Inequalities on graded groups

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

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 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 Schrodinger 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 H\”older spaces from stratified groups to general homogeneous groups.

Ruzhansky M., Taranto C., Time-dependent wave equations on graded groups, arxiv

Abstract. In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent H\”older 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-evolution equations for higher order operators, 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 loss of regularity phenomenon depending on the step of the group and on the order of the considered operator.

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 ℝn, 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 ℝn, 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., Tokmagambetov N., Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups, arxiv

Abstract. In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on $\mathbb{R}^n$ and on the Heisenberg group, such as powers of the Laplacian or of the sub-Laplacian. In addition, we establish a new family of Gagliardo-Nirenberg inequalities on graded Lie groups that play a crucial role in the proof but which are also of interest on their own: if G is a graded Lie group of homogeneous dimension Q and $a>0, 1<r<\frac{Q}{a}$ , and $1\leq p \leq q \leq \frac{rQ}{Q-\alpha r}$ , then we have the following Gagliardo-Nirenberg type inequality

$\| u\|_{L^q(G)} \leq \| u \|^s_{\dot{L}_a(G)}\|u\|^{1-s}_{L^p(G)}$

for $s = \left( \frac{1}{p}-\frac{1}{q} \right) \left( \frac{a}{Q} + \frac{1}{p}-\frac{1}{r} \right)^{-1} \in [0,1]$ provided that $\frac{a}{Q}+\frac{1}{p}-\frac{1}{r} \neq 0$ , where $\dot{L}_a^r$ is the homogeneous Sobolev space of order a over $L^r$ . If $\frac{a}{Q}+\frac{1}{p}-\frac{1}{r}=0$ , we have $p=q=\frac{rQ}{Q-ar}$ , and then the above inequality holds for any $0 \leq s \leq 1$ .

Cardona D., Ruzhansky M., Littlewood-Paley theorem, Nikolskii inequality, Besov spaces, Fourier and spectral multipliers on graded Lie groups, arxiv

Abstract. In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups.

Fischer V., Ruzhansky M., Fourier multipliers on graded Lie groups, arxiv

Abstract. We study the Lp-boundedness of Fourier multipliers defined on graded nilpotent Lie groups via their group Fourier transform. We show that H\”ormander type conditions on the Fourier multipliers imply Lp-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper.

Fischer V., Ruzhansky M., Sobolev spaces on graded groups, Ann. Inst. Fourier, 67 (2017), 1671-1723. offprint (open access), arxiv, link

Abstract. We study the Lp-properties of positive Rockland operators and define Sobolev spaces on general graded groups. This generalises the case of sub-Laplacians on stratified groups studied by G. Folland. We show that the defined Sobolev spaces are actually independent of the choice of a positive Rockland operator. Furthermore, we show that they are interpolation spaces and establish duality and Sobolev embedding theorems in this context.

Cardona D., Ruzhansky M., Multipliers for Besov spaces on graded Lie groups, C. R. Acad. Sci. Paris, 355 (2017), 400-405. offprint (open access), link

Abstract. In this note, we give embeddings and other properties of Besov spaces, as well as spectral and Fourier multiplier theorems, in the setting of graded Lie groups. We also present a Nikolskii-type inequality and the Littlewood–Paley theorem that play a role in this analysis and are also of interest on their own.

Fischer V., Ruzhansky M., A pseudo-differential calculus on graded nilpotent Lie groups, in Fourier Analysis, pp. 107-132, Trends in Mathematics, Birkhauser, 2014. arxiv, link

Abstract. In this paper, we present first results of our investigation regarding symbolic pseudo-differential calculi on nilpotent Lie groups. On any graded Lie group, we define classes of symbols using difference operators. The operators are obtained from these symbols via the natural quantisation given by the representation theory. They form an algebra of operators which shares many properties with the usual Hormander calculus.

Fischer V., Ruzhansky M., Lower bounds for operators on graded Lie groups, C. R. Acad. Sci. Paris, Ser I, 351 (2013), 13-18. arxiv, link

Abstract. In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.

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