Volume 30, Issue 1

In this paper, using the method in [1], i.e., reduce Moebius measures μxn{\mu_{x}^{n}} indexed by |x|<1{|x|<1} on spheres Sn-1{S^{n-1}} (n≥3{n\geq 3}) to one-dimensional diffusions on [0,π]{[0,\pi]}, we obtain that the optimal Poincaré constant is not greater than 2n-2{\frac{2}{n-2}} and the optimal logarithmic Sobolev constant denoted by CLS⁢(μxn){C_{\rm LS}(\mu_{x}^{n})} behaves like 1n⁢log⁡(1+11-|x|){\frac{1}{n}\log(1+\frac{1}{1{-}|x|})}. As a consequence, we claim that logarithmic Sobolev inequalities are strictly stronger than L2{L^{2}}-transportation-information inequalities.

We present a fast algorithm to compute the Delta set of a nonsymmetric numerical semigroup with embedding dimension three. We also characterize the sets of integers that are the Delta set of a numerical semigroup of this kind.

The Pseudo-differential operators (p.d.o.) L⁢(x,Ax){L(x,A_{x})} and ℒ⁢(x,Ax){\mathcal{L}(x,A_{x})} involving the Kontorovich–Lebedev transform are defined. An estimate for these operators in the Hilbert space L2⁢(ℝ+;d⁢xx){L^{2}(\mathbb{R}_{+};\frac{dx}{x})} is obtained. A symbol class Λ is defined and it is shown that the product of any two symbols from this class is again in Λ. At the end, commutators for the p.d.o. and their boundedness results are discussed.

This paper comprises two parts. We first investigate an Lp{L^{p}}-type of limiting absorption principle for Schrödinger operators H=-Δ+V{H=-\Delta+V} on ℝn{\mathbb{R}^{n}} (n≥3{n\geq 3}), i.e., we prove the ϵ-uniform L2⁢(n+1)/(n+3){L^{{2(n+1)}/({n+3})}}–L2⁢(n+1)/(n-1){L^{{2(n+1)}/({n-1})}}-estimates of the resolvent (H-λ±i⁢ϵ)-1{(H-\lambda\pm i\epsilon)^{-1}} for all λ>0{\lambda>0} under the assumptions that the potential V belongs to some integrable spaces and a spectral condition of H at zero is satisfied. As applications, we establish a sharp Hörmander-type spectral multiplier theorem associated with Schrödinger operators H and deduce Lp{L^{p}}-bounds of the corresponding Bochner–Riesz operators. Next, we consider the fractional Schrödinger operator H=(-Δ)α+V{H=(-\Delta)^{\alpha}+V} (0<2⁢α<n{0<2\alpha<n}) and prove a uniform Hardy–Littlewood–Sobolev inequality for (-Δ)α{(-\Delta)^{\alpha}}, which generalizes the corresponding result of Kenig–Ruiz–Sogge [20].

Suppose that B is a Brauer p -block of a finite group G with a unique modular character φ. We prove that φ is liftable to an ordinary character of G (which moreover is p -rational for odd p ). This confirms the basic set conjecture for these blocks.

We apply the framework of Rosendal to study the large-scale geometry of the topological groups Diff+k⁡(M1){\operatorname{Diff}_{+}^{k}(M^{1})}, consisting of orientation-preserving Ck{C^{k}}-diffeomorphisms (for 1≤k≤∞{1\leq k\leq\infty}) of a compact 1-manifold M1{M^{1}} (=I{=I} or 𝕊1{\mathbb{S}^{1}}). We characterize the relative property (OB) in such groups: A⊆Diff+k⁡(M1){A\subseteq\operatorname{Diff}_{+}^{k}(M^{1})} has property (OB) relative to Diff+k⁡(M1){\operatorname{Diff}_{+}^{k}(M^{1})} if and only if supf∈Asupx∈M1|logf′(x)|<∞{\sup_{f\in A}\sup_{x\in M^{1}}\lvert\log f^{\prime}(x)|<\infty} and supf∈A⁡supx∈M1⁡|f(j)⁢(x)|<∞{\sup_{f\in A}\sup_{x\in M^{1}}|f^{(j)}(x)|<\infty} for every integer j with 2≤j≤k{2\leq j\leq k}. We deduce that Diff+k⁡(M1){\operatorname{Diff}_{+}^{k}(M^{1})} has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if k<∞{k<\infty}. We show that the groups Diff+1⁡(I){\operatorname{Diff}_{+}^{1}(I)} and Diff+1⁡(𝕊1){\operatorname{Diff}_{+}^{1}(\mathbb{S}^{1})} are quasi-isometric to the infinite-dimensional Banach space C⁢[0,1]{C[0,1]}.

In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: T⁢(f)⁢(x)=1(2⁢π)2⁢n⁢∫ℝ2⁢nei⁢φ⁢(x,ξ,η)⋅a⁢(x,ξ,η)⋅f^⁢(ξ,η)⁢𝑑ξ⁢𝑑η,T(f\/)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}% \cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta, where x=(x1,x2)∈ℝn×ℝn{x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}} and ξ,η∈ℝn∖{0}{\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}}, a⁢(x,ξ,η)∈L∞⁢B⁢Sρm{a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} is the amplitude, and the phase function is of the form φ⁢(x,ξ,η)=φ1⁢(x1,ξ)+φ2⁢(x2,η)\varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta), with φ1,φ2∈L∞⁢Φ2⁢(ℝn×ℝn∖{0})\varphi_{1},\varphi_{2}\in L^{\infty}\Phi^{2}(\mathbb{R}^{n}\times\mathbb{R}^{% n}\setminus\{0\}), and satisfies a certain rough non-degeneracy condition (see (2.2)). The study of these operators are motivated by the Lp{L^{p}} estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the Lp{L^{p}} boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the Lp{L^{p}} boundedness of the more general FIOs with amplitude a⁢(x,ξ,η)∈L∞⁢B⁢Sρm{a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} and non-smooth phase function φ⁢(x,ξ,η){\varphi(x,\xi,\eta)} on x satisfying a rough non-degeneracy condition.

