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We study the gradient flow of an energy with mixed homogeneity, which is at the interface of Finsler and sub-Riemannian geometry.

The existence and the uniqueness of solutions to some semilinear parabolic equations on homogeneous Lie groups, namely, the Fokker–Planck equation and the Hamilton–Jacobi equation, are addressed. The anisotropic geometry of the state space plays a crucial role in our analysis and creates several issues that need to be overcome. Indeed, the ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space. Finally, the above results are used to obtain the short-time existence of classical solutions to the mean field games system defined on an homogenous Lie group. [ABSTRACT FROM AUTHOR] more...

In this paper we study the asymptotic behavior of solutions to the subelliptic p-Poisson equation as p → + ∞ in Carnot-Carathéodory spaces. In particular, introducing a suitable notion of differentiability, extend the celebrated result of Bhattacharya et al. (Rend Sem Mat Univ Politec Torino Fascicolo Speciale 47:15–68, 1989) and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the ∞ -Laplacian and the Eikonal equation. [ABSTRACT FROM AUTHOR] more...

We investigate the interior regularity to nonlinear subelliptic systems in divergence form with drift term for the case of superquadratic controllable structure conditions in the Heisenberg group. On the basis of a generalization of the A -harmonic approximation technique, C 1 -regularity is established for horizontal gradients of vector-valued solutions to the subelliptic systems with drift term. Specially, our result is optimal in the sense that in the case of Hölder continuous coefficients we directly attain the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set. [ABSTRACT FROM AUTHOR] more...

We provide a representation of the weak solution of the continuity equation on the Heisenberg group H 1 with periodic data (the periodicity is suitably adapted to the group law). This solution is the push forward of a measure concentrated on the flux associated with the drift of the continuity equation. Furthermore, we shall use this interpretation for proving that weak solutions to first order Mean Field Games on H 1 are also mild solutions. [ABSTRACT FROM AUTHOR] more...

In this paper, we prove new Liouville-type properties of fully nonlinear subelliptic equations modelled on the Heisenberg group. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential equations. [ABSTRACT FROM AUTHOR] more...

In this paper, when 1 < p < 2 , we establish the C loc 1 , α -regularity of weak solutions to the degenerate subelliptic p-Laplacian equation ▵ H , p u (x) = ∑ i = 1 6 X i ∗ (| ∇ H u | p - 2 X i u) = 0 on SU(3) endowed with the horizontal vector fields X 1 , ⋯ , X 6 . The result can be extended to a class of compact connected semi-simple Lie group. [ABSTRACT FROM AUTHOR] more...

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 with G being the Heisenberg group. [ABSTRACT FROM AUTHOR] more...

This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group H n . Consequently, several weighted Hardy type, Heisenberg–Pauli–Weyl uncertainty principle and Hardy–Rellich type inqualities are established on H n . Moreover, new weighted Sobolev type embeddings are derived. Finally, an integral inequality for vector fields in a domain of the Heisenberg group is obtained, leading to several specific weighted Hardy type inequalities by making careful choices of vector fields. [ABSTRACT FROM AUTHOR] more...

In this paper, we study the semilinear subelliptic equation where is the self-adjoint Hörmander operator associated with the vector fields X = (Xl, X2, ..., Xm) satisfying the Hörmander's condition, is a Carathéodory function on Ω × ℝ, and Ω is an open bounded domain in ℝn with smooth boundary. Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values, we obtain two kinds of existence results for multiple weak solutions to the problem above. Furthermore, we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations. Compared with the classical elliptic cases, both approaches here have their own strengths in the degenerate cases. This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases. [ABSTRACT FROM AUTHOR] more...

We prove a sharp Gevrey hypoellipticity for the operator D x 2 + x 2 n + 1 D y 2 + x n y m D y 2 , in Ω open neighborhood of the origin in R 2 , where n and m are positive integers. The operator is a non trivial generalization of the Métivier operator studied in Métivier (C R Acad Sci Paris 292:401–404, 1981). However it has a symplectic characteristic manifold and a non symplectic stratum according to the Poisson–Treves stratification. According to Treves conjecture it turns out not to be analytic hypoelliptic. [ABSTRACT FROM AUTHOR] more...

In this paper, the following conformable fractional sub-elliptic equation is investigated: \[ \begin{cases} -\Delta_H^{\alpha} u = f(\xi,u,\nabla_H^\alpha u), & {\rm in} \ \Omega,\\ u > 0, &amp; {\rm in} \ \Omega,\\ u=0, & {\rm on} \ \partial \Omega. \end{cases} \] { − Δ H α u = f (ξ , u , ∇ H α u) , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω. Ω represents the bounded domain in space $ \mathbb {H}^n $ H n , $ \alpha \in (0,1) $ α ∈ (0 , 1) , and f>0 is a bounded function. This paper is organized into three sections. In the first section, we present some of the features of the Heisenberg Laplacian. To prove the solution to the above equation, we use the Arzelà–Ascoli theorem and Bolzano–Weierstrass theorem in the second section. We then prove the uniqueness and cylindrical symmetry of the solution in the third section. [ABSTRACT FROM AUTHOR] more...

