Recent zbMATH articles in MSC 43https://zbmath.org/atom/cc/432023-12-07T16:00:11.105023ZWerkzeugBook review of: K. Juschenko, Amenability of discrete groups by exampleshttps://zbmath.org/1522.000282023-12-07T16:00:11.105023Z"Bon, N. Matte"https://zbmath.org/authors/?q=ai:bon.n-matteReview of [Zbl 1500.43001].Robert Israel ``Bob'' Jewett (1937--2022)https://zbmath.org/1522.010542023-12-07T16:00:11.105023Z"Bloom, Walter R."https://zbmath.org/authors/?q=ai:bloom.walter-r"Gardner, Richard J."https://zbmath.org/authors/?q=ai:gardner.richard-j"Hales, Al"https://zbmath.org/authors/?q=ai:hales.al"Spencer, Joel"https://zbmath.org/authors/?q=ai:spencer.joel-h"Tao, Terence"https://zbmath.org/authors/?q=ai:tao.terence-c"Weiss, Benjamin"https://zbmath.org/authors/?q=ai:weiss.benjamin.1|weiss.benjamin|weiss.benjamin-l(no abstract)Gelfand pairs associated with the action of graph automaton groupshttps://zbmath.org/1522.200992023-12-07T16:00:11.105023Z"Cavaleri, Matteo"https://zbmath.org/authors/?q=ai:cavaleri.matteo"D'Angeli, Daniele"https://zbmath.org/authors/?q=ai:dangeli.daniele"Donno, Alfredo"https://zbmath.org/authors/?q=ai:donno.alfredoIn this paper, the authors pursue a construction they introduced in [Adv. Group Theory Appl. 11, 75--112 (2021; Zbl 1495.20038)]. This construction begins with a finite graph and results in an invertible automaton. The group generated by the transformations associated with the states of the automaton is an automorphism group of the rooted regular tree whose valency is the order of the input graph. Properties of these particular automata were studied in more detail in the original paper.
In this paper, the authors show that the group \(G_n\) restriction of the action of this group to the \(n\)th level of the rooted tree, together with the stabiliser \(K_n\) in \(G_n\) of a fixed vertex in that level, are a symmetric ``Gelfand pair''. That is, the algebra of bi-\(K_n\)-invariant functions on \(G_n\) is commutative under convolution. They explicitly determine the associated spherical functions.
Reviewer: Joy Morris (Lethbridge)Hypergeometric functions of type \(BC\) and standard multiplicitieshttps://zbmath.org/1522.330092023-12-07T16:00:11.105023Z"Narayanan, Eravimangalam Krishnan"https://zbmath.org/authors/?q=ai:narayanan.eravimangalam-krishnan"Pasquale, Angela"https://zbmath.org/authors/?q=ai:pasquale.angelaSummary: We study the Heckman-Opdam hypergeometric functions associated with a root system of type \(BC\) and a multiplicity function that is allowed to assume some nonpositive values (a standard multiplicity function). For such functions, we obtain positivity properties and sharp estimates that imply a characterization of the bounded hypergeometric functions. As an application, our results extend the known properties of Harish-Chandra's spherical functions on Riemannian symmetric spaces of the non-compact type \(G/K\) to spherical functions over homogeneous vector bundles on \(G/K\), which are associated with certain small \(K\)-types.Bohr sets in sumsets II: countable abelian groupshttps://zbmath.org/1522.370072023-12-07T16:00:11.105023Z"Griesmer, John T."https://zbmath.org/authors/?q=ai:griesmer.john-t"Le, Anh N."https://zbmath.org/authors/?q=ai:le.anh-ngoc"Lê, Thái Hoàng"https://zbmath.org/authors/?q=ai:le.thai-hoangSummary: We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting \(G\) be a countable discrete abelian group and \(\phi_1, \phi_2, \phi_3: G \to G\) be commuting endomorphisms whose images have finite indices, we show that \begin{itemize} \item[(1)] If \(A \subset G\) has positive upper Banach density and \(\phi_1 + \phi_2 + \phi_3 = 0\), then \(\phi_1(A) + \phi_2(A) + \phi_3(A)\) contains a Bohr set. This generalizes a theorem of
\textit{V. Bergelson} and \textit{I. Z. Ruzsa} [Isr. J. Math. 174, 1--18 (2009; Zbl 1250.11009)]
