Recent zbMATH articles in MSC 43https://zbmath.org/atom/cc/432024-08-28T19:40:24.813883ZWerkzeugContinuity of universally measurable homomorphismshttps://zbmath.org/1539.031592024-08-28T19:40:24.813883Z"Rosendal, Christian"https://zbmath.org/authors/?q=ai:rosendal.christianSummary: Answering a longstanding problem originating in \textit{J. P. R. Christensen}'s seminal work on Haar null sets [Math. Scand. 28, 124--128 (1971; Zbl 0217.08502); Isr. J. Math. 13, 255--260 (1972; Zbl 0249.43002); Topology and Borel structure. Descriptive topology and set theory with applications to functional analysis and measure theory. Elsevier, Amsterdam (1974; Zbl 0273.28001)], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo \(\text{ZF}+\text{DC}\), the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on \(\{0,1\}^{\mathbb{N}}\) has finite chromatic number.Prime power order circulant determinantshttps://zbmath.org/1539.110532024-08-28T19:40:24.813883Z"Mossinghoff, Michael J."https://zbmath.org/authors/?q=ai:mossinghoff.michael-j"Pinner, Christopher"https://zbmath.org/authors/?q=ai:pinner.christopher-gLet \(\mathcal{S}(\mathbb{Z}_n)=\{D(a_0,\ldots,a_{n-1}):a_i\in\mathbb{Z}\}\) be the set of all the determinants of integral \(n\times n\) circulant matrices, i.e.,
\[
D(a_0,\ldots,a_{n-1})=\det \begin{pmatrix}
a_0&a_{n-1}&\cdots &a_1\\
a_1&a_0&\ldots&a_2\\
\vdots & \vdots &\ddots &\vdots\\
a_{n-1}& a_{n-2}&\ldots &a_0
\end{pmatrix}.
\]
The set \(\mathcal{S}(\mathbb{Z}_n)\) is closed under multiplication and can be defined more broadly for a finite group \(G\). Here, \(G=\mathbb{Z}_n\) is a cyclic group of order \(n\). The complete description of \(\mathcal{S}(G)\) is known only for groups of order less than \(16\). By the results of \textit{H. T. Laquer} [in: The Fibonacci sequence, Collect. Manuscr., 18th anniv. Vol., The Fibonacci Assoc., 212--217 (1980; Zbl 0524.15007)] and \textit{M. Newman} [Ill. J. Math. 24, 156--158 (1980; Zbl 0414.15007)], we have \(n^2\mathbb{Z}\) and \(\{m\in \mathbb{Z}:\gcd(m,n)=1\}\) are both contained in \( \mathcal{S}(\mathbb{Z}_n)\). Furthermore, certain divisibility conditions for \(m\in\mathcal{S}(\mathbb{Z}_n)\) are known.
The current paper describes the integers contained in \(\mathcal{S}(\mathbb{Z}_{p^t}),\) where \(p\ge 5\) is a prime number and \(t\) is an integer greater or equal to two, or \(p=3\) and \(t\ge 3.\) The complete description of \(\mathcal{S}(\mathbb{Z}_{25})\) is given in Theorem 1.2, with representation including the use of special prime numbers congruent to 1 modulo 5, so-called \textit{perissads}. Similarly, the description of \(\mathcal{S}(\mathbb{Z}_{27})\) is given (in Theorem 1.3), invoking a certain family of primes congruent to 1 modulo 3 and 9. Finally, in Proposition 1.4, it is shown that \(\mathcal{S}(\mathbb{Z}_{p^t})\subseteq \mathcal{S}(\mathbb{Z}_{p^{t-1}})\) for \(t\ge 2.\)
Reviewer: Pavlo Yatsyna (Espoo)Properties of pure pseudorepresentations of commutative groupshttps://zbmath.org/1539.200042024-08-28T19:40:24.813883Z"Shtern, A. I."https://zbmath.org/authors/?q=ai:shtern.alexander-iSummary: We prove that a bounded pure pseudorepresentation of the group of integers \(\mathbb{Z}\) with an arbitrarily small defect need not be an ordinary representation of the group. We also prove that a bounded pure pseudorepresentation with a sufficiently small defect of a commutative group on a dual Banach space is an ordinary representation if and only if the image of the pseudorepresentation forms a commutative family of operators.Group pseudo representations that are trivial on a normal