Schrödinger Operators with Reverse Hölder Class Potentials in the Dunkl Setting and Their Hardy Spaces

For a normalized root system R in $${\mathbb {R}}^N$$ R N and a multiplicity function $$k\ge 0$$ k ≥ 0 let $${\mathbf {N}}=N+\sum _{\alpha \in R} k(\alpha )$$ N = N + ∑ α ∈ R k ( α ) . We denote by $$dw({\mathbf {x}})=\varPi _{\alpha \in R}|\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}\,d{\mathbf {x}}$$ d w ( x ) = Π α ∈ R | ⟨ x , α ⟩ | k ( α ) d x the associated measure in $${\mathbb {R}}^N$$ R N . Let $$L=-\varDelta +V$$ L = - Δ + V , $$V\ge 0$$ V ≥ 0 , be the Dunkl–Schrödinger operator on $${\mathbb {R}}^N$$ R N . Assume that there exists $$q >\max (1,\frac{{\mathbf {N}}}{2})$$ q > max ( 1 , N 2 ) such that V belongs to the reverse Hölder class $$\mathrm{{RH}}^{q}(dw)$$ RH q ( d w ) . We prove the Fefferman–Phong inequality for L. As an application, we conclude that the Hardy space $$H^1_{L}$$ H L 1 , which is originally defined by means of the maximal function associated with the semigroup $$e^{-tL}$$ e - t L , admits an atomic decomposition with local atoms in the sense of Goldberg, where their localizations are adapted to V.

Boundary value problems for the Dunkl Laplacian

Let Δk be the Dunkl Laplacian on ℜd associated with a reflection group W and a multiplicity function k. The purpose of this paper is to establish the existence and the uniqueness of a positive solution on the unit ball B of ℜd to the following boundary value problem:Δku = φu in B and u = ƒ on ∂BWe distinguish two cases of nonnegative perturbation φ: trivial and nontrivial.

Almost Gentle Algebras and their Trivial Extensions

In this paper we define almost gentle algebras, which are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and, as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one is the trivial extension of an almost gentle algebra. We show that a hypergraph is associated with every almost gentle algebra A, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Among other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.

Corwin–Greenleaf multiplicity function for compact extensions of the Heisenberg group

Let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group and [Formula: see text] a closed subgroup of [Formula: see text] acting on [Formula: see text] by automorphisms such that [Formula: see text] is a Gelfand pair. Let [Formula: see text] be the semidirect product of [Formula: see text] and [Formula: see text]. Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text], and [Formula: see text] the natural projection. For coadjoint orbits [Formula: see text] and [Formula: see text], we denote by [Formula: see text] the number of [Formula: see text]-orbits in [Formula: see text], which is called the Corwin–Greenleaf multiplicity function. In this paper, we give two sufficient conditions on [Formula: see text] in order that [Formula: see text] For [Formula: see text], assuming furthermore that [Formula: see text] and [Formula: see text] are admissible and denoting respectively by [Formula: see text] and [Formula: see text] their corresponding irreducible unitary representations, we also discuss the relationship between [Formula: see text] and the multiplicity [Formula: see text] of [Formula: see text] in the restriction of [Formula: see text] to [Formula: see text]. Especially, we study in Theorem 4 the case where [Formula: see text]. This inequality is interesting because we expect the equality as the naming of the Corwin–Greenleaf multiplicity function suggests.

Uniform bounds in F-finite rings and their applications

This dissertation establishes uniform bounds in characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the Fsignature function. From this we establish that the F-signature function is lower semicontinuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, a result originally proven by Ilya Smirnov.

Corwin–Greenleaf multiplicity function for compact extensions of ℝn

Let G = K ⋉ ℝn, where K is a compact connected subgroup of O(n) acting on ℝn by rotations. Let 𝔤 ⊃ 𝔨 be the respective Lie algebras of G and K, and pr : 𝔤* → 𝔨* the natural projection. For admissible coadjoint orbits [Formula: see text] and [Formula: see text], we denote by [Formula: see text] the number of K-orbits in [Formula: see text], which is called the Corwin–Greenleaf multiplicity function. Let π ∈ Ĝ and [Formula: see text] be the unitary representations corresponding, respectively, to [Formula: see text] and [Formula: see text] by the orbit method. In this paper, we investigate the relationship between [Formula: see text] and the multiplicity m(π, τ) of τ in the restriction of π to K. If π is infinite-dimensional and the associated little group is connected, we show that [Formula: see text] if and only if m(π, τ) ≠ 0. Furthermore, for K = SO(n), n ≥ 3, we give a sufficient condition on the representations π and τ in order that [Formula: see text].