in \(\mathbb{Z}\) and a recent result of the first author. \item[(2)] For any partition \(G = \bigcup_{i=1}^r A_i\), there exists an \(i \in \{1, \ldots, r\}\) such that \(\phi_1(A_i) + \phi_2(A_i) - \phi_2(A_i)\) contains a Bohr set. This generalizes a result of the second and third authors from \(\mathbb{Z}\) to countable abelian groups. \item[(3)] If \(B, C \subset G\) have positive upper Banach density and \(G = \bigcup_{i=1}^r A_i\) is a partition, \(B + C + A_i\) contains a Bohr set for some \(i \in \{1, \ldots, r\}\). This is a strengthening of a theorem of
\textit{V. Bergelson} et al. [Algorithms Comb. 26, 13--37 (2006; Zbl 1114.37008)]. \end{itemize} All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices \([G:\phi_j(G)]\), the upper Banach density of \(A\) (in (1)), or the number of sets in the given partition (in (2) and (3)).Norms of certain functions of a distinguished Laplacian on the \(ax+b\) groupshttps://zbmath.org/1522.430012023-12-07T16:00:11.105023Z"Akylzhanov, Rauan"https://zbmath.org/authors/?q=ai:akylzhanov.r-kh"Kuznetsova, Yulia"https://zbmath.org/authors/?q=ai:kuznetsova.yulia-n"Ruzhansky, Michael"https://zbmath.org/authors/?q=ai:ruzhansky.michael-v"Zhang, Haonan"https://zbmath.org/authors/?q=ai:zhang.haonanSummary: The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator \({\mathcal{L}}\) on the `\(ax+b\)' groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type \(\psi (\sqrt{{\mathcal{L}}})\exp (it \sqrt{{\mathcal{L}}})\), with \(\psi \in C_0({\mathbb{R}})\). We show that for \(t\rightarrow +\infty \), the convolution kernel \(k_t\) of this operator satisfies
\[
\|k_t\|_1\asymp t, \qquad \|k_t\|_\infty \asymp 1,
\] so that the upper estimates of
\textit{D.~Müller} and \textit{C.~Thiele} [Stud. Math. 179, No.~2, 117--148 (2007; Zbl 1112.43002)]
are sharp. As a necessary component, we recall the Plancherel density of \({\mathcal{L}}\) and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator \({\tilde{\Delta}}\), closely related to \({\mathcal{L}} \). The functions include in particular \(\exp (-t{\tilde{\Delta}}^\gamma)\), \(t>0\), \(\gamma >0\), and \(({\tilde{\Delta}}-z)^s\), with complex \(z, s\).An update on the \(L^p\)-\(L^q\) norms of spectral multipliers on unimodular Lie groupshttps://zbmath.org/1522.430022023-12-07T16:00:11.105023Z"Rottensteiner, David"https://zbmath.org/authors/?q=ai:rottensteiner.david"Ruzhansky, Michael"https://zbmath.org/authors/?q=ai:ruzhansky.michael-vSummary: This note gives a wide-ranging update on the multiplier theorems by \textit{R.~Akylzhanov} and the second author [J. Funct. Anal. 278, No.~3, Article ID 108324, 49~p. (2020; Zbl 1428.43008)].
The proofs of the latter crucially rely on \(L^p\)-\(L^q\) norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians, and Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the \(L^p\)-\(L^q\) norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The comparison with previously established bounds by the authors show that the heat kernel estimates are sharp. As an application, it is shown that several consequences of the multiplier theorems, such as time asymptotics for the \(L^p\)-\(L^q\) norms of the heat kernels and Sobolev-type embeddings, are then automatic for the considered operators.Sobolev spaces on compact groupshttps://zbmath.org/1522.430032023-12-07T16:00:11.105023Z"Kumar, Manoj"https://zbmath.org/authors/?q=ai:kumar.manoj.10"Kumar, N. Shravan"https://zbmath.org/authors/?q=ai:kumar.nageswaran-shravanLet \(G\) be a compact group. Denote by \(\widehat{G}\) its dual object, that is, the equivalence classes of unitary irreducible representations of \(G\). The Fourier transform of a complex Haar-integrable function \(f\) on \(G\) is the operator defined on the representation space \(H_\pi\) by
\[
\widehat{f}(\pi)=\displaystyle\int_G f(x)\pi(x)^*dx,\quad \pi\in \widehat{G}.
