Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

Borderline gradient estimates at the boundary in Carnot groups

In this article, we prove the continuity of the horizontal gradient near a C1,Dini non-characteristic portion of the boundary for solutions to $\Gamma ^{0,{\rm Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below, where the scalar term is in scaling critical Lorentz space L(Q, 1) with Q being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the $\Gamma ^{1,\alpha }$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.

A two-weight Sobolev inequality for Carnot-Carathéodory spaces

Let $$X = \{X_1,X_2, \ldots ,X_m\}$$ X = { X 1 , X 2 , … , X m } be a system of smooth vector fields in $${{\mathbb R}^n}$$ R n satisfying the Hörmander’s finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory space $$\mathbb G$$ G associated to system X$$\begin{aligned} \left( \frac{1}{\int _{B_R} K(x)\; dx} \int _{B_R} |u|^{t} K(x) \; dx \right) ^{1/t} \le C\, R \left( \frac{1}{\int _{B_R}\frac{1}{K(x)} \; dx} \int _{B_R} \frac{|X u|^2}{K(x)} \; dx \right) ^{1/2}, \end{aligned}$$ 1 ∫ B R K ( x ) d x ∫ B R | u | t K ( x ) d x 1 / t ≤ C R 1 ∫ B R 1 K ( x ) d x ∫ B R | X u | 2 K ( x ) d x 1 / 2 , where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt’s class $$A_2$$ A 2 and Gehring’s class $$G_{\tau }$$ G τ , where $$\tau $$ τ is a suitable exponent related to the homogeneous dimension.

Approximation of critical regularity functions on stratified homogeneous groups

Let [Formula: see text] be a stratified homogeneous group with homogeneous dimension [Formula: see text] and whose Lie algebra is generated by the left-invariant vector fields [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text]. We prove that for any function [Formula: see text] there exists a function [Formula: see text] such that [Formula: see text] [Formula: see text] where [Formula: see text] is the largest integer smaller than [Formula: see text] and [Formula: see text] is a positive constant depending only on [Formula: see text]. Here, [Formula: see text] is a homogeneous Triebel–Lizorkin type space adapted to [Formula: see text]. This generalizes earlier results of Bourgain, Brezis [New estimates for eliptic equations and Hodge type systems, J. Eur. Math. Soc. 9(2) (2007) 277–315] and of Bousquet, Russ, Wang, Yung [Approximation in fractional Sobolev spaces and Hodge systems, J. Funct. Anal. 276(5) (2019) 1430–1478] in the Euclidean case and answers an open problem in the latter reference.

Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups

Let {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} be a Schrödinger operator on the stratified Lie group {\mathbb{G}}, where {{\Delta}_{\mathbb{G}}} is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class {B_{q_{0}}} with {q_{0}>\mathcal{Q}/2} and {\mathcal{Q}} is the homogeneous dimension of {\mathbb{G}}. In this article, by Campanato type spaces {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}, we introduce Hardy type spaces associated with {\mathcal{L}} denoted by {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} and prove the atomic characterization of {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})}. Further, we obtain the following duality relation:\Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{\mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)<p<1\quad\text{for}\ \delta=\min\{1,2-\mathcal{Q}/q_{0}\}.The above relation enables us to characterize {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup, respectively. Also, we obtain two classes of perturbation formulas associated with the semigroups related to {\mathcal{L}}. As applications, we obtain the boundedness of the Littlewood–Paley function and the Lusin area function on {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}.

Translation-like actions of nilpotent groups

We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion-free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other. As Lipschitz injections need not be bi-Lipschitz embeddings, this is a strengthening of a classical result of Pansu in the context of groups of the same homogeneous dimension.

Nodal solutions for the fractional Yamabe problem on Heisenberg groups

We prove that the fractional Yamabe equation ${\rm {\cal L}}_\gamma u = \vert u \vert ^{((4\gamma )/(Q-2\gamma ))}u$ on the Heisenberg group ℍn has [n + 1/2] sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where ${\rm {\cal L}}_\gamma $ denotes the CR fractional sub-Laplacian operator on ℍn, Q = 2n + 2 is the homogeneous dimension of ℍn, and $\gamma \in \bigcup\nolimits_{k = 1}^n [k,((kQ)/Q-1)))$. Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere S2n + 1 combined with a suitable Hebey-Vaugon-type compactness result and group-theoretical constructions for special subgroups of the unitary group U(n + 1).

The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

The role of negative emotionality and impulsivity in depressive/anxiety disorders and alcohol dependence

BackgroundMuch is still unclear about the role of personality in the structure of common psychiatric disorders such as depressive/anxiety disorders and alcohol dependence. This study will therefore examine whether various traits of negative emotionality and impulsivity showed shared or specific associations with these disorders.MethodCross-sectional data were used from the Netherlands Study of Depression and Anxiety (NESDA), including individuals with no DSM-IV psychiatric disorder (n = 460), depressive/anxiety disorder only (i.e. depressive and/or anxiety disorder; n = 1398), alcohol dependence only (n = 32) and co-morbid depressive/anxiety disorder plus alcohol dependence (n = 358). Aspects of negative emotionality were neuroticism, hopelessness, rumination, worry and anxiety sensitivity, whereas aspects of impulsivity included disinhibition, thrill/adventure seeking, experience seeking and boredom susceptibility.ResultsAspects of negative emotionality formed a homogeneous dimension, which was unrelated to the more heterogeneous construct of impulsivity. Although all aspects of negative emotionality were associated with alcohol dependence only, associations were much stronger for depressive/anxiety disorder only and co-morbid depressive/anxiety disorder with alcohol dependence. The results for impulsivity traits were less profound and more variable, with disinhibition and boredom susceptibility showing modest associations with both depressive/anxiety disorder and alcohol dependence, whereas low thrill/adventure seeking and high disinhibition were more strongly related with the first and the latter, respectively.ConclusionsOur results suggest that depressive/anxiety disorder and alcohol dependence result from shared as well as specific aetiological pathways as they showed the same associations with all aspects of negative emotionality, disinhibition and boredom susceptibility as well as specific associations with thrill/adventure seeking and disinhibition.