Volume 26, Issue 5

Let (𝒳,d,μ)$({\mathcal {X}},d,\mu )$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. Under this assumption, the authors establish some weighted norm inequalities for the multilinear Calderón–Zygmund operators via first obtaining some weighted estimates involving the John–Strömberg maximal operators and the John–Strömberg sharp maximal operators.

We characterize all cusp forms among the degree two Siegel modular forms by the growth of their Fourier coefficients. We also give a similar result for Jacobi forms over the group SL 2(ℤ)⋉ℤ2$\mathrm {SL}_{2}(\mathbb {Z}) \ltimes \mathbb {Z}^2$.

Let (𝔤,[p])$(\mathfrak {g},[p])$ be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p > 0. The scheme of tori of maximal dimension of 𝔤 gives rise to a finite group S (𝔤) that coincides with the Weyl group of 𝔤 in case 𝔤 is a Lie algebra of classical type. In this paper, we compute the group S (𝔤) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.

In this paper we present an L p -theory for a class of stochastic partial differential equations (SPDEs in abbreviation) driven by Lévy processes. The SPDEs under consideration can have random coefficients that depend both on the time and space variable. Existence and uniqueness of solutions in various Sobolev spaces are obtained. These Sobolev spaces describe the regularity of the solutions of the SPDEs.

In this paper, we study Clifford–Wolf translations of Finsler spaces. We give a characterization of those Clifford–Wolf translations generated by Killing vector fields. In particular, we show that there is a natural interrelation between the local one-parameter groups of Clifford–Wolf translations and the Killing vector fields of constant length. In the special case of homogeneous Randers spaces, we give some explicit sufficient and necessary conditions for a Killing vector field to have a constant length, in which case the local one-parameter group of isometries generated by the Killing field consist of Clifford–Wolf translations. Finally, we construct explicit examples to explain some of the results of this paper.

In this paper, we use the discrete Littlewood–Paley–Stein analysis to get the duality result of the weighted product Hardy space for arbitrary number of parameters under a rather weak condition on the product weight w∈A∞(ℝn1×⋯×ℝnk)$w \in A_\infty ( {\mathbb {R}}^{n_1} \times \cdots \times {\mathbb {R}}^{n_k})$. We will show that for any k ≥ 2, (Hwp(ℝn1×⋯×ℝnk))*=CMOwp(ℝn1×⋯×ℝnk)${(H^p_w({\mathbb {R}}^{n_1} \times \cdots \times {\mathbb {R}}^{n_k}))^\ast =\textup {CMO}^p_w({\mathbb {R}}^{n_1} \times \cdots \times {\mathbb {R}}^{n_k})}$ (a generalized Carleson measure), and obtain the boundedness of singular integral operators on BMO w$\mathrm {BMO}_w$. Our theorems even when the weight function w = 1 extend the H 1 -BMO duality of Chang–R. Fefferman for the non-weighted two-parameter Hardy space H1(ℝn×ℝm)$H^1({\mathbb {R}}^n\times {\mathbb {R}}^m)$ to Hp(ℝn1×⋯×ℝnk)$H^p({\mathbb {R}}^{n_1} \times \cdots \times {\mathbb {R}}^{n_k})$ for all 0<p≤1$0&lt;p\le 1$ and our weighted theory extends the duality result of Krug–Torchinsky on weighted Hardy spaces Hwp(ℝn×ℝm)$H^p_w({\mathbb {R}}^n\times {\mathbb {R}}^m)$ for w∈Ar(ℝn×ℝm)${w\in A_r({\mathbb {R}}^n\times {\mathbb {R}}^m)}$ with 1≤r≤2$1\le r\le 2$ and r/2<p≤1${r/2&lt;p\le 1}$ to Hwp(ℝn1×⋯×ℝnk)$H^p_w({\mathbb {R}}^{n_1} \times \cdots \times {\mathbb {R}}^{n_k})$ with w∈A∞(ℝn1×⋯×ℝnk)${w \in A_\infty ( {\mathbb {R}}^{n_1} \times \cdots \times {\mathbb {R}}^{n_k})}$ for all 0<p≤1${0&lt;p\le 1}$.

Let F be a field of characteristic p ≠ 2 and G a group without 2-elements having an involution *. Write (FG)+$(FG)^+$ for the set of elements in the group ring F G that are symmetric with respect to the induced involution. In the present note, we show that if G is finite and (FG)+$(FG)^+$ is Lie metabelian, then G is nilpotent. Based on this result, we deduce that if G is torsion, p ≠ 3 and (FG)+$(FG)^+$ is Lie metabelian, then G must be abelian. This extends a result of Levin and Rosenberger.

I present a construction of connected affine algebraic supergroups 𝐆 V associated with simple Lie superalgebras 𝔤 of Cartan type and with 𝔤-modules V . Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups 𝐆 V that I introduced. In particular, the supergroup associated in this way with 𝔤=W(n)$ \mathfrak {g}= W(n) $ and its standard representation is described.

We investigate the algebraic K - and L -theory of the group ring RG , where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit.