Group Hn Research Articles

Spherical means on the Heisenberg group: Stability of a maximal function estimate

Consider the surface measure μ on a sphere in a nonvertical hyperplane on the Heisenberg group ℍn, n ≥ 2, and the convolution f * μ. Form the associated maximal function Mf = supt>0 ∣f * μt∣ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain—Demeter to prove Lp-boundedness of M in an optimal range.

Multicritical hypercubic models

We study renormalization group multicritical fixed points in the ϵ-expansion of scalar field theories characterized by the symmetry of the (hyper)cubic point group HN. After reviewing the algebra of HN-invariant polynomials and arguing that there can be an entire family of multicritical (hyper)cubic solutions with ϕ2n interactions in d=frac{2n}{n-1}-epsilon dimensions, we use the general multicomponent beta functionals formalism to study the special cases d = 3 − ϵ and d=frac{8}{3}-epsilon , deriving explicitly the beta functions describing the flow of three- and four-critical (hyper)cubic models. We perform a study of their fixed points, critical exponents and quadratic deformations for various values of N, including the limit N = 0, that was reported in another paper in relation to the randomly diluted single-spin models, and an analysis of the large N limit, which turns out to be particularly interesting since it depends on the specific multicriticality. We see that, in general, the continuation in N of the random solutions is different from the continuation coming from large-N, and only the latter interpolates with the physically interesting cases of low-N such as N = 3. Finally, we also include an analysis of a theory with quintic interactions in d=frac{10}{3}-epsilon and, for completeness, the NNLO computations in d = 4 − ϵ.

Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group

Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree 0 cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group Hn and Weyl multipliers on Cn acting on Sobolev Spaces.

Effects of high-fat diet intake during perinatal period on reflex-ontogeny and intestinal morphometry of rat offspring

Reflex-ontogeny and intestinal morphometrics were evaluated in Wistar rats whose mothers were fed on a high-fat diet during the perinatal period. Male pups (n=52) formed four experimental groups: NN (pups from mothers with lab chow diet during gestation and lactation); NH (pups from mothers with lab chow diet during pregnancy and high-fat in lactation); HH (pups from mothers with high-fat diet during gestation and lactation); HN (pups from mothers with high-fat diet during pregnancy and lab chow in lactation). The reflex ontogeny, the maturation of physical characteristics and parameters of somatic growth were evaluated during lactation. In addition, the body mass index (BMI), the specific rate of weight gain (SRWG), the Lee index, the weight of the brain and intestinal parameters were analyzed after weaning. High-fat diet during pregnancy (HH and HN groups) delayed the maturation of reflexes and physical characteristics. The high-fat diet affected somatic growth differently, reducing somatic growth parameters in the groups NH and HH and increasing in the HN group. The highest SRWG was found in group HN. SRWG and BMI were reduced in the groups NH and HH. The relative intestinal weight was reduced in the groups NH, HH and HN. The relative length of small intestine was longer in group HN than in group NN. The total height of the mucosa and size of the villous were lower in group HH than in group NN. In conclusion, high-fat diet promoted negative consequences for the development of the nervous and enteric systems of the offspring.

Estimates for the linear viscoelastic damped wave equation on the Heisenberg group

In the paper, we investigate the linear wave equation with viscoelastic damping on the Heisenberg group Hn. Basing on the group Fourier transform on Hn and on the properties of the Hermite functions, we derive some L2(Hn)−L2(Hn) estimates with additional L1(Hn) regularity on initial data for the solution and its higher-order horizontal gradients.

Invariant metric on the extended Siegel–Jacobi upper half space

The real Jacobi group GnJ(R), defined as the semidirect product of the Heisenberg group Hn(R) with the symplectic group Sp(n,R), admits a matrix embedding in Sp(n+1,R). The modified pre-Iwasawa decomposition of Sp(n,R) allows us to introduce a convenient coordinatization Sn of GnJ(R), which for G1J(R) coincides with the S-coordinates. Invariant one-forms on GnJ(R) are determined. The formula of the 4-parameter invariant metric on G1J(R) obtained as sum of squares of 6 invariant one-forms is extended to GnJ(R), n∈N. We obtain a three parameter invariant metric on the extended Siegel–Jacobi upper half space X̃nJ≈XnJ×R by adding the square of an invariant one-form to the two-parameter balanced metric on the Siegel–Jacobi upper half space XnJ=GnJ(R)U(n)×R.

Notion of [formula omitted]-orientability for surfaces in the Heisenberg group [formula omitted

This paper aims to define and study a notion of orientability in the Heisenberg sense (H-orientability) for the Heisenberg group Hn. In particular, we define such notion for H-regular 1-codimensional surfaces. Analysing the behaviour of a Möbius Strip in H1, we find a 1-codimensional H-regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, H-orientability implies Euclidean-orientability. As a consequence, we conclude that non-H-orientable H-regular surfaces exist in H1.

Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups

In this paper, assume that L=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.

Decay estimates for the linear damped wave equation on the Heisenberg group

This paper is devoted to the derivation of L2 - L2 decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group Hn, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further L1(Hn) regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of Hn and on the properties of the Hermite functions that form a maximal orthonormal system for L2(Rn) of eigenfunctions of the harmonic oscillator.

