-1}\|\colon T\colon E\to F\text{ is a isomorphism}\}.
\]
The authors provide two different proofs, one for the two-dimensional case (Theorem 3) and the other for \(n\geq 3\) (Theorem 2).
Reviewer: Jacopo Somaglia (Milano)Fixed point theorems with applicationshttps://zbmath.org/1530.540012024-04-15T15:10:58.286558Z"Mebarki, Karima"https://zbmath.org/authors/?q=ai:mebarki.karima"Georgiev, Svetlin G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Djebali, Smail"https://zbmath.org/authors/?q=ai:djebali.smail"Zennir, Khaled"https://zbmath.org/authors/?q=ai:zennir.khaledThe present book consists of the following chapters: 1. Preliminaries. This is a course on operators in Banach spaces with particular emphasis on non-compactness measures and selected compactness criteria. 2. Fixed-point index for sums of two operators. In this chapter, the definition of a generalized fixed point index as well as some of its properties are presented. 3. Positive fixed points for sums of two operators. Some versions of Krasnoselskii's compression/expansion fixed point theorem and Leggett-Williams fixed point theorem are presented. 4. Applications to ODEs. 5. Applications to parabolic equations. 6. Applications to hyperbolic equations. The last three chapters are exclusively concerned with some applications of the theory developed in Chapter 2, 3 to some ordinary and partial differential equations. Studying this book requires some knowledge of mathematics, particularly a basic course in functional analysis and differential equations. The book can be used in graduate or postgraduate programs where knowledge of mathematics describing nonlinear phenomena is required.
Reviewer: Jarosław Górnicki (Rzeszów)Half-space depth of log-concave probability measureshttps://zbmath.org/1530.600112024-04-15T15:10:58.286558Z"Brazitikos, Silouanos"https://zbmath.org/authors/?q=ai:brazitikos.silouanos"Giannopoulos, Apostolos"https://zbmath.org/authors/?q=ai:giannopoulos.apostolos-a"Pafis, Minas"https://zbmath.org/authors/?q=ai:pafis.minasSummary: Given a probability measure \(\mu\) on \({{\mathbb{R}}}^n\), Tukey's half-space depth is defined for any \(x\in{{\mathbb{R}}}^n\) by \(\varphi_{\mu}(x)=\inf \{\mu(H):H\in{{{\mathcal{H}}}}(x)\} \), where \(\mathcal{H}(x)\) is the set of all half-spaces \(H\) of \({{\mathbb{R}}}^n\) containing \(x\). We show that if \(\mu\) is a non-degenerate log-concave probability measure on \({{\mathbb{R}}}^n\) then
\[
e^{-c_1n}\leqslant \int_{{\mathbb{R}}^n}\varphi_{\mu}(x)\,d\mu (x) \leqslant e^{-c_2n/L_{\mu}^2}
\]
where \(L_{\mu}\) is the isotropic constant of \(\mu\) and \(c_1,c_2>0\) are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of \(L_q\)-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.