We show obstructions to the existence of a coclosed G2{\mathrm{G}_{2}}-structure on a Lie algebra 𝔤{\mathfrak{g}} of dimension seven with non-trivial center. In particular, we prove that if there exists a Lie algebra epimorphism from 𝔤{\mathfrak{g}} to a six-dimensional Lie algebra 𝔥{\mathfrak{h}}, with the kernel contained in the center of 𝔤{\mathfrak{g}}, then any coclosed G2{\mathrm{G}_{2}}-structure on 𝔤{\mathfrak{g}} induces a closed and stable three form on 𝔥{\mathfrak{h}} that defines an almost complex structure on 𝔥{\mathfrak{h}}. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed G2{\mathrm{G}_{2}}-structures. We also prove that each one of these Lie algebras has a coclosed G2{\mathrm{G}_{2}}-structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed G2{\mathrm{G}_{2}}-structures. The existence of contact metric structures is also studied.

We prove the existence of a solution to the semirelativistic Hartree equation -Δ+m2⁢u+V⁢(x)⁢u=A⁢(x)⁢(W*|u|p)⁢|u|p-2⁢u\sqrt{-\Delta+m^{2}}u+V(x)u=A(x)(W*|u|^{p})|u|^{p-2}u under suitable growth assumption on the potential functions V and A . In particular, both can be unbounded from above.

Let E/F{E/F} be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the irreducible smooth representations of GLn⁢(E){\mathrm{GL}_{n}(E)} that are distinguished by its subgroup GLn⁢(F){\mathrm{GL}_{n}(F)}. One relates this class to representations which come as base change lifts from a quasi-split unitary group over F , while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps, we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose GLn⁢(F){\mathrm{GL}_{n}(F)}-distinction, and thus exhibit the limitations of these two approaches.

We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains. The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in ℝn{\mathbb{R}^{n}} with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.

We give a second main theorem of Nevanlinna theory on complete non-positively curved Kähler manifolds. Its remainder term depends only on Ricci curvature of the manifolds except for the terms depending only on the characteristic functions.

Let p be an odd prime and BP the classifying space of a p -group P with rankp⁢(P)=2{\mathrm{rank}_{p}(P)=2}. Using stable homotopy splitting of BP , we study the decomposition of cohomology Heven⁢(P;ℤ)/p{H^{\mathrm{even}}(P;{\mathbb{Z}})/p}.

Let G be a group, F a field and FG the corresponding group algebra. We consider an involution on FG which is the linear extension of an involution of G , e.g., g*=g-1{g^{*}=g^{-1}} for g∈G{g\in G}. This paper is focused on the characterization of a non-torsion group G provided the group of units U⁢(F⁢G){U(FG)} satisfies a *{*}-group identity. The torsion case was studied in [7], and when *{*} is the classical involution, this problem was solved in the case of symmetric units in [21].

In the last decade the growth in groups focused the interest of many researchers in the domain of additive combinatorics. In this note, we investigate the special case of the Heisenberg group ℋ{\mathcal{H}} of order 3 on different fields R , namely, R=𝐑{R=\mathbf{R}} and R=𝐅p{R=\mathbf{F}_{p}}. Denoting by [x,y,z]{[x,y,z]} the elements of ℋ{\mathcal{H}}, a cube is a subset of the type [A,B,C]{[A,B,C]}, with A,B,C⊂R{A,B,C\subset R}. This study relies to the well-known sum-product problem and its extensions, from which we obtain strong lower bounds for the size of the square [A,B,C]2{[A,B,C]^{2}} of cubes in ℋ{\mathcal{H}}. In the final section, we also consider the problem of counting the subsets of ℋ{\mathcal{H}} of the type [A,B,C]2{[A,B,C]^{2}}.

We consider a semilinear elliptic problem, driven by the Laplacian with Robin boundary condition. We consider a reaction term which is resonant at ±∞{\pm\infty} and at 0. Using variational methods and critical groups, we show that under resonance conditions at ±∞{\pm\infty} and at zero the problem has at least two nontrivial smooth solutions.

Given any finitely aligned higher-rank graph Λ and any unital commutative ring R , the Kumjian–Pask algebra KPR⁢(Λ){\mathrm{KP}_{R}(\Lambda)} is known as the higher-rank generalization of Leavitt path algebras. After the characterization of simple Kumjian–Pask algebras by Clark and Pangalela among others, in this article we focus on the purely infinite simple ones. Briefly, we show that if KPR⁢(Λ){\mathrm{KP}_{R}(\Lambda)} is simple and every vertex of Λ is reached from a generalized cycle with an entrance, then KPR⁢(Λ){\mathrm{KP}_{R}(\Lambda)} is purely infinite. We also prove a dichotomy for simple Kumjian–Pask algebras: If each vertex of Λ is reached only from finitely many vertices and KPR⁢(Λ){\mathrm{KP}_{R}(\Lambda)} is simple, then KPR⁢(Λ){\mathrm{KP}_{R}(\Lambda)} is either purely infinite or locally matritial. This result covers all unital simple Kumjian–Pask algebras.