Volume 28, Issue 1

The theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars and shown to imply novel properties of Barnes beta distributions. The applications are given to the meromorphic extension of the Selberg integral as a function of its dimension and the scaling invariance of the underlying probability distribution. This probability distribution in the critical case is described and conjectured to be the distribution of the derivative martingale. The Jacobi triple product is interpreted probabilistically resulting in an approximation of the Riemann xi function by the Mellin transform of the logarithm of a limit of Barnes beta distributions.

Atomic decomposition plays an important role in establishing the boundedness of operators on function spaces. Let 0<p,q<∞${0&lt;p, q&lt;\infty }$ and α=(α1,α2)∈ℝ2${\alpha =(\alpha _1,\alpha _2)\in \mathbb {R}^2}$. In this paper, we introduce multi-parameter Triebel–Lizorkin spaces F˙pα,q(ℝm)${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ associated with different homogeneities arising from the composition of two singular integral operators whose weak ( 1 , 1 ) boundedness was first studied by Phong and Stein [Amer. J. Math. 104 (1982), 141–172]. We then establish its atomic decomposition which is substantially different from that for the classical one-parameter Triebel–Lizorkin spaces. As an application of our atomic decomposition, we obtain the necessary and sufficient conditions for the boundedness of an operator T on the multi-parameter Triebel–Lizorkin type spaces. In the special case of α1=α2=0${\alpha _1=\alpha _2=0}$, q = 2 and 0<p≤1${0&lt;p\le 1}$, our spaces F˙pα,q(ℝm)${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ coincide with the Hardy spaces H com p${H^p_{\mathrm {com}}}$ associated with the composition of two different singular integrals (see []). Therefore, our results also give an atomic decomposition of H com p${H^p_{\mathrm {com}}}$. Our work appears to be the first result of atomic decomposition in the Triebel–Lizorkin spaces in the multi-parameter setting.

We prove L p estimates for a class of Littlewood–Paley functions on homogeneous groups under a sharp integrability condition of the kernel. The results obtained in the present paper essentially improve some known results.

We establish the Triebel–Lizorkin boundedness for a class of singular integral operators associated to surfaces of revolution, {(t,ψ(t)):t∈ℝn}${\lbrace {(t,\psi (t))} : t\in \mathbb {R}^n\rbrace }$, with rough kernels Ω, provided that the corresponding maximal function along the plane curve {(t,ψ(t)):t∈ℝ}${\lbrace {(t,\psi (t))} : t\in \mathbb {R} \rbrace }$ is bounded on Lp(ℝ2)${L^p(\mathbb {R}^2)}$. We treat kernel functions belonging to a generalized function space relating to Grafakos and Stefanov's ℱα(Sn-1)${\mathcal {F}_\alpha ({S}^{n-1})}$ spaces.

We describe the cohomology ring H*(J2;𝔽3)${H^*(J_2;\mathbb {F}_3)}$ both as a subring of H*(3+1+2;𝔽3)${H^*(3^{1+2}_+;\mathbb {F}_3)}$ and with an abstract presentation. We also give its Poincaré series. Our calculational tool is the first author's spectral sequence for the cohomology of a finite group with a strongly closed p -subgroup.

Actions of the modular group on categories are constructed. A hyperelliptic involution is used to convert the braid representations underlying Khovanov homology to representations of the modular group.

We motivate and study the reduced Koszul map, relating the invariant bilinear maps on a Lie algebra and the third homology. We show that it is concentrated in degree 0 for any grading in a torsion-free abelian group, and in particular it vanishes whenever the Lie algebra admits a positive grading. We also provide an example of a 12-dimensional nilpotent Lie algebra whose reduced Koszul map does not vanish. In an appendix, we reinterpret the results of Neeb and Wagemann about the second homology of current Lie algebras, which are closely related to the reduced Koszul map.

We prove a twisted Bass–Heller–Swan decomposition for both the connective and the non-connective K -theory spectrum of additive categories.

Let A be a finite-dimensional algebra with two simple modules. It is shown that if the derived category of A admits a stratification with simple factors being the base field k , then A is derived equivalent to a quasi-hereditary algebra. As a consequence, if further k is algebraically closed and A has finite global dimension, then A is either derived simple or derived equivalent to a quasi-hereditary algebra.

A locally compact group G is said to be approximable by discrete subgroups if there exists a sequence of discrete subgroups (Hn)n∈ℕ$(H_n)_{n\in \mathbb {N}}$ of G such that for any nonempty open set O of G , there exists an integer k such that O∩Hn≠⌀$O\cap H_n \ne \varnothing $, for every n ≥ k . It has been proved that the identity component of a Lie group approximable by discrete subgroups is nilpotent. In this paper we shall extend this result to pro-Lie groups. In particular, every connected locally compact group approximable by discrete subgroups is nilpotent. An application of discrete approximation is given.

Interchanging the character and co-character groups of a torus T over a field k introduces a contravariant functor T → T ∨ . Interpreting ρ : T (ℂ) → ℂ × , half the sum of positive roots for T , a maximal torus in a simply connected semi-simple group G (over ℂ) using this duality, we get a co-character ρ ∨ : ℂ × → T ∨ (ℂ) for which ρ ∨ ( e 2π i / h ) ( h the Coxeter number) is the Coxeter conjugacy class of the dual group G ∨ (ℂ). This point of view gives a rather transparent proof of a theorem of Kostant on the character values of irreducible finite-dimensional representations of G (ℂ) at the Coxeter conjugacy class: the proof amounting to the fact that in G ∨ sc (ℂ), the simply connected cover of G ∨ (ℂ), there is a unique regular conjugacy class whose image in G ∨ (ℂ) has order h (which is the Coxeter conjugacy class).