$$\alpha $$-Modulation Spaces for Step Two Stratified Lie Groups

We define and investigate $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }({\mathbb {R}}^n)$$ M p , q s , α ( R n ) that act as intermediate spaces between the modulation spaces ($$\alpha = 0$$ α = 0 ) in time-frequency analysis and the Besov spaces ($$\alpha = 1$$ α = 1 ) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases $$\alpha = 0,1$$ α = 0 , 1 where the spaces $$M_{p,q}^{s}(G)$$ M p , q s ( G ) and $${\mathcal {B}}_{p,q}^{s}(G)$$ B p , q s ( G ) have non-standard translation and dilation symmetries. Moreover, we show that the spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings $${\mathcal {Q}}(G)$$ Q ( G ) underlying the $$\alpha = 0$$ α = 0 case $$M_{p,q}^{s}(G)$$ M p , q s ( G ) allows for the existence of geometric embeddings $$\begin{aligned} F:M_{p,q}^{s}({\mathbb {R}}^k) \longrightarrow {} M_{p,q}^{s}(G), \end{aligned}$$ F : M p , q s ( R k ) ⟶ M p , q s ( G ) , as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper.

Critical Gagliardo–Nirenberg, Trudinger, Brezis–Gallouet–Wainger inequalities on graded groups and ground states

In this paper, we investigate critical Gagliardo–Nirenberg, Trudinger-type and Brezis–Gallouet–Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of [Formula: see text], Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo–Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo–Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland’s analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

Time-Dependent Wave Equations on Graded Groups

In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder (or more regular) non-negative propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$ p -evolution equations for higher order operators on ${{\mathbb{R}}}^{n}$ R n or on groups, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In particular, we describe an interesting local loss of regularity phenomenon depending on the step of the group (for stratified groups) and on the order of the considered operator.

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})}.

Weighted higher order exponential type inequalities in metric spaces and applications

In this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.