Boundedness of operators generated by fractional heat semigroups related to Schrödinger operators on stratified Lie groups via T1 theorem

The classical Caffarelli–Kohn–Nirenberg inequalities, originally established in Euclidean space in the 1980s, provide a unified framework for interpolation between Sobolev and Hardy inequalities. Their extension to stratified (or homogeneous Carnot) Lie groups began in the early 2000s, motivated by subelliptic analysis and geometric measure theory, revealing rich interactions between group structure, dilation symmetry, and functional inequalities. In this paper, we establish the weighted and logarithmic Caffarelli–Kohn–Nirenberg type inequalities on a stratified Lie group. As a consequence, we can apply them to prove the weighted ultracontractivity of positive strong solutions to the equation: dα · ∂u/∂t = ℒₚ((dα · u)m), where ℒₚ f = ∇H ( |∇H f|p−2 · ∇H f ) is a p-sub-Laplacian, d is a homogeneous norm associated with a fundamental solution of the sub-Laplacian, α ∈ ℝ, and 1 < p < Q.

In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional p-sublaplacian (1<p<infty ) on stratified Lie groups. We also prove the existence of ground state (least energy) solutions to nonlinear subelliptic fractional Schrödinger equation on stratified Lie groups. Different from the proofs of analogous results in the setting of classical Sobolev spaces on Euclidean spaces given by Weinstein (Comm. Math. Phys. 87(4):576-676, 1982/1983) using the rearrangement inequality which is not available in stratified Lie groups, we apply a subelliptic version of vanishing lemma due to Lions extended in the setting of stratified Lie groups combining it with the compact embedding theorem for subelliptic fractional Sobolev spaces obtained in our previous paper (Math. Ann. 388(4):4201-4249, 2024). We also present subelliptic fractional logarithmic Sobolev inequalities with explicit constants on stratified Lie groups. The main results are new for p=2 even in the context of the Heisenberg group.

In this paper, generalised weighted L^p-Hardy, L^p-Caffarelli–Kohn–Nirenberg, and L^p-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg–Pauli–Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted L^2-Rellich type inequality with the boundary term is obtained.

We consider a class of spectral multipliers on stratified Lie groups which generalise the class of Hormander multipliers and include multipliers with an oscillatory factor. Oscillating multipliers have been examined extensively in the Euclidean setting where sharp, endpoint Lp estimates are well known. In the Lie group setting, corresponding Lp bounds for oscillating spectral multipliers have been established by several authors but only in the open range of exponents. In this paper we establish the endpoint Lp(G) bound when G is a stratified Lie group. More importantly we begin to address whether these estimates are sharp.

The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional p-sub-Laplacian over the fractional Folland–Stein–Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the fractional p-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional p-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document} with G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {G}$$\\end{document} being the Heisenberg group.

This paper is devoted to present a version of horizontal weighted Hardy–Rellich type inequality on stratified Lie groups and study some of its consequences. In particular, Sobolev type spaces are defined on stratified Lie groups and proved embedding theorems for these functional spaces.

Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

Jet spaces over \mathbb{R}^n have been shown to have a canonical structure of stratified Lie groups (also known as Carnot groups). We construct jet spaces over stratified Lie groups adapted to horizontal differentiation and show that these jet spaces are themselves stratified Lie groups. Furthermore, we show that these jet spaces support a prolongation theory for contact maps, and in particular, a Bäcklund type theorem holds. A byproduct of these results is an embedding theorem that shows that every stratified Lie group of step s + 1 can be embedded in a jet space over a stratified Lie group of step s .

In this paper, we investigate the subelliptic nonlocal Brezis–Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, [Formula: see text] [Formula: see text] where [Formula: see text] is the fractional [Formula: see text]-sub-Laplacian on a stratified Lie group [Formula: see text] with homogeneous dimension [Formula: see text] [Formula: see text] is an open bounded subset of [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text] is subelliptic fractional Sobolev critical exponent, [Formula: see text] are real parameters and [Formula: see text] is a lower order perturbation of the critical power [Formula: see text]. Utilizing direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter [Formula: see text] is sufficiently small. Additionally, we examine the problem for [Formula: see text], representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for [Formula: see text] even for the Heisenberg group.

Let be a Schrödinger operator on stratified Lie groups, where is the sub‐Laplacian on and V belongs to the reverse Hölder class. In this paper, we introduce a new Campanato‐type space of vanishing mean oscillation associated with L. By Carleson measures related to fractional heat semigroups, we establish an equivalent characterization of . As an application, we prove that the dual of is , where is the completeness of .

Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups

Let \(\mathcal {L}=-{\Delta } + V\) be a Schrodinger operator on stratified Lie group G, where V is a nonnegative potential satisfying the suitable reverse Holder’s inequality. In this paper, we study the boundedness of the second-order Riesz transforms such as \(\mathcal {L}^{-1} \nabla ^{2}\) and \(\mathcal {L}^{-1}(-{\Delta })\) on the spaces of BMO type for weighted case. We generalized the known results to the weighted case and the case of the stratified Lie group. So our results are new, even in the case \(\mathbb {R}^{n}\), and they have some interest in its own right.

We propose analogues of Green’s and Picone’s identities for the p-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Diaz-Saa inequality. Using these we derive a comparison principle and uniqueness of positive solutions to nonlinear hypoelliptic equations on general stratified Lie groups extending to this setting previously known results on Euclidean and Heisenberg groups.

In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in Ruzhansky and Suragan (J Differ Eq 262:1799–1821, 2017) for stratified groups.

We announce the optimal C1+α regularity of the gradient of weak solutions to a class of quasilinear degenerate elliptic equations in nilpotent stratified Lie groups of step two. As a consequence we also prove a Liouville type theorem for 1-quasiconformal mappings between domains of the Heisenberg group Hn. Statement of the results Consider the quasilinear elliptic equation n ∑ i=1 ∂xiAi(x,∇u) = 0, (1) where Ai(x, ξ) : R → R, i = 1, . . . , n, are differentiable functions satisfying λ|η| ≤ ∑n i,j=1 ∂ξjAi(x, ξ)ηiηj ≤ λ−1|η|2, and ∑n i,j=1 ∂xjAi(x, ξ) ≤ C(1 + |ξ|), for every η ∈ R, and almost every x, ξ ∈ R. The sharp C regularity of weak solutions to (1) is one of the pillars on which the modern theory of quasilinear partial differential equations rests. The ideas on which its proof is based form a recurring theme in nonlinear analysis: first use difference quotients to prove that the weak solutions admit second (weak) derivatives, then differentiate the equations and observe that the derivatives of the solution are themselves solutions to some linear partial differential equations, whose coefficients are not very regular. At this point the regularity theory for linear equations with nonsmooth coefficients provides the final step in the proof of the Holder continuity of the gradient of weak solutions to (1). The observation that one could reduce the study of quasilinear equations to studying linear equations with “bad” coefficients goes back to the pioneering work of Morrey, and has been developed by many mathematicians in the last thirty years. Although (1) is a relatively simple elliptic equation, the regularity theorem has far-reaching applications in calculus of variations (see, for instance, [Gi]) and in the theory of quasiconformal mappings in space [G1]. Two natural and inter-related questions arise: Can the ellipticity hypothesis in (1) be somewhat weakened, and still expect regularity of the gradient of the solutions? Is this “new” problem of some geometric relevance? In this announcement we provide positive answers to both questions. Received by the editors March 15, 1996. 1991 Mathematics Subject Classification. Primary 35H05. Alfred P. Sloan Doctoral Dissertation Fellow. c ©1996 American Mathematical Society 60 OPTIMAL REGULARITY 61 In the proof of his famous rigidity theorem [Mo], Mostow introduced quasiconformal mappings in the setting of stratified nilpotent Lie groups [F]. With this name one refers to the class of simply connected Lie groups G endowed with a stratification of the Lie algebra g = V 1 ⊕ · · · ⊕ V , with r ≥ 1 (the step of the group), such that [V , V j ] = V , j = 1, . . . , r − 1, and [V , V ] = 0. The simplest example of a stratified Lie group is the Euclidean space R, with r = 1. A less trivial, and genuinely non-Euclidean example is provided by the Heisenberg group H, n ≥ 1, whose Lie algebra is h = R ⊕ R. The central role played by the Heisenberg group in many problems of complex geometry, representation theory and partial differential equations makes H the prototype par excellence of stratified nilpotent Lie groups. The theory of quasiconformal mappings between domains of the Heisenberg group has been developed recently in a series of papers by Koranyi and Reimann [KR1]–[KR3] and by Pansu [P]. As in the Euclidean case (see [G1] and [R]), this development led to various questions concerning the regularity of weak solutions to a class of quasilinear equations similar to (1). The notion of quasiconformality is a metric one, and in this setting it is related to the Carnot-Caratheodory metric associated to a basis X 1 , . . . , X 1 m, m = m =dim(V ) of V 1 (with our notation we do not distinguish between elements of g and left invariant vector fields). Since this metric is nonisotropic, it is natural to expect some nonisotropic structure in the relevant equations. In order to be more precise we need to recall that the exponential mapping exp : g → G is a diffeomorphism, and so we can use exponential coordinates p = (p1, . . . , p 1 m, p 2 1, . . . , p 2 dim(V 2), . . . ) on G. Let Xu = (X 1u, . . . ,X 1 mu) denote the horizontal gradient of the function u. Consider the equation