Riesz Means of Dirichlet Eigenvalues for the Sub-Laplace Operator on the Engel Group

In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of eigenvalues are given.


Introduction
The Engel group is a Carnot group of step G 3 r  (see [1]), its Lie algebra is generated by the left-invariant vector fields where is a point of .It is easy to see that , , , where and The sub-Laplace operator on G is of the form The Riesz means of Dirichlet eigenvalues for the Laplace operator in the Euclidean space have been extensively studied(see [3][4][5]).In recent years, E. M. Harrell II and L. Hermi in [6] treated the Riesz means and pointed out that: for 0 2 is a nondecreasing function of z, and then the Weyl-type estimates of means of eigenvalues is derived.Jia et al. in [7] extended (1.2), (1.3) to the Heisenberg group.
The main results of this paper are the following.Theorem 1.1 For 0 2 and and therefore Moreover, for all , we have the upper bound Authors in [6] combined the Weyl-type estimates of means of eigenvalues established in [6] and the result in [8] to obtain the Weyl-type estimates of eigenvalues.But it is not easy to extend the result in [8] to the Engel group.The Weyl-type estimates of eigenvalues for (1.1) is the ramp function. Clearly, Similarly to Theorem 1 of [9], we immediately have Lemma 2.2 Denoting the -normalized eigenfunctions of (1.1) by .

The Proof of Theorem 1.1
In this section, we prove Theorem 1.1 and two corollaries.
Proof.Let us use (2.2) and denote the first term on the right-hand side of (2.2) by 4 here we used the symmetry on and in the last step.

j m
Putting the above estimate into (2.2),we have j j q j q j j q j j j m j z m j j q j q z j jq j q z q j j j m j z m j q j jq j j q z q j where we denote , , .
The remainders are discussed similarly to 1).
is a nondecreasing function of .

Corollary 3.1 For all 2
  and where Since (3.7) holds for arbitrary 0 z and the inequality in the left-hand side of (3.6) is valid.
2) By the Berezin-Lieb inequality (see [11]), we have and the inequality in the right-hand side of (3.6) is proved.
2) For 0    and Proof. 1) By Corollary 3.1 we know that for 1 2 Using Theorem 1.1, we have and (3.8) is proved.
2) By Corollary 3.1, it shows that for 0 1 From Theorem 1.1, we see that for 0 1 In the light of (3.12) and (3.13), it obtains and (3.9) is proved.Remark 3.3 Specially, we have and let

 
ind z be the greatest integer i such that For any integer j and , : Using Theorem 1.1, we have that for By the Cauchy-Schwarz inequality, it follows

Proof of Theorem 1.3
We first recall the following definition before proving Theorem 1.3.
; Since the maximizing value of in (5.1) is a nondecreasing function of , it follows that for sufficiently large, the maximizing exceeds is piecewise linear function of , it implies that the maximizing value of in the Legendre transform of is attained at one of the critical values.
Noting that the maximizing value of is a nondecreasing function of , we see , therefore the critical value We point out that (5.7) is sharper than (4.4).In fact, we get from (4.4) that   paper, we investigate the Riesz means of the Dirichlet problem group .Here  is a bounded and noncharacteristics domain in , with smooth boundary G G  .The existence of eigenvalues for (1.1) is from [2].Let us by   R z  denote the Riesz means of order  of the sequence   k  of eigenvalues of (1.1).