Inequalities on stratified groups

Ruzhansky M., Suragan D. Layer potentials, Green formulae, Kac problem, and refined Hardy inequality on homogeneous Carnot groups, Adv. Math., 308 (2017), 483-528.  offprint (open access)

Abstract. We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green’s first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac’s “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy’s inequality and of the uncertainty principle

Ruzhansky M., Suragan D. On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and p-sub-Laplacian inequalities on stratified groups, J. Differential Equations, 262 (2017), 1799-1821. offprint (open access)

Abstract. In this paper, we present a version of horizontal weighted Hardy–Rellich type and Caffarelli–Kohn–Nirenberg type inequalities on stratiﬁed groups and study some of their consequences. Our results reﬂect on many results previously known in special cases. Moreover, anew simple proof of the Badiale–Tarantello conjecture on the best constant of a Hardy type inequality is provided. We also show a family of Poincaré inequalities as well as inequalities involving the weighted and unweighted p -sub-Laplacians.

Sabitbek B., Suragan D. Horizontal Weighted Hardy-Rellich Type Inequalities on Stratified Lie Groups. Complex Analysis and Operator Theory.  DOI 10.1007/s11785-017-0650-z

Abstract. 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.

Ruzhansky M., Sabitbek B., Suragan D.  Weighted Lp-Hardy and Lp-Rellich inequalities with boundary terms on stratified Lie groupsarxiv

Abstract. In this paper, generalised weighted Lp-Hardy, Lp-Caffarelli-Kohn-Nirenberg, and Lp-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 L2-Rellich type inequality with the boundary term is obtained.

Ruzhansky M.,  Suragan D.,  Yessirkegenov N. Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups, NoDEA Nonlinear Differential Equations Application, 24 (2017), no. 5, 24:56. offprint (open access)

Abstract. In this short paper, we establish a range of Caffarelli–Kohn– Nirenberg and weighted Lp-Sobolev type inequalities on stratified Lie groups. All inequalities are obtained with sharp constants. Moreover, the equivalence of the Sobolev type inequality and Hardy inequality is shown in the L2-case.

Ruzhansky M., Tokmagambetov N., Yessirkegenov N., Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations, arxiv

Abstract. In this paper the dependence of the best constants in Sobolev and Gagliardo-Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of ${\mathbb R}^n,$ Heisenberg, and more general stratified Lie groups. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The orders of these equations can be high depending on the Sobolev space order in the Sobolev or Gagliardo-Nirenberg inequalities, or may be fractional. Applications are obtained also to equations with lower order terms given by different hypoelliptic operators. Already in the case of ${\mathbb R}^n,$ the obtained results extend the classical relations by Weinstein to a wide range of nonlinear elliptic equations of high orders with elliptic low order terms and a wide range of interpolation inequalities of Gagliardo-Nirenberg type. However, the proofs are different from those ivy Weinstein because of the impossibility of using the rearrangement inequalities already in the setting of the Heisenberg group. The considered class of graded groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations.

Ruzhansky M., Yessirkegenov N., Critical Sobolev, Gagliardo-Nirenberg, Trudinger and Brezis-Gallouet-Wainger inequalities, best constants, and ground states on graded groupsarxiv

Abstract. In this paper we investigate critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded groups, which includes the cases of ${\mathbb R}^n,$ 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 groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland’s analysis of Hölder spaces from stratified groups to general homogeneous groups.

Ruzhansky M., Yessirkegenov N., Factorizations and Hardy-Rellich inequalities on stratified groups, arxiv

In this paper, we obtain Hardy, Hardy-Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by the recent work of Gesztesy and Littlejohn. We note that some of the obtained inequalities are new also in the usual Euclidean setting. We also obtain analogues of Gesztesy and Littlejohn’s 2-parameter version of the Rellich inequality on stratified groups and on the Heisenberg group, and a new two-parameter estimate on ${\mathbb R}^n,$  which can be regarded as a counterpart to the Gesztesy and Littlejohn’s estimate.