\]
The authors define a class of Sobolev spaces \(H_\gamma^s (G)\) based on the Plancherel type formula
\[
\|f\|_2^2=\sum\limits_{\pi\in\widehat{G}}d_\pi \mbox{tr}(\widehat{f}(\pi)^*\widehat{f}(\pi))
\]
where \(\widehat{f}(\pi)^*\) is the adjoint of the operator \(\widehat{f}(\pi)\) in the Hilbert representation space of \(\pi\).
Denote by \(\Gamma\) the space consisting of all sequences with non-negative terms indexed over \(\widehat{G}\). Let \(\gamma\in \Gamma\) and let \(s\geq 0\). The Sobolev space \(H_\gamma^s (G)\) is defined to be the set
\[
H_\gamma^s (G)=\left\lbrace f\in L^2 (G) : \sum\limits_{\pi\in\widehat{G}}d_\pi (1+\gamma (\pi)^2)^s\mbox{tr}(\widehat{f}(\pi)^*\widehat{f}(\pi))<\infty\right\rbrace
\]
endowed with the norm
\[
\|f\|_{H_\gamma^s }=\left(\sum\limits_{\pi\in\widehat{G}}d_\pi (1+\gamma (\pi)^2)^s\mbox{tr}(\widehat{f}(\pi)^*\widehat{f}(\pi))\right)^{\frac{1}{2}}.
\]
Then, the authors prove several continuous embedding theorems and they use the results to solve a generalized bosonic string equation. They provide a concrete example with the compact group \(SO(3)\).
Reviewer: Yaogan Mensah (Lomé)A corollary to a triviality theorem for quasirepresentations of an amenable group in reflexive Banach spaceshttps://zbmath.org/1522.430042023-12-07T16:00:11.105023Z"Shtern, A. I."https://zbmath.org/authors/?q=ai:shtern.alexander-iSummary: As is known, for a sufficiently small defect of a (not necessarily bounded) quasirepresentation of an amenable group in a reflexive Banach space \(E\) with dense set of bounded orbits, there is an extension of this quasirepresentation for which there is a close ordinary representation of the group in the space of this extension. In the present note it is proved that, if the original quasirepresentation \(\pi\) of an amenable group \(G\) in a reflexive Banach space \(E\) is a pseudorepresentation, then an ordinary representation of \(G\), in the vector subspace \(L\) of \(E\) formed by vectors with bounded orbits and equipped with a natural Banach norm, which is close to \(\pi |_L\) (this ordinary representation exists if the defect of \(\pi\) is sufficiently small) is equivalent to \(\pi |_L\).Fractional differential operators, fractional Sobolev spaces and fractional variation on homogeneous Carnot groupshttps://zbmath.org/1522.430052023-12-07T16:00:11.105023Z"Zhang, Tong"https://zbmath.org/authors/?q=ai:zhang.tong.3|zhang.tong|zhang.tong.1|zhang.tong.2"Zhu, Jie-Xiang"https://zbmath.org/authors/?q=ai:zhu.jie-xiang(no abstract)A theorem of Chernoff on quasi-analytic functions for Riemannian symmetric spaceshttps://zbmath.org/1522.430062023-12-07T16:00:11.105023Z"Bhowmik, Mithun"https://zbmath.org/authors/?q=ai:bhowmik.mithun"Pusti, Sanjoy"https://zbmath.org/authors/?q=ai:pusti.sanjoy"Ray, Swagato K."https://zbmath.org/authors/?q=ai:ray.swagato-kAuthors' abstract: An \(L^2\) version of the classical Denjoy-Carleman theorem regarding quasi-analytic functions was proved by \textit{P.~R. Chernoff} [Bull. Am. Math. Soc. 81, 637--646 (1975; Zbl 0304.47032)]
on \(\mathbb{R}^n\) using iterates of the Laplacian. We give a simple proof of this theorem that generalizes the result on \(\mathbb{R}^n\) for any \(p\in [1,2]\). We then extend this result to Riemannian symmetric spaces of compact and noncompact type for \(K\)-biinvariant functions.