subgrouphttps://zbmath.org/1539.200052024-08-28T19:40:24.813883Z"Shtern, A. I."https://zbmath.org/authors/?q=ai:shtern.alexander-iSummary: Continuing the study of general group pseudorepresentations, we prove that, if \(\pi\) is a pseudorepresentation of a group \(G\) in a Banach space \(E\) with sufficiently small defect and if \(N\) is a normal subgroup of \(G\) for which \(\pi(n)=1|_E\) for all \(n\in N\), then there is a pseudorepresentation \(\rho\) of the quotient group \(G/N\) such that the pseudorepresentation \(\pi\) is locally equivalent to the pseudorepresentation of \(G\) defined by the rule \(g\mapsto\rho(gN)\), \(g\in G\).A decomposition of the space of pseudocharacters on a group with respect to a normal subgrouphttps://zbmath.org/1539.200062024-08-28T19:40:24.813883Z"Shtern, A. I."https://zbmath.org/authors/?q=ai:shtern.alexander-iSummary: We prove that the vector space of nontrivial pseudocharacters on a group having a normal subgroup admits a natural direct decomposition.On the hyperbolic Bloch transformhttps://zbmath.org/1539.352002024-08-28T19:40:24.813883Z"Nagy, Ákos"https://zbmath.org/authors/?q=ai:nagy.akos"Rayan, Steven"https://zbmath.org/authors/?q=ai:rayan.stevenThe authors of this interesting paper study the noncommutative Bloch transform of Fuchsian groups which is called the hyperbolic Bloch transform. It concerns some mathematical models in the theory of the hyperbolic crystals. To recall the Bloch transform one has to consider a group \(G\) with Haar measure \(\mu_H\). Assume that the canonical action of \(G\) on the Hilbert space \(\ell^2(\Gamma )\) is such that there exists the direct integral decomposition
\[
\ell^2(\Gamma ) = \int\limits_{\hat{G}}^{\oplus} \mathcal{H}_{[\varrho ]} d\hat{\mu}([\varrho ]),
\]
where \(\hat{G}\) is the space of irreducible finite dimensional representations of \(G\), \(\mathcal{H}_{[\varrho ]}\) is the representation space of \(\varrho \), and \(\hat{\mu }\) is the canonical measure on \(\hat{G}\). The Bloch transform of \(\Psi\in\ell^2(\Gamma )\otimes \mathcal{H}_{0}\) at \([\varrho ]\in \hat{G}\) is
\[
\mathcal{B}(\Psi )([\varrho ]) = \int\limits_{G}\Psi (\gamma ) \varrho (\gamma )d\mu_{H}(\gamma )\in \mathrm{End}(\mathcal{H}_{[\varrho ]}).
\]
Here, \(\mathcal{H}_{0}\) is an auxiliary Hilbert space. Actually the definition of the Bloch transform includes the idea that \(\mathcal{B}(\Psi )([\varrho ])\) is the \([\varrho ]\)-quasiperiodic (isotypical) component of \(\Psi \),
\[
\mathcal{B}(\gamma\cdot\Psi ) ([\varrho ])=\varrho (\gamma )^{-1} \mathcal{B}(\Psi ) ([\varrho ]).
\]
This can be used to study spectra of operators, such as Schrödinger operators on Euclidean spaces. Thus it allows to decompose partial differential operators on noncompact domains into `easier-to-understand' operators on compact domains with quasiperiodic boundary conditions. The authors prove that the hyperbolic Bloch transform is injective and asymptotically unitary in the simplest case. This occurs in the case when the Hilbert space is the regular representation of the Fuchsian group \(\Gamma \). It turns out that when \(\Gamma\subset PSU(1, 1)\) acts isometrically on the hyperbolic plane \(\mathbb{H}\) and the Hilbert space is \(L^2(\mathbb{H})\), then one can define a modified geometric Bloch transform that `sends' wave functions to sections of irreducible and flat Hermitian vector bundles over \(\Sigma = \mathbb{H}/\Gamma \). Moreover, it transforms the hyperbolic Laplacian into the covariant one. Further the authors study the case when a periodic gauge field is presented on the hyperbolic plane. Finally, they reinterpret the hyperbolic Bloch transform from noncommutative geometric point of view.