In this paper, we study a class of dispersive wave equations on the Heisenberg group H n . Based on the group Fourier transform on H n , the properties of the Laguerre functions and the stationary phase lemma, we establish the decay estimates for a class of dispersive semigroup on H n given by e it ϕ (L) , where ϕ : R + → R is smooth, and L is the sub-Laplacian on H n . Finally, using the duality arguments, we apply the obtained results to derive the Strichartz inequalities for the solutions of some specific equations, such as the fractional Schrödinger equation, the fractional wave equation and the fourth-order Schrödinger equation. [ABSTRACT FROM AUTHOR] more...

In this work, we study the following sub-elliptic equations on Carnot group G with Hardy-type singularity and critical Sobolev–Hardy exponents − Δ G u = λ ψ α | u | 2 ∗ (α) − 2 u d (z) α + β f (z) | u | p − 2 u , z ∈ G , where − Δ G stands for the sub-Laplacian operator on Carnot group G , 0 < α ≤ 2 , and 2 ∗ (α) = 2 (Q − α) Q − 2 is the critical Sobolev–Hardy exponent, Q is the homogeneous dimension with respect to the dilation δ γ naturally associated with − Δ G , d is the natural gauge associated with the fundamental solution of − Δ G on G , ψ = | ∇ G d | and ∇ G is the horizontal gradient associated with Δ G . Through variational methods combined with the theory of genus, we prove that our problems admit infinitely many solutions. [ABSTRACT FROM AUTHOR] more...

A flag domain in ℝ 3 is a subset of ℝ 3 of the form { (x , y , t) : y < A ⁢ (x) } , where A : ℝ → ℝ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn–Laplacian △ ♭ = X 2 + Y 2 in flag domains Ω ⊂ ℝ 3 , with L 2 -boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order L 2 -Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries. [ABSTRACT FROM AUTHOR] more...

In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset Ω\Omega of the CR manifold S3{{\mathbb{S}}}^{3} bounded by the Clifford torus Σ\Sigma is discussed. The Yamabe-type problem of finding a contact form on Ω\Omega which has zero Tanaka-Webster scalar curvature and for which Σ\Sigma has a constant pp-mean curvature is also discussed. more...

The partial and the microlocal regularity are provided via L 2 -estimate and via FBI-transform respectively, for the following generalization of the Métivier operator, Métivier (C R Acad Sci Paris 292:401–404, 1981), D x 2 + x 2 n + 1 D y 2 + x n y m D y 2 , in Ω open neighborhood of the origin in R 2 , where n and m are positive integers. [ABSTRACT FROM AUTHOR] more...

In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group G for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp vanishing order estimate for solutions to stationary Schrödinger equations. [ABSTRACT FROM AUTHOR] more...

Consider the equation { - Δ H 3 ⁢ u = f ⁢ (ξ , u , ∇ H ⁡ u , ∇ H 2 ⁡ u , ∇ H 3 ⁡ u , ∇ H 4 ⁡ u , ∇ H 5 ⁡ u) in ⁢ Ω , u > 0 in ⁢ Ω , u = ∂ ∂ ⁡ ν ⁢ (∇ H 2 ⁡ u) = ∂ ∂ ⁡ ν ⁢ (∇ H 3 ⁡ u) = 0 on ⁢ ∂ ⁡ Ω , where Ω is a domain of the finite-dimensional space ℍ n and f is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution. [ABSTRACT FROM AUTHOR] more...

In this paper, we obtain a finite-time non-blow-up result for the sub-Laplacian heat equations with logarithmic nonlinearity on stratified groups. In our proof, the logarithmic Sobolev-Folland-Stein inequality plays a key role. We also establish a blow-up result at infinite time on stratified groups. [ABSTRACT FROM AUTHOR] more...

We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type, thus providing a microlocal version, in the analytic category, of a result due to Derridj (Pac J Math 302(2):511–543, 2019) concerning the problem of the local regularity for the Gevrey vectors for sums of squares of vector fields with real-valued real analytic/Gevrey coefficients. [ABSTRACT FROM AUTHOR] more...

We present self-contained proofs of the stability of the constants in the volume doubling property and the Poincaré and Sobolev inequalities for Riemannian approximations in Carnot groups. We use an explicit Riemannian approximation based on the Lie algebra structure that is suited for studying nonlinear subelliptic partial differential equations. Our approach is independent of the results obtained in [11]. [ABSTRACT FROM AUTHOR] more...