Hyperprogressive disease and its clinical impact in patients with recurrent and/or metastatic head and neck squamous cell carcinoma treated with immune-checkpoint inhibitors: Korean cancer study group HN 18-12.

Although immune-checkpoint inhibitors (ICIs) have emerged as therapeutic options for recurrent and/or metastatic head and neck squamous cell carcinoma (R/M-HNSCC), concerns have been raised on exceptional acceleration of tumor growth during treatment with ICIs, a condition described as hyperprogressive disease (HPD). This study examined the incidence, potential predictors, and clinical impact of HPD in R/M-HNSCC. We retrospectively collected data of patients with R/M-HNSCC treated with ICIs between January 2013 and June 2018 from 11 medical centers in Korea. HPD was defined as tumor growth kinetics ratio (TGKr) > 2, which was calculated by comparing TGK on ICIs with that before treatment with ICIs. Of 125 patients, 68 (54.4%) obtained progressive disease as their best responses (progressors). HPD was identified in 18 (26.5% of progressors, 14.4% of total) patients. Relatively younger age, primary tumor of oral cavity, and previous locoregional irradiation were significant predictors of HPD according to multivariable analysis (p = 0.040, 0.027, and 0.015, respectively). Compared to patients without HPD, patients with HPD had significantly shorter median progression-free survival (PFS) (1.2 vs. 3.4months, p < 0.001) and overall survival (OS) (3.4 vs. 10.7months, p = 0.047). However, interestingly, HPD did not significantly affect the therapeutic benefit of post-ICIs chemotherapy. Younger patients with oral cavity cancer or prior treatment with locoregional radiotherapy could be regarded potential risk groups for HPD in patients with R/M-HNSCC treated with ICIs. Although HPD could consistently predict poorer survival outcomes, patients who experienced HPD with ICIs should not be excluded from the subsequent salvage chemotherapy treatments.

Existence and nonexistence results for Kohn Laplacian with Hardy-Littlewood-Sobolev critical exponents

In this article, we are study the following Dirichlet problem with Choquard type non linearity−ΔHu=au+(∫Ω|u(η)|Qλ⁎|η−1ξ|λdη)|u|Qλ⁎−2uinΩ,u=0 on ∂Ω, where Ω is a smooth bounded subset of the Heisenberg group HN,N∈N with C2 boundary and ΔH is the Kohn Laplacian on the Heisenberg group HN. Here, Qλ⁎=2Q−λQ−2,Q=2N+2 and a is a positive real parameter. We derive the Brezis-Nirenberg type result for the above problem. Moreover, we also prove the regularity of solutions and nonexistence of solutions depending on the range of a.

Fractional Adams–Moser–Trudinger type inequality on Heisenberg group

The aim of this paper is to establish singular fractional Adams–Moser–Trudinger inequality for both bounded and unbounded domains in the Heisenberg group Hn. We first establish fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.12) and then using this inequality and Hardy–Littlewood–Sobolev inequality adapted to the result of R.O’Neil(1963), we establish singular fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.13). We also establish singular fractional Adams–Moser–Trudinger type inequality in Hn, using the approach of N.Lam and G.Lu (2012, 2013). The main idea of our approach is that any function in higher order fractional Sobolev space in Heisenberg group can be represented in terms of Riesz potential and then using techniques from harmonic analysis and kernel properties of the associated operator, we establish fractional Adams–Moser–Trudinger type inequality in Hn. In this paper, our approach is quite simple and free from symmetrization arguments. As an applications of our theorems, we establish the existence of solution to the following class of problems Tαu=f(ξ,u)|ξ|a+b(ξ)|u|γ−1uinΩ,u=0inHn∖Ω,where Ω is a bounded subset of Hn of class C0,1 with bounded boundary and 0≤a<Q,f satisfies either the subcritical exponential growth or critical exponential growth condition and b is a small L2-perturbation, that is, there exists a small η>0 with 0<‖b‖L2(Ω)<η,0≤γ<1 and α=Q2.

Fredholm Property of Integral Operators with Homogeneous Kernels of Compact Type in the L2 Space on the Heisenberg Group

We consider the Heisenberg group ℍn with Koranyi norm. In the space L2(ℍn), we introduce integral operators with homogeneous kernels of compact type and multiplicatively weakly oscillating coefficients. For the unital C*-algebra $$\mathfrak{W}$$(ℍn) generated by such operators, we construct a symbolic calculus and in terms of this calculus formulate necessary and sufficient conditions for an operator in $$\mathfrak{W}$$(ℍn) to be a Fredholm operator.

Globalization of group cohomology in the sense of Alvares-Alves-Redondo

Recently E.R. Alvares, M.M. Alves and M.J. Redondo introduced a cohomology for a group G with values in a module over the partial group algebra Kpar(G), which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action α of G on a (unital) algebra A we consider A as a Kpar(G)-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in A. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that A is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group Hparn(G,A) is isomorphic to the usual cohomology group Hn(G,M(B)), where M(B) is the multiplier algebra of B and B is the algebra under the enveloping action of α.