Reviewer: Dimitar A. Kolev (Sofia)Generalized polynomials on semigroupshttps://zbmath.org/1539.390202024-08-28T19:40:24.813883Z"Ebanks, Bruce"https://zbmath.org/authors/?q=ai:ebanks.bruce-rThis paper deals with generalized polynomials defined on noncommutative semigroups. Let \(S\) be a semigroup and \(T\) a commutative semigroup. For \(j\in \mathbb{N}\cup \{0\}\) a function \(f:S\to T\) is a generalized monomial of degree \(j\) if there exists a \(j\)-additive function \(F:S^j\to T\) such that \(f(x)=F(x,\dots,x)\) for all \(x\in S\). For \(n\in \mathbb{N}\cup \{0\}\) a generalized polynomial of degree at most \(n\) is a function \(f:S\to T\) of the form \[f=\sum_{j}^n f_j,\] where \(f_j:S\to T\) is a generalized monomial of degree \(j\).
If \(S\) be a sub-semigroup of a group \(G\) such that \(G=S\cdot S^{-1}=\{xy^{-1}: x,y\in S\}\), we say that \(S\) generates \(G\). The main result of the paper is given by the following:
Theorem. Let \(G\) be a group, \(S\) a semigroup that generates \(G\), and \(H\) an Abelian group. Then every generalized polynomial \(f:S\to H\) of degree at most \(n\) can be extended to a generalized polynomial \(\tilde{f}:G\to H\) of degree at most \(n\). Furthermore, if either \(S\) is uniquely divisible by \(n!\) or multiplication by \(n!\) is bijective in \(H\) then the extension is unique.
In the last section of the paper the author discusses the extendability of exponentials and generalized exponential polynomials on semigroups.
Reviewer: Gian Luigi Forti (Milano)Restriction theorem for the Fourier-Dunkl transform and its applications to Strichartz inequalitieshttps://zbmath.org/1539.420072024-08-28T19:40:24.813883Z"Senapati, P. Jitendra Kumar"https://zbmath.org/authors/?q=ai:senapati.p-jitendra-kumar"Boggarapu, Pradeep"https://zbmath.org/authors/?q=ai:boggarapu.pradeep"Mondal, Shyam Swarup"https://zbmath.org/authors/?q=ai:mondal.shyam-swarup"Mejjaoli, Hatem"https://zbmath.org/authors/?q=ai:mejjaoli.hatemIn the paper under review, the authors introduce a so-called Fourier-Dunkl transform, for \(f\in L^1_\kappa(\mathbb{R}^n\times\mathbb{R}^d)\), by
\[
\widehat{f}(\xi,\zeta)=\frac{1}{c_\kappa\,(2\pi)^{n/2}}\int_{\mathbb{R}^n}\hskip-1mm\int_{\mathbb{R}^d}f(x,y)E_\kappa(-i\zeta,y)e^{-ix.\xi}h^2_\kappa(y)dydx\,,\ \ (\xi,\zeta)\in\mathbb{R}^n\times\mathbb{R}^d\,,
\]
\(\kappa\) being a positive multiplicity function associated with a root system \(\mathcal{R}\), \(h_\kappa^2\) and \(E_\kappa\) being, respectively, the weight function and the Dunkl kernel associated with the root system \(\mathcal{R}\), \(c_\kappa\) is a positive constant.
This paper aims to study the Strichartz restriction problem for the Fourier-Dunkl transform as follows:
Let \(S\) be a surface embedded in \(\mathbb{R}^n\times\mathbb{R}^d\) with \(n+d\geq2\), for what values of \(1\leq p<2\), can we have
\[
\left(\int_S|\widehat{f}(\xi,\zeta)|^2h_\kappa^2(\zeta)d\sigma(\xi,\zeta)\right)^{\frac{1}{2}}\leq C\|f\|_{L^p_\kappa(\mathbb{R}^n\times\mathbb{R}^d)}\,?
\]
here \(d\sigma\) denotes surface measure on \(S\) or \((n+d-1)\)-dimensional Lebesgue measure endowed with the surface \(S\).
