Band 35, Heft 1

Let 𝖤 {\mathsf{E}} be a flat Lorentzian space of signature ( 2 , 1 ) {(2,1)} . A Margulis space-time is a noncompact complete Lorentz flat 3-manifold 𝖤 / Γ {\mathsf{E}/\Gamma} with a free holonomy group Γ of rank 𝗀 {\mathsf{g}} , 𝗀 ≥ 2 {\mathsf{g}\geq 2} . We consider the case when Γ contains a parabolic element. We obtain a characterization of proper Γ-actions in terms of Margulis and Charette–Drumm invariants. We show that 𝖤 / Γ {\mathsf{E}/\Gamma} is homeomorphic to the interior of a compact handlebody of genus 𝗀 {\mathsf{g}} generalizing our earlier result. Also, we obtain a bordification of the Margulis space-time with parabolics by adding a real projective surface at infinity giving us a compactification as a manifold relative to parabolic end neighborhoods. Our method is to estimate the translational parts of the affine transformation group and use some 3-manifold topology.

Let Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} be open and c k ⁢ l ∈ L ∞ ⁢ ( Ω , ℂ ) {c_{kl}\in L_{\infty}(\Omega,\mathbb{C})} with Im ⁡ c k ⁢ l = Im ⁡ c l ⁢ k {\operatorname{Im}c_{kl}=\operatorname{Im}c_{lk}} for all k , l ∈ { 1 , … , d } {k,l\in\{1,\ldots,d\}} . Assume that C = ( c k ⁢ l ) 1 ≤ k , l ≤ d {C=(c_{kl})_{1\leq k,l\leq d}} satisfies ( C ⁢ ( x ) ⁢ ξ , ξ ) ∈ Σ θ {(C(x)\xi,\xi)\in\Sigma_{\theta}} for all x ∈ Ω {x\in\Omega} and ξ ∈ ℂ d {\xi\in\mathbb{C}^{d}} , where Σ θ {\Sigma_{\theta}} is the closed sector with vertex 0 and semi-angle θ in the complex plane. We emphasize that Ω is an arbitrary domain and C need not be symmetric. We show that C is (degenerate) p -elliptic for all p ∈ ( 1 , ∞ ) {p\in(1,\infty)} with | 1 - 2 p | < cos ⁡ θ {|1-\frac{2}{p}|<\cos\theta} in the sense of Carbonaro and Dragičević. As a consequence, we obtain the consistent holomorphic extension for the C 0 {C_{0}} -semigroup generated by the second-order differential operator in divergence form associated with C . The core property for this operator is also investigated.

This article deals with various kinds of quantitative results about the comparison between the normalized Hecke eigenvalues of two distinct Siegel cuspidal Hecke eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts. We also prove some simultaneous sign change results for their eigenvalues.

In this paper, we provide some isomorphism preserving conditions for (weighted) Orlicz spaces, and as a main result, it is proved that if there exist a bicontinuous linear operator T : L w 1 Φ ⁢ ( G 1 ) → L w 2 Φ ⁢ ( G 2 ) {T\colon L^{\Phi}_{w_{1}}(G_{1})\rightarrow L^{\Phi}_{w_{2}}(G_{2})} and a mapping a ↦ ( ξ ⁢ ( a ) , h ⁢ ( a ) ) {a\mapsto(\xi(a),h(a))} from G 1 {G_{1}} to ℂ × G 2 {\mathbb{C}\times G_{2}} with T ⁢ λ a = ξ ⁢ ( a ) ⁢ λ h ⁢ ( a ) ⁢ T {T\lambda_{a}=\xi(a)\lambda_{h(a)}T} for all a ∈ G 1 {a\in G_{1}} , then G 1 {G_{1}} and G 2 {G_{2}} are isomorphic, where Φ is a Δ 2 {\Delta_{2}} -regular Young function, G 1 {G_{1}} and G 2 {G_{2}} are locally compact groups and w 1 {w_{1}} and w 2 {w_{2}} are weight functions. Also, for a class of Young functions Φ, we show that if C ⁢ V Φ ⁢ ( G 1 ) {CV_{\Phi}(G_{1})} and CV Φ ⁢ ( G 2 ) {\mathrm{CV}_{\Phi}(G_{2})} are isometrically isomorphic, then G 1 {G_{1}} and G 2 {G_{2}} are isomorphic, CV Φ ⁢ ( G i ) {\mathrm{CV}_{\Phi}(G_{i})} is the space of all convolution operators on the Orlicz space L Φ ⁢ ( G i ) {L^{\Phi}(G_{i})} for i = 1 , 2 {i=1,2} .

