H-type Groups Research Articles

Centered Hardy–Littlewood maximal functions on H-type groups revisited

Centered Hardy–Littlewood maximal functions on H-type groups revisited

Injectivity of Spherical Means on H-Type Groups

Injectivity of Spherical Means on H-Type Groups

Schauder Estimates up to the boundary on H-type groups: an approach via the double layer potential

Schauder Estimates up to the boundary on H-type groups: an approach via the double layer potential

Riesz means and bilinear Riesz means on H-type groups

In this paper, we investigate the Riesz means Sδ and the bilinear Riesz means Sα associated to the sublaplacian on H-type groups. We obtain the Lp-boundedness of Sδ by using the restriction theorem on H-type groups. Our result is different from that on Heisenberg groups. We prove that Sα is bounded from Lp1×Lp2 into Lp for 1≤p1,p2≤∞ and 1/p=1/p1+1/p2 when α is larger than a suitable smoothness index α(p1,p2). Because we consider H-type groups with the center dimension larger than one, it is necessary to use some different techniques from that for Heisenberg groups.

The optimal constant in the L2 Folland-Stein inequality on the H-type group

We determine the optimal constant in the L2 Folland-Stein inequality on the H-type group, which partially confirms the conjecture given by Garofalo and Vassilev (2001) [14]. The proof is inspired by the work of Frank and Lieb (2012) [12] and Hang and Wang.

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

This paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper, we proved an abstract result and discussed operators on the Heisenberg group. Here, we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and 0-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

Gelfand pairs and spherical means on H-type groups

Gelfand pairs and spherical means on H-type groups

H-type foliations

With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K-contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound on these structures, we prove a sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian. Then, using a result by Moroianu-Semmelmann [38], we classify the H-type foliations that carry a parallel horizontal Clifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature of these spaces in codimension more than 2.

CT-based Analysis of Vascular Tree Abnormalities in Different Phenotypes of COVID-19 Pneumonia

In this paper, Computed Tomography (CT) images of six confirmed COVID-19 patients were analyzed in order to investigate the physiological abnormalities in the vascular tree in response to the disease-induced hypoxia. The CT images were classified into an L-type and H-type groups based on the cumulative voxel distribution of the CT scan. A 3-Dimentional model of the vascular tree was reconstructed out of each CT image following a computational framework. Then, the Cross-Sectional Area (CSA) of the vessels belonging to each vascular tree was computed. The acquired results were compared against averaged measurements of three healthy subjects that were computed following the same approach. The results showed that as the severity of COVID-19 lean towards H-type phenotype, signs of vasoconstriction in small blood vessels with a CSA less than 10 mm2 tend to decreases, whereas signs of vasodilation in medium to large blood vessels increases. The intensity of dilated blood vessels proximal to consolidated areas of the lung was found to increase significantly as the disease progresses in the lungs. Furthermore, signs of vasoconstriction and vasodilation in the vascular tree were observed in all lobes of the lung in both phenotypes including seemingly healthy lobes. The results in this paper are suggestive of intrapulmonary blood flow shunting towards unaerated areas of the lungs which may lead to a ventilation/perfusion mismatch even at minor cases of COVID-19. The results also suggest that subject-specific regulated use of vasodilating medication may reduce the number of cases that require mechanical ventilation.

Resolvent kernel on H-type groups and a Green kernel for fractional powers of its sub-Laplacian

Resolvent kernel on H-type groups and a Green kernel for fractional powers of its sub-Laplacian

The association between retinal vessel abnormalities and H-type hypertension

BackgroundThis study aimed to investigate the relationship between H-type hypertension and retinal vessel abnormalities.MethodsHypertensive patients were retrospectively enrolled in this study. According to plasma homocysteine (HCY), patients were divided into isolated hypertension and H-type hypertension groups. The diameter of retinal vessels and retinopathy were evaluated by retinal fundus photography. The differences of retinal vessel abnormalities between H-type hypertension and isolated hypertension were investigated by univariate and multivariate regression.ResultsA total of 191 hypertensive patients were included, of which 86 were with isolated hypertension and 105 with H-type hypertension. The H-type hypertension group had a higher ratio of retinopathy(P = 0.004) and higher degree of retinal arteriosclerosis (P = 0.005) than the isolated hypertension group. CRAE (107.47 ± 13.99µ m vs. 113.49 ± 11.72µ m, P = 0.002) and AVR (0.55 ± 0.06 vs. 0.58 ± 0.06, P = 0.001) were smaller in H-type hypertension group than those in isolated hypertension group. Multivariate analysis showed that after adjusting for age, sex, course of hypertension and diabetes, H-type hypertension was an independent risk factor of retinopathy (OR, 2.259; 95%CI, 1.165—4.378; P = 0.016), CRAE (β=-5.669; 95%CI, -9.452—-1.886; P = 0.004), and AVR (β=-0.023; 95%CI, -0.039—-0.007; P = 0.005).ConclusionsH-type hypertension is closely related to more retinal vessel abnormalities than isolated hypertension. Controlling H-type hypertension may reduce the risk of small vascular damage.