In this paper, we present blow-up results for solutions to the Dirichlet initial value problem for the p-sup-Laplacian heat operators on the stratified (Lie) groups. A stratified group version of the Poincaré inequality from Ruzhansky and Suragan (J Differ Equ 262:1799–1821, 2017) plays a key role in our proofs. [ABSTRACT FROM AUTHOR] more...

In this paper, we establish some new L 2 − L 2 Carleman estimates for the Baouendi–Grushin operators B γ , in Equation (1). We apply such estimates to obtain: (i) an extension of the Bourgain–Kenig quantitative unique continuation and (ii) the strong unique continuation property for some degenerate sublinear equations. [ABSTRACT FROM AUTHOR] more...

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries. [ABSTRACT FROM AUTHOR] more...

In this article we study singular subelliptic p-Laplace equations and best constants in Sobolev inequalities on Carnot groups. We prove solvability of these subelliptic p-Laplace equations and existence of the minimizer of the corresponding variational problem. It leads to existence of the best constant in the corresponding (q, p)-Sobolev inequality, 0 < q < 1 , 1 < p < ν . [ABSTRACT FROM AUTHOR] more...

In a previous work we proved a spectral multiplier theorem of Mihlin–Hörmander type for two-dimensional Grushin operators - ∂ x 2 - V (x) ∂ y 2 , where V is a doubling single-well potential, yielding the surprising result that the optimal smoothness requirement on the multiplier is independent of V. Here we refine this result, by replacing the L ∞ -Sobolev condition on the multiplier with a sharper L 2 -Sobolev condition. As a consequence, we obtain the sharp range of L 1 -boundedness for the associated Bochner–Riesz means. The key new ingredient of the proof is a precise pointwise estimate in the transition region for eigenfunctions of one-dimensional Schrödinger operators with doubling single-well potentials. [ABSTRACT FROM AUTHOR] more...

We consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primary model covered by our analysis is the non-degenerate sub-elliptic p-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms. more...

In this paper, we discuss two Liouville-type theorems for subelliptic harmonic maps from sub-Riemannian manifolds to Riemannian manifolds. One is the Dirichlet version which states that two subelliptic harmonic maps from a sub-Riemannian manifold with boundary to a regular ball must be same if their restrictions on boundary are same; it is generalized to complete noncompact domains as well. The other is the vanishing-type theorem for finite L p -energy subelliptic harmonic maps on complete noncompact totally geodesic Riemannian foliations which are special sub-Riemannian manifolds. [ABSTRACT FROM AUTHOR] more...

We study semigroups generated by general fractional Ornstein–Uhlenbeck operators acting on L 2 (R n) . We characterize geometrically the partial Gevrey-type smoothing properties of these semigroups, and we sharply describe the blow-up of the associated seminorms for short times, generalizing the hypoelliptic and the quadratic cases. As a byproduct of this study, we establish partial subelliptic estimates enjoyed by fractional Ornstein–Uhlenbeck operators on the whole space by using interpolation theory. [ABSTRACT FROM AUTHOR] more...

Let H n = C n × R be the n-dimensional Heisenberg group, Q = 2 n + 2 be the homogeneous dimension of H n . We establish in this paper that the following sharpened Trudinger–Moser inequalities on the Heisenberg group H n under the homogeneous constraints of the Sobolev norm: sup ‖ ∇ H u ‖ Q Q + ‖ u ‖ Q Q ≤ 1 ∫ H n Φ Q α Q 1 + α ‖ u ‖ Q Q 1 Q - 1 | u | Q Q - 1 d ξ < + ∞ , holds if and only if α < 1 , where Φ Q (t) = e t - ∑ 0 Q - 2 t j j ! . Unlike all the proofs in the literature even in the Euclidean spaces, our proof avoids using the complicated blow-up analysis of the Euler–Lagrange equation associated with the Moser functional. In fact, our proof reveals a surprising fact that the known critical Trudinger–Moser inequality on the entire space is equivalent to seemingly much stronger sharpened Trudinger–Moser inequality on the entire space. Furthermore, we obtain the critical Trudinger–Moser inequality and the Concentration-Compactness Principle under the inhomogeneous constraints on the entire Heisenberg group. Finally, using the method of scaling again, we obtain improved Trudinger–Moser inequality under the inhomogeneous constraints. Our approach is surprisingly simple and general and can be easily applied to the all stratified nilpotent groups and other settings. In particular, our method also gives an alternative and much simpler proof of the corresponding results in the Euclidean space. [ABSTRACT FROM AUTHOR] more...

We use the representation theory of nilpotent Lie groups of step two to find the fundamental solution for the Paneitz operator on the anisotropic quaternionic Heisenberg group. For the isotropic case, we also construct the closed-form of this solution. This result is the extension of the conclusion on the Heisenberg group. [ABSTRACT FROM AUTHOR] more...

This 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. more...