In particular, the authors prove the Strichartz restriction theorem for the Fourier-Dunkl transform for three surfaces namely, paraboloid:
\[
S_1=\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^d\,,\ \text{such that}\ \ x_n=x_1^2+\dots+x_{n-1}^2-y_1^2-\dots-y_d^2\}\,,
\]
sphere:
\[
S_2=\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^d\,,\ \text{such that}\ x_1^2+\dots+x_n^2+y_1^2+\dots+y_d^2=1\}\,,
\]
and two sheeted hyperboloids:
\[
S_3=\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^d\,,\ \text{such that}\ x_1^2+\dots+x_n^2-y_1^2-\dots-y_d^2=1\}\,.
\]
On the other hand, for each surface mentioned above, the authors provide restriction estimates for orthonormal functions. The authors at the end of the paper, as an application of the previous theorems, establish various Strichartz inequalities.
Reviewer: Lotfi Kamoun (Monastir)Riesz transforms and commutators in the Dunkl settinghttps://zbmath.org/1539.420112024-08-28T19:40:24.813883Z"Han, Yongsheng"https://zbmath.org/authors/?q=ai:han.yongsheng"Lee, Ming-Yi"https://zbmath.org/authors/?q=ai:lee.ming-yi"Li, Ji"https://zbmath.org/authors/?q=ai:li.ji.1"Wick, Brett D."https://zbmath.org/authors/?q=ai:wick.brett-duaneSummary: In this paper we characterise the pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where these two metrics are not equivalent. We further establish a suitable version of the pointwise kernel lower bound of the Dunkl Riesz transform via the Euclidean metric only. Then we show that the lower bound of commutator of the Dunkl Riesz transform is with respect to the BMO space associated with the Euclidean metric, and that the upper bound is respect to the BMO space associated with the Dunkl metric. Moreover, the compactness and the two types of VMO are also addressed.On generalized Hardy spaces associated with singular partial differential operatorshttps://zbmath.org/1539.420172024-08-28T19:40:24.813883Z"Ghandouri, Amal"https://zbmath.org/authors/?q=ai:ghandouri.amal"Mejjaoli, Hatem"https://zbmath.org/authors/?q=ai:mejjaoli.hatem"Omri, Slim"https://zbmath.org/authors/?q=ai:omri.slimSummary: We define and study the Hardy spaces associated with singular partial differential operators. Also, a characterization by mean of atomic decomposition is investigated.Function spaces via fractional Poisson kernel on Carnot groups and applicationshttps://zbmath.org/1539.430012024-08-28T19:40:24.813883Z"Maalaoui, Ali"https://zbmath.org/authors/?q=ai:maalaoui.ali"Pinamonti, Andrea"https://zbmath.org/authors/?q=ai:pinamonti.andrea"Speight, Gareth"https://zbmath.org/authors/?q=ai:speight.garethThe methods in this paper give a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. Moreover, several applications to estimates for commutators of fractional powers of the sub-Laplacian are provided.
Though technically not difficult and the proofs may follow the classical results, these results are worthy to be written up in the literature for future reference.
Reviewer: Jinsen Xiao (Guangzhou)Product formula for the one-dimensional \((k,a)\)-generalized Fourier kernelhttps://zbmath.org/1539.430022024-08-28T19:40:24.813883Z"Amri, Béchir"https://zbmath.org/authors/?q=ai:amri.bechirSummary: In this paper, a product formula for the one-dimensional \((k,a)\)-generalized Fourier kernel is given for \(k \geq 0\), \(a>0\) and \(2k > 1 - \frac{a}{2}\), extending the special case of
[\textit{M.~A. Boubatra} et al., Integral Transforms Spec. Funct. 33, No.~3, 247--263 (2022; Zbl 1515.33008)]
when \(a=\frac{2}{n}\), \(n \in \mathbb{N}^*\).Double preconditioning for Gabor frame operators: algebraic, functional analytic and numerical aspectshttps://zbmath.org/1539.430032024-08-28T19:40:24.813883Z"Feichtinger, Hans G."https://zbmath.org/authors/?q=ai:feichtinger.hans-georg"Balazs, Peter"https://zbmath.org/authors/?q=ai:balazs.peter.2"Haider, Daniel"https://zbmath.org/authors/?q=ai:haider.danielThe motivation for this paper is the observation that the potentially computationally expensive task of computing the exact canonical dual or tight Gabor atom for a given time-frequency lattice can be replaced by the very simple procedure of double preconditioning of the Gabor frame operators. The authors have tried to move the idea of double or even multiple preconditioning (i.e., concatenation of various preconditioners) for Gabor families from a purely experimental observation to a more detailed analysis, by discussing algebraic, numerical and functional analytic aspects. In this paper, the formulation of the algebraic and analytic background is done and also the numerical arguments are provided to justify that this technique is also more efficient.