Reviewer: Arash Ghaani Farashahi (Wien)Elements of mathematics. Spectral theories. Chapters 3--5https://zbmath.org/1522.460012023-12-07T16:00:11.105023Z"Bourbaki, N."https://zbmath.org/authors/?q=ai:bourbaki.nIt seems that the Bourbaki collective has resurrected; in the last few years they have produced a number of new or substantially revised volumes [\textit{N.~Bourbaki}, Éléments de mathématique. Algèbre. Chapitre~8. Modules et anneaux semi-simples. 2nd revised ed. of the 1958 original. Berlin: Springer (2012; Zbl 1245.16001); Éléments de mathématique. Topologie algébrique. Chapitres~1 à~4. Berlin: Springer (2016; Zbl 1355.55001); Éléments de mathématique. Théories spectrales. Chapitres~1 et~2. 2nd revised and updated edition. Cham: Springer (2019; Zbl 1417.46001)]. For some background on the Bourbaki project, the reader might wish to consult W.~Kleinert's splendid zbMATH reviews of the books mentioned above.
The first two chapters of ``Théories spectrales'' (note the plural) first appeared in 1967 [\textit{N.~Bourbaki}, Éléments de mathématique. Fasc. XXXII: Théories spectrales. Chap. 1 et 2: Algèbres normées. Groupes localement compacts commutatifs. Paris: Hermann \& Cie (1967; Zbl 0152.32603)], a reprint with Springer in 2007 [\textit{N.~Bourbaki}, Éléments de mathématique. Théories spectrales. Chapitres 1 et 2. Reprint of the 1967 original. Berlin: Springer (2007; Zbl 1106.46004)], and a revised and updated second edition came out in 2019 [\textit{N.~Bourbaki}, Éléments de mathématique. Théories spectrales. Chapitres 1 et 2. 2nd revised and updated edition. Cham: Springer (2019; Zbl 1417.46001)]. Now we have Chapters~3 to~5 dealing with compact and Fredholm operators (Ch.~3), spectral theory of operators in Hilbert space (both bounded and unbounded) (Ch.~4), and representation theory of noncommutative groups (Ch.~5) -- the commutative case was discussed in Chapter~2. There is also a very readable ``Note historique'' at the end of the book.
Notably, Chapter~3 contains L.~Schwartz's perturbation theorem to the effect that the sum of a Fredholm operator \(u\) and a compact operator on a locally convex space is again Fredholm with the same index as~\(u\), which is rarely found in textbooks. The text also offers an example of a Banach space without the approximation property in an exercise, indicating the construction step by step; the first such example was famously provided by Per Enflo. All the classical results, like Riesz's theory of compact operators and the Fredholm alternative, are of course also presented, sometimes with an unexpected twist due to the top-down approach of this work.
Chapter 4 discusses compact operators on Hilbert spaces and the spectral theorem for normal operators in both the multiplication operator form and the spectral measure form. Eventually, this theorem is extended to the case of unbounded normal or self-adjoint operators. Most often in the literature, this is done by reducing the unbounded case to the bounded one by means of the Cayley transform. Bourbaki, however, employs a different method called \textit{la bornification} of an unbounded operator~\(u\); cf. p.~262. Von Neumann's theory of defect indices is elaborated as well as the Friedrichs extension. In addition, there is a section on tempered distributions and Sobolev spaces that come in handy when describing self-adjoint extensions of the Laplacian.
The theme of the final chapter is unitary group representations culminating in the Peter-Weyl theorem for compact groups. Stone's theorem on unitary groups of operators is proved as well, providing a link with Chapter~4.
The style of presentation is what has indeed become known as the Bourbaki style: on the one hand, the arguments are presented in a crystal-clear manner, but no motivation is given, nor are there any indications of the intuition behind the results. The reader should be aware that Bourbaki tends to use slightly idiosyncratic notation that appears to be little known outside his (?) sphere: an ``algèbre stellaire'' is a \(C^*\)-algebra, an ``espace de type dénombrable'' is a separable space, an ``espace d'approximation'' is one with the approximation property, and a ``base banachique'' is a Schauder basis. The reader should also be cautioned that an inner product in this volume is linear in the second variable and anti-linear in the first one.
The typesetting is almost flawless if it were not for the incorrect use of delimiters in formulas like ``\(p\in ]2,+\infty[\)'' (p.~115), which visibly leads to wrong spacing, and of the five typos that I have spotted only one is of some consequence: in the definition of singular values on p.~156, a square root is missing.
Finally, every serious reader should be advised to have other Bourbaki volumes at hand, preferably the most recent editions, in order to track down references to previous work (of which there are many); in particular, the volumes on Topological Vector Spaces, Integration, and, of course, the first two chapters of Spectral Theories are cited on a regular basis.