Gabor Transform and Donoho–Stark’s U.P. in NA-Groups

Using the basic properties of Damek–Ricci spaces and its Fourier transform, we introduce the Gabor transform and study some of its properties. We will generalize the uncertainty principle of Donoho–Stark to this solvable extension of H-type group.

Geodesic graphs in Randers g.o. spaces

The concept of geodesic graph is generalized from Riemannian geometry to Finsler geometry, in particular to homogeneous Randers g.o. manifolds. On modified H-type groups which admit a Riemannian g.o. metric, invariant Randers g.o. metrics are determined and geodesic graphs in these Finsler g.o. manifolds are constructed. New structures of geodesic graphs are observed.

Well-posedness of a Neumann-type problem on a gauge ball in H-type groups

We discuss the existence and uniqueness of solution for the second boundary value problem of potential theory often referred to as the Neumann problem, on a gauge ball for the canonical sub-Laplacian in H-type groups. In this way we extend the classical results of the problem as well as its generalization to the Heisenberg group.

Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups

Pseudo H-type Lie groups Gr,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓr,s on a vector space V≅R2n. They form a subclass of all 2-step nilpotent Lie groups. Based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let Nr,s denote the Lie algebra corresponding to Gr,s. In the case s>0 a choice of left-invariant vector fields [X1,…,X2n] which generate a complement of the center of Nr,s gives rise to a second order differential operatorΔr,s:=(X12+…+Xn2)−(Xn+12+…+X2n2), which we call ultra-hyperbolic. We prove that Δr,s is locally solvable if and only if r=0. In particular, it follows that Δr,s does not admit a fundamental solution in the space D′(Gr,s) of Schwartz distributions whenever r>0. In terms of classical special functions we present families of fundamental solutions of Δ0,s in the class of tempered distributions S′(G0,s) and study their properties.

AN UNCERTAINTY PRINCIPLE FOR SOLUTIONS OF THE SCHRÖDINGER EQUATION ON -TYPE GROUPS

In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$ -type groups. We first prove that, on $H$ -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$ -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3)95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc.142 (2014), 2101–2118].

Structure of geodesics in weakly symmetric Finsler metrics on H-type groups

Structure of geodesics in weakly symmetric Finsler metrics on H-type groups

Radial processes for sub-Riemannian Brownian motions and applications

We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Ito’s formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng’s type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.

Semi-classical analysis on H-type groups

In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodi fferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account thefinite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.

Correlations of plasma homocysteine with cerebral white matter lesion severity and cognitive impairment in elderly patients with hypertension

Objective To investigate the correlations of plasma homocysteine(Hcy)levels with cerebral white matter lesion(WML)severity and cognitive impairment in elderly hypertension patients. Methods A retrospective case-control study was conducted.Ninty-eight elderly patients with hypertension were enrolled.Based on the presence or absence of hyperhomocysteinemia(hypertension with plasma Hcy levels ≥10 μmol/L was defined as H-type hypertension), patients were divided into two groups: a control group(n=48)and an H-type hypertension group(n=50). The degree of WML was rated by the Fazekas scale using magnetic resonance imaging(MRI)analysis.Neuropsychological examinations including mini-mental state examination(MMSE)and Montreal cognitive assessment(MoCA)were taken to assess cognitive function.The degree of WML and cognitive impairment were compared between the two groups.Correlations of plasma Hcy levels with the degree of WML and cognitive impairment were analyzed. Results Compared with the control group, scores of the Fazekas scale(3.2±1.0 vs.2.5±1.0, P<0.05)and deep white matter hyperintensity on MRI(1.7±0.8 vs.1.3±0.7, P<0.05)increased, and scores of MMSE(24.9±3.3 vs.27.7±1.8, P<0.05)and MoCA(18.6±3.9 vs.25.0±3.0, P<0.05)decreased in the H-type hypertension group.Plasma Hcy levels were positively correlated with the degree of WML(r=0.430, P<0.01)and negatively correlated with cognitive function(r=-0.406 in MMSE and-0.663 in MoCA, P<0.01). Multivariate logistic regression showed that Hcy, fasting blood-glucose, deep WML, low-density lipoprotein cholesterolin, age and systolic pressure were influencing factors for cognitive impairment(P<0.01). Conclusions WML and cognitive function are worse in elderly H-type hypertension patients than in hypertension patients with normal Hcy levels.WML and cognitive impairment are aggravated with increased plasma Hcy levels.The plasma Hcy level may be an effective clinical indicator of cognitive function in elderly people with hypertension. Key words: Hypertension; Cysteine; Leukoencephalopathies; Cognitive disorders