Making use of Banach-Gelfand triples, based on the Segal algebra, the definition of the approach is extended to the continuous setting. The continuous dependency of the double preconditioning operators on their parameters is established. The generalization allows to investigate the influence of the order of the two main single preconditioners, viz., diagonal and convolutional.
In the last two sections, further numerical insights are provided regarding the influence of the redundancy on the quality of preconditioning approaches and applications: finding an (approximate) tight window, improving the convergence of an iterative approach, and using it as a fast reconstruction option for varying lattices.
Reviewer: Ashish Bansal (Delhi)Sufficient conditions for the weighted integrability of Fourier-Laguerre transformshttps://zbmath.org/1539.430042024-08-28T19:40:24.813883Z"Tyr, Othman"https://zbmath.org/authors/?q=ai:tyr.othman"Daher, Radouan"https://zbmath.org/authors/?q=ai:daher.radouanSummary: The problem of weighted integrability of the Fourier-Laguerre transform in terms of the moduli of smoothness related to generalized translations is considered. Sufficient conditions are given to solve this problem. These results generalize a famous Titchmarsh's theorem and Younis' theorem, due to
\textit{S.~Negzaoui} [Mediterr. J. Math. 14, No.~5, Paper No.~191, 12~p. (2017; Zbl 1376.43008)]
in the Laguerre hypergroup. Also some results connected with the integrability of Fourier-Laguerre transforms of Laguerre convolutions are given.Pettis-integration approach for characterizing almost periodic functions and flowshttps://zbmath.org/1539.430052024-08-28T19:40:24.813883Z"Amini, Fardin"https://zbmath.org/authors/?q=ai:amini.fardin"Saeidi, Shahram"https://zbmath.org/authors/?q=ai:saeidi.shahramThis paper concerns almost periodic functions on semigroups \(S\) with values in a topological vector space (TVS) \(E\) whose topological dual \(E^\prime,\) consisting of continuous linear functionals on \(E,\) separates points. It uses an extension of Pettis integration. We'll summarize the paper's results below.
Section 2 defines standard functional analysis concepts and notation.
Section~3 considers a set \(S\) and a subspace \(X \subset \ell^\infty(S) := \ell^\infty (S,\mathbb R)\) containing constant functions. \(\mu \in X^\prime\) is called a mean on \(X\) if
\[
\inf_{s \in S} x(s) \leq \mu(x) \leq \sup_{s \in S} x(s)
\]
and \(M(X)\) denotes the set of means on \(X.\) Assume that \(E\) is a TVS whose dual \(E^\prime\) separates points, \(f : S \rightarrow E,\) and \(X\) contains all functions \(s \mapsto \langle f(s),x^\prime\rangle\), \( x^\prime \in X^\prime.\) Then for each \(\mu \in X^{\prime}\) we define the linear functional \(\tau(\mu)f \in E^{\prime *}\) on \(E^\prime\) by
\[
(\tau(\mu)f)(x^\prime) := \mu (s \rightarrow \langle f(s),x^\prime \rangle).
\]
Theorem 3.1 characterizes those \(f\) such that \(\tau(\mu)f \in E\) for all \(\mu \in M(X)\). If \(S\) is a compact Hausdorff topological space and \(X = C(S)\) is the Banach space of continuous complex valued functions on \(S\) with the \(\| \cdot \|_\infty\) norm, then the Banach space of Borel measures \(\mathcal M(X) = C(X)^\prime\) and the \(M(X) = \mathcal P(X),\) the set of probability measures on \(S.\) Therefore Theorem~3.1 characterizes the \(E\)-valued functions on \(S\) that are Pettis integrable with respect to probability measures.