Existence for (p,q) critical systems in the Heisenberg group

AbstractThis paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (𝓢) in the Heisenberg group ℍn, driven by general (p,q) elliptic operators of Marcellini types. The study of (𝓢) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p,q) systems.

Morrey spaces for Schrödinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groups

Let L=−ΔHn+V be a Schrödinger operator on the Heisenberg group Hn, where ΔHn is the sublaplacian on Hn and the nonnegative potential V belongs to the reverse Hölder class RHs with s∈[Q/2,∞). Here Q=2n+2 is the homogeneous dimension of Hn. For given α∈(0,Q), the fractional integral operator associated with the Schrödinger operator L is defined by Iα=L−α/2. In this article, the author introduces the Morrey space Lρ,∞p,κ(Hn) and weak Morrey space WLρ,∞p,κ(Hn) associated with L, where (p,κ)∈[1,∞)×[0,1) and ρ(⋅) is an auxiliary function related to the nonnegative potential V. The relation between the fractional integral operator and the maximal operator on the Heisenberg group is established. From this, the author further obtains the Adams (Morrey-Sobolev) inequality on these new spaces. It is shown that the fractional integral operator Iα=L−α/2 is bounded from Lρ,∞p,κ(Hn) to Lρ,∞q,κ(Hn) with 0<α<Q, 1<p<Q/α, 0<κ<1−(αp)/Q and 1/q=1/p−α/Q(1−κ), and bounded from Lρ,∞1,κ(Hn) to WLρ,∞q,κ(Hn) with 0<α<Q, 0<κ<1−α/Q and 1/q=1−α/Q(1−κ). Moreover, in order to deal with the extreme cases κ≥1−(αp)/Q, the author also introduces the spaces BMOρ,∞(Hn) and Cρ,∞β(Hn), β∈(0,1] associated with L. In addition, it is proved that Iα is bounded from Lρ,∞p,κ(Hn) to BMOρ,∞(Hn) under κ=1−(αp)/Q, and bounded from Lρ,∞p,κ(Hn) to Cρ,∞β(Hn) under κ>1−(αp)/Q and β=α−(1−κ)Q/p.

Higher orbitals of quizzy quantum group actions

The hyperoctahedral group HN is known to have two natural liberations: the “good” one HN+, which is the quantum symmetry group of N segments, and the “bad” one O¯N, which is the quantum symmetry group of the N-hypercube. We study here this phenomenon, in the general “quizzy” framework, which covers the various liberations and twists of HN,ON. Our results include: (1) an interpretation of the embedding O¯N⊂S2N+, as corresponding to the antisymmetric representation of ON, (2) a study of the liberations of HN, notably with the result <HN+,O¯N>=ON+, and (3) a comparison of the k-orbitals for the inclusions HN⊂HN+ and HN⊂O¯N, for k∈N small.

Higher Triple Massey products and symbols

Fix an arbitrary prime p. Let F be a field containing a primitive p-th root of unity, with absolute Galois group GF, and let Hn denote its mod p cohomology group Hn(GF,Z/pZ). The triple Massey product of weight (n,k,m)∈N3 is a partially defined, multi-valued function 〈⋅,⋅,⋅〉:Hn×Hk×Hm→Hn+k+m−1. In this work we prove that for an arbitrary prime p, any defined 3MP of weight (n,1,m), where the first and third entries are symbols, contains zero; and that any defined 3MP of weight (1,k,1), where the middle entry is a symbol, contains zero.

1123TiP - A randomized phase II study of cisplatin plus radiotherapy versus durvalumab plus radiotherapy followed by adjuvant durvalumab versus durvalumab plus radiotherapy followed by adjuvant tremelimumab and durvalumab in intermediate risk, HPV-positive, locoregionally advanced oropharyngeal squamous cell cancer (LA-OSCC) (Canadian Cancer Trials Group HN.9)

1123TiP - A randomized phase II study of cisplatin plus radiotherapy versus durvalumab plus radiotherapy followed by adjuvant durvalumab versus durvalumab plus radiotherapy followed by adjuvant tremelimumab and durvalumab in intermediate risk, HPV-positive, locoregionally advanced oropharyngeal squamous cell cancer (LA-OSCC) (Canadian Cancer Trials Group HN.9)

Spherical analysis on homogeneous vector bundles

Given a Lie group G, a compact subgroup K and a representation τ∈Kˆ, we assume that the algebra of End(Vτ)-valued, bi-τ-equivariant, integrable functions on G is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the rôle of the algebra of G-invariant differential operators on the homogeneous bundle Eτ over G/K. In particular, we observe that, under the above assumptions, (G,K) is a Gelfand pair and show that the Gelfand spectrum for the triple (G,K,τ) admits homeomorphic embeddings in some Cn.In the second part, we develop in greater detail the spherical analysis for G=K⋉H with H nilpotent. In particular, for H=Rn and K⊂SO(n) and for H equal to the Heisenberg group Hn and K⊂U(n), we characterize the representations τ∈Kˆ giving a commutative algebra.