For a finite group G , we investigate the directed graph Γ ⁢ ( G ) {\Gamma(G)} , whose vertices are the non-hypercentral elements of G and where there is an edge x ↦ y {x\mapsto y} if and only if [ x , n y ] = 1 {[x,_{n}y]=1} for some n ∈ ℕ {n\in\mathbb{N}} . We prove that Γ ⁢ ( G ) {\Gamma(G)} is always weakly connected and is strongly connected if G / Z ∞ ⁢ ( G ) {G/Z_{\infty}(G)} is neither Frobenius nor almost simple.

In this paper, we investigate the nonnegative solutions of the differential inequality u p ≤ ℒ s ⁢ u u^{p}\leq\mathcal{L}^{s}u on the Grushin space 𝔾 α n {\mathbb{G}_{\alpha}^{n}} for ( p , s , α ) ∈ ( 1 , ∞ ) × ( 0 , 1 ) × ( 0 , ∞ ) {(p,s,\alpha)\in(1,\infty)\times(0,1)\times(0,\infty)} , where the ℒ s {\mathcal{L}^{s}} are the fractional powers of the Grushin operator ℒ {\mathcal{L}} . We show that any nonnegative solution of the fractional order differential inequality displayed above is zero if and only if p ≤ Q Q - 2 ⁢ s {p\leq\frac{Q}{Q-2s}} , where Q is the homogeneous dimension of 𝔾 α n {\mathbb{G}_{\alpha}^{n}} . Moreover, we also consider the similar problems of nonnegative weak solutions of the fractional sub-Laplacian differential inequality on nilpotent Lie groups.

In this paper, we generalize Schur–Weyl duality and Morita Theorem on associative algebras to those on associative H -pseudoalgebras. Meanwhile, we get a plenty of associative H -pseudoalgebras over a cocommutative Hopf algebra H

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in ℝ n {\mathbb{R}^{n}} for all n ≥ 2 {n\geq 2} . Our result generalizes and improves the inequalities previously obtained in [M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Convex components, Commun. Contemp. Math. 21 2019, 6, Article ID 1850036] and [M. La Civita and F. Leonetti, Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 2008/09, 71–78].

A Borel probability measure on ℝ n {\mathbb{R}^{n}} is called a spectral measure if the Hilbert space L 2 ⁢ ( μ ) {L^{2}(\mu)} has an orthonormal basis consisting of exponentials. In the present paper we show that, under a mild condition, compatible pairs and lacunary sequences generate homogeneous Moran spectral measures μ on ℝ n {\mathbb{R}^{n}} . As a consequence we show that compatible pairs are necessary and sufficient conditions for certain planar homogeneous Moran measures to become spectral ones.

The supercharacter theory of a finite group was introduced by Diaconis and Isaacs as an alternative to the usual irreducible character theory, and illustrated with a construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with a characterization of standard supercharacters of the group of upper unitriangular matrices over an algebraically closed field of prime characteristic.

Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider L 0 ⁢ u = - div ⁢ ( A 0 ⁢ ∇ ⁡ u ) {L_{0}u=-\mathrm{div}(A_{0}\nabla u)} , L ⁢ u = - div ⁢ ( A ⁢ ∇ ⁡ u ) {Lu=-\mathrm{div}(A\nabla u)} , two real (non-necessarily symmetric) uniformly elliptic operators in Ω, and write ω L 0 {\omega_{L_{0}}} , ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\mathrm{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper we establish that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to ω L 0 {\omega_{L_{0}}} , then ω L ∈ A ∞ ⁢ ( ω L 0 ) {\omega_{L}\in A_{\infty}(\omega_{L_{0}})} . Additionally, we can prove that ω L ∈ RH q ⁢ ( ω L 0 ) {\omega_{L}\in\mathrm{RH}_{q}(\omega_{L_{0}})} for some specific q ∈ ( 1 , ∞ ) {q\in(1,\infty)} , by assuming that such Carleson condition holds with a sufficiently small constant. This “small constant” case extends previous work of Fefferman–Kenig–Pipher and Milakis–Pipher together with the last author of the present paper who considered symmetric operators in Lipschitz and bounded chord-arc domains, respectively. Here we go beyond those settings, our domains satisfy a capacity density condition which is much weaker than the existence of exterior Corkscrew balls. Moreover, their boundaries need not be Ahlfors regular and the restriction of the n -dimensional Hausdorff measure to the boundary could be even locally infinite. The “large constant” case, that is, the one on which we just assume that the discrepancy of the two matrices satisfies a Carleson measure condition, is new even in the case of nice domains (such as the unit ball, the upper-half space, or non-tangentially accessible domains) and in the case of symmetric operators. We emphasize that our results hold in the absence of a nice surface measure: all the analysis is done with the underlying measure ω L 0 {\omega_{L_{0}}} , which behaves well in the scenarios we are considering. When particularized to the setting of Lipschitz, chord-arc, or 1-sided chord-arc domains, our methods allow us to immediately recover a number of existing perturbation results as well as extend some of them.