Section 4 characterizes vector-valued almost periodic functions. Consider a semigroup \(S,\) \(f \in \ell^\infty(S,E)\) and a subspace of \(\ell^\infty(S)\) containing constants and functions \( \langle f,x^\prime\rangle \) for all \(x^\prime \in E^\prime\). Lemma~4.1 says that in the product space topology \(p\) on \({E^{\prime *}}^S,\)
\[
\overline {\mathrm{co}}^p\, R(S)f = \{\tau(\mu)L(\cdot)f : \mu \in M(X)\}.
\]
Here \(\mathrm{co}\) denotes the convex hull, and \(R(S)f\) denotes the set of right translates on \(f,\) and \(L(\cdot)f\) denotes the function \(s \rightarrow L(s)f\). Other results in this section follow from Lemma~4.1, and some results, such as Theorem~4.8 and Theorem~4.13, relate weak almost periodicity to uniform almost periodicity.
Section 5 considers flows \(S \times \mathcal X \mapsto \mathcal X\) on a topological space \(\mathcal X\) where \(S\) is a semigroup. Lemma~5.1 gives a relation between flows and almost periodicity. If \(\mathcal X\) is compact and the flow is continuous in each variable and equicontinuous and \(y \in \mathcal X\) and \(h \in C(\mathcal X, E)\), then the function \(s \mapsto h(sy)\) is almost periodic. Other results in this section also relate equicontinuity and almost periodicity.
The paper is well referenced. It emphasizes derivation of equations more than concrete examples.
Reviewer: Wayne M. Lawton (Krasnoyarsk)\(L^{p}\)-\(L^{q}\) multipliers on commutative hypergroupshttps://zbmath.org/1539.430062024-08-28T19:40:24.813883Z"Kumar, Vishvesh"https://zbmath.org/authors/?q=ai:kumar.vishvesh"Ruzhansky, Michael"https://zbmath.org/authors/?q=ai:ruzhansky.michael-vSummary: The main purpose of this paper is to prove Hörmander's \(L^p\)-\(L^q\) boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff-Young-Paley inequality for commutative hypergroups. We show the \(L^p\)-\(L^q\) boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli-Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the \(L^p\)-\(L^q\) norms of the heat kernel for generalised radial Laplacian.Spectral summability for the quartic oscillator with applications to the Engel grouphttps://zbmath.org/1539.430072024-08-28T19:40:24.813883Z"Bahouri, Hajer"https://zbmath.org/authors/?q=ai:bahouri.hajer"Barilari, Davide"https://zbmath.org/authors/?q=ai:barilari.davide"Gallagher, Isabelle"https://zbmath.org/authors/?q=ai:gallagher.isabelle"Léautaud, Matthieu"https://zbmath.org/authors/?q=ai:leautaud.matthieuSummary: In this article, we investigate spectral properties of the sublaplacian \(-\Delta_G\) on the Engel group, which is the main example of a Carnot group of step 3. We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set.
This enables us to give fine estimates on the convolution kernel satisfying \(F(-\Delta_G) u=u\star k_F\), for suitable scalar functions \(F\), and in turn to obtain proofs of classical functional embeddings, via Fourier techniques.