Reviewer: Dirk Werner (Berlin)\(L^p\)-boundedness of a Hausdorff operator associated with change of variable and weightshttps://zbmath.org/1522.470722023-12-07T16:00:11.105023Z"Daher, Radouan"https://zbmath.org/authors/?q=ai:daher.radouan"Kawazoe, Takeshi"https://zbmath.org/authors/?q=ai:kawazoe.takeshi"Saadi, Faouaz"https://zbmath.org/authors/?q=ai:saadi.faouazThe Hausdorff operator \(H_\psi\) is defined by
\[
H_\psi(f)(x)=\int_{\mathbb{R}^n}\psi (t)f(A(t)x)\,d\mu (t)
\]
where \(A(t)\) is a matrix which depends on a parameter \(t\). Conditions on the function \(\psi\) that provide the boundedness of the operator \(H_\psi\) on \(H^p\), BMO, Herz-type spaces, Morrey-type spaces, etc., have been studied by several authors. In this article, the Hausdorff operator \(H_\psi\) is modified and it is given the form
\[
H_\psi(f)(x)=\int_{U}\psi (u)f(F_t(u))w_U(t)\,dt
\]
where \(F_t\) is a change of variables which depends on a parameter \(t\) and \(w_U\) is a weight function. Conditions to have \(L^p\)-boundedness of \(H_\psi\) are provided. By varying the weight, conditions on \(L^p\)-boundedness of \(H_\psi\) in link with the Euclidean space, the Dunkl hypergroup and the Jacobi hypergroup are obtained.
Reviewer: Yaogan Mensah (Lomé)Rough Hausdorff operators on Lebesgue spaces with variable exponenthttps://zbmath.org/1522.470732023-12-07T16:00:11.105023Z"Li, Ziwei"https://zbmath.org/authors/?q=ai:li.ziwei"Zhao, Jiman"https://zbmath.org/authors/?q=ai:zhao.jimanSummary: In this paper, we study rough Hausdorff operators on variable exponent Lebesgue spaces in the setting of the Heisenberg group. We prove the boundedness of rough Hausdorff operators by giving some sufficient conditions.From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodieshttps://zbmath.org/1522.520192023-12-07T16:00:11.105023Z"Kotrbatý, Jan"https://zbmath.org/authors/?q=ai:kotrbaty.jan"Wannerer, Thomas"https://zbmath.org/authors/?q=ai:wannerer.thomasA remarkable theorem by \textit{S. Alesker} et al. [Geom. Funct. Anal. 21, No. 4, 751--773 (2011; Zbl 1228.53088)] states that each irreducible representation of the special orthogonal group \(\mathrm{SO}(n)\) appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity.
In the paper under review the authors refine the Alesker-Bernig-Schuster theorem by constructing, in an explicit way, a non-trivial highest weight vector for each \(\mathrm{SO}(n)\)-type in the subspace of \(r\)-homogeneous valuations \(\mathrm{Val}_r(\mathbb{R}^n)\). Then, they describe the action of a number of important natural operations on valuations (pullback, pushforward, Fourier transform, Lefschetz operator and Alesker-Poincaré pairing) on these highest weight vectors. Moreover, they prove the so-called Hodge-Riemann relations for valuations in the case of Euclidean balls as reference bodies.
Reviewer: Maria A. Hernández Cifre (Murcia)On a characterization theorem in the space \(\mathbb{R}^n\)https://zbmath.org/1522.600242023-12-07T16:00:11.105023Z"Feldman, G. M."https://zbmath.org/authors/?q=ai:feldman.gennadiy-mThe author establishes an analogue in \(\mathbb{R}^n\) of Heyde's theorem characterizing Gaussian distributions on the real line by the symmetry of one linear form of independent random variables conditioned on another. In particular, let \(\alpha\) be an invertible linear operator in \(\mathbb{R}^n\), and \(\xi_1\) and \(\xi_2\) independent random vectors. The author shows that if the distribution of \(\xi_1+\alpha\xi_2\) conditional on \(\xi_1+\xi_2\) is symmetric, then the distributions of the \(\xi_j\) are shifts of convolutions of symmetric Gaussian distributions supported on an \(\alpha\)-invariant subspace \(G\), and a distribution supported on \(K=\text{Ker}(I+\alpha)\), where \(I\) is the identity. It also holds that \(G\cap K=\{0\}\). This result is of principal interest in the case where \(I+\alpha\) is not invertible.
Reviewer: Fraser Daly (Edinburgh)