This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest.Distribution vectors on \(SU (p, q)/ SO (p, q)\)https://zbmath.org/1539.430082024-08-28T19:40:24.813883Z"Fan, Xingya"https://zbmath.org/authors/?q=ai:fan.xingyaSummary: In this paper, we consider Flensted-Jensen's discrete series on semisimple symmetric space \(SU(p, q) / SO(p, q)\). The main idea is to construct the \((\mathfrak{g}_+, K \cap SO(p, q))\)-module, where \(K = S(U(p) \times U(q)), \mathfrak{g}_+\) is the Lie algebra of \(G_+\) and \(G_+\) denotes a connected linear reductive subgroup of \(S U(p, q)\) with the maximal compact subgroup \(K \cap S O(p, q)\).Pointwise multipliers for Triebel-Lizorkin and Besov spaces on Lie groupshttps://zbmath.org/1539.460212024-08-28T19:40:24.813883Z"Bruno, Tommaso"https://zbmath.org/authors/?q=ai:bruno.tommaso"Peloso, Marco M."https://zbmath.org/authors/?q=ai:peloso.marco-maria"Vallarino, Maria"https://zbmath.org/authors/?q=ai:vallarino.mariaSummary: On a general Lie group \(G\) endowed with a sub-Riemannian structure and of local dimension \(d\), we characterize the pointwise multipliers of Triebel-Lizorkin spaces \(F_\alpha^{p,q}\) for \(p,q \in(1,\infty)\) and \(\alpha>d/p\), and those of Besov spaces \(B_\alpha^{p,q}\) for \(q\in[1,\infty]\), \(p>d\) and \(d/p<\alpha<1\). When \(G\) is stratified, we extend the latter characterization to all \(p,q\in [1,\infty]\) and \(\alpha>d/p\).Boundedness and nuclearity of pseudo-differential operators on homogeneous treeshttps://zbmath.org/1539.470882024-08-28T19:40:24.813883Z"Mondal, Shyam Swarup"https://zbmath.org/authors/?q=ai:mondal.shyam-swarupThe paper is devoted to studying pseudo-differential operators on a special graph \(\mathfrak{X}\) in which every vertex is adjacent to \(q+1\) other vertices, which is called a homogeneous tree, \(q> 2\). The main problem is to correctly introduce an analogue of a pseudo-differential operator on such set \(\mathfrak{X}\). The author suggests such a construction using the Fourier-Helgason transform \(\mathcal{H}\) which permits to state a duality and an analogue of the Plancherel theorem. This Fourier-Helgason transform \(\mathcal{H}\) can be extended to an isometry from \(L^2(\mathfrak{X})\) to its image in \(L^2(\Omega\times\mathbb{T})\), where \(\Omega\) is the boundary of \(\mathfrak{X}\). Further, choosing a symbol as a measurable function \(\sigma:\mathfrak{X}\times\Omega\times[0,\tau]\rightarrow\mathbb{C}\), the author defines the pseudo-differential operator \(T_{\sigma}\) using the Fourier-Helgason transform \(\mathcal{H}\) and certain characteristics of the tree \(\mathfrak{X}\).
The other sections are devoted to studying such pseudo-differential operators in spaces \(L^p(\mathfrak{X})\), including \(L^2\)-estimates and Schatten-von Neumann classes, weak \(l^p\)-estimates and \(L^p\)-\(L^q\)-estimates. Also, the author gives a characterization of nuclear operators and presents a formula for an adjoint operator, which is very heavy.
Reviewer: Vladimir Vasilyev (Belgorod)Characterization of generalized periodic vector fields on hyperbolic spacehttps://zbmath.org/1539.530892024-08-28T19:40:24.813883Z"Volchkova, Natal'ya Petrovna"https://zbmath.org/authors/?q=ai:volchkova.natalya-petrovna"Volchkov, Vitaliĭ Vladimirovich"https://zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: We study vector fields which have zero flux through every sphere of fixed radius in a ball on a real hyperbolic space. For fields in such classes a description in the form of a series in special functions is obtained.B-type anomaly coefficients for the D3-D5 domain wallhttps://zbmath.org/1539.810962024-08-28T19:40:24.813883Z"de Leeuw, Marius"https://zbmath.org/authors/?q=ai:de-leeuw.marius"Kristjansen, Charlotte"https://zbmath.org/authors/?q=ai:kristjansen.charlotte-f"Linardopoulos, Georgios"https://zbmath.org/authors/?q=ai:linardopoulos.georgios"Volk, Matthias"https://zbmath.org/authors/?q=ai:volk.matthiasSummary: We compute type-B Weyl anomaly coefficients for the domain wall version of \(\mathcal{N} = 4\) SYM that is holographically dual to the D3-D5 probe-brane system with flux. Our starting point is the explicit expression for the improved energy momentum tensor of \(\mathcal{N} = 4\) SYM. We determine the two-point function of this operator in the presence of the domain wall and extract the anomaly coefficients from the result. In the same process we determine the two-point function of the displacement operator.