ABSTRACT

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.

1. Introduction

The Engel group is a Carnot group of step (see [1]), its Lie algebra is generated by the left-invariant vector fields

where is a point of. It is easy to see that

and. So the Lie algebra of is

where and

. The sub-Laplace operator on is of the form.

In the paper, we investigate the Riesz means of the Dirichlet problem

(1.1)

in the Engel group. Here is a bounded and noncharacteristics domain in, with smooth boundary. The existence of eigenvalues for (1.1) is from [2]. Let us by denote the Riesz means of order of the sequence of eigenvalues of (1.1).

The Riesz means of Dirichlet eigenvalues for the Laplace operator in the Euclidean space have been extensively studied(see [3-5]). In recent years, E. M. Harrell II and L. Hermi in [6] treated the Riesz means of order of on the bounded domain and pointed out that: for and,

(1.2)

and is a nondecreasing function of; for

and,

(1.3)

and 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 and, we have

(1.4)

(1.5)

and is a nondecreasing function of z; for

and, we have

(1.6)

(1.7)

and is a nondecreasing function of z.

Theorem 1.2 Suppose that, then

(1.8)

and therefore

(1.9)

(1.10)

Moreover, for all, we have the upper bound

(1.11)

Theorem 1.3 For, we have

(1.12)

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) still are open questions.

This paper is arranged as follows. In Section 2 the definition of Riesz means and Lemmas are described; Section 3 is devoted to the proof of Theorem 1.1. The proof of Theorem 1.2 is appeared in Section 4. In Section 5 the proof of Theorem 1.3 is given.

2. Preliminaries

Definition 2.1 For an increasing sequence of real numbers and, the Riesz means of order of is defined by

where is the ramp function.

Clearly,

(2.1)

Similarly to Theorem 1 of [9], we immediately have

Lemma 2.2 Denoting the -normalized eigenfunctions of (1.1) by, let

for Then for each fixed, we have

(2.2)

Lemma 2.3 ([10]) Let and, then

where

3. 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. Applying Lemma 2.3 it follows

here we used the symmetry on and in the last step.

Putting the above estimate into (2.2), we have

(3.1)

where we denote

(3.2)

Since is a complete orthonormal set, it follows

and

Returning to (3.1) with them, it yields

(3.3)

Since

we have

namely,

(3.4)

We consider three cases: 1); 2) and 3).

1). In this case, it sees and

Since, it follows

and therefore

Substituting this into (3.4), we obtain

and

Now (1.4) is proved.

Using (2.1), we have

and (1.5) is proved.

Since

it follows that is a nondecreasing function of.

2). Now, so and

(3.5)

Then

and

Substituting this into (3.4), we obtain

namely,

and (1.4) is proved.

The remainders are discussed similarly to 1).

3). In this case, so and

Substituting this into (3.4), we have

and (1.6) is proved.

Noting (2.1), it implies

and (1.7) is proved.

Similarly,

thus is a nondecreasing function of.

Corollary 3.1 For all and,

(3.6)

where.

Proof. 1) Notingfor any, it follows from Theorem 1.1 that for all,

So

(3.7)

Since (3.7) holds for arbitrary, it yields

Due to

we see that when , it gets

For, we have

and the inequality in the left-hand side of (3.6) is valid.

2) By the Berezin-Lieb inequality (see [11]), we have

Notice that is nondecreasing to, it follows

and the inequality in the right-hand side of (3.6) is proved.

Corollary 3.2 1) For and,

(3.8)

2) For and,

(3.9)

Proof. 1) By Corollary 3.1 we know that for and, it holds

(3.10)

Using Theorem 1.1, we have

(3.11)

Combining (3.10) and (3.11), it follows

and (3.8) is proved.

2) By Corollary 3.1, it shows that for and, it holds

(3.12)

From Theorem 1.1, we see that for,

(3.13)

In the light of (3.12) and (3.13), it obtains

Noting that, forwe have

and (3.9) is proved.

Remark 3.3 Specially, we have

(3.14)

(3.15)

4. Proof of Theorem 1.2

Denote

and let be the greatest integer such that.

Let, it implies that and, so

(4.1)

For any integer j and, it implies, and

Using Theorem 1.1, we have that for,

or

(4.2)

By the Cauchy-Schwarz inequality, it follows

and

(4.3)

Proof of Theorem 1.2 1) Substituting into (4.2) and noticing (4.3), we have

and (1.8) is proved.

2) We take (1.8) into (3.14) to obtain

and (1.9) is proved.

3) Combining (1.8) and (3.15), it implies

and (1.10) is proved.

4) If, then (1.11) is clearly valid; if

, then (1.10) shows by letting that

So (1.11) is proved and Theorem 1.2 is proved.

Corollary 4.1 We have

and

(4.4)

5. Proof of Theorem 1.3

We first recall the following definition before proving Theorem 1.3.

Definition 5.1 If is superlinear in z as, then its Legendre transform is defined by

(5.1)

Remark 5.2 If for all, then for all; Since the maximizing value of in (5.1) is a nondecreasing function of, it follows that for sufficiently large, the maximizing exceeds.

Proof of Theorem 1.3 From (1.9), we have

(5.2)

Now let us calculate. Since

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.

In fact if, then

Noting that the maximizing value of is a nondecreasing function of, we see, therefore the critical value.

It is easy to check and

(5.3)

Next we calculate. Noting

and letting

we know. By, it solves

(5.4)

Therefore

(5.5)

Taking (5.3) and (5.5) into (5.2), we have

(5.6)

By (5.4), it has

From Theorem 1.2, , so.

Then it follows that if w is restricted to the valuethen (5.6) is valid.

Meanwhile, for any, we can always find an integer such that and

If and approaches to from belowthen we obtain from (5.5) that

Therefore

and Theorem 1.3 is proved.

Remark 5.3 If we let, then

(5.7)

We point out that (5.7) is sharper than (4.4). In fact, we get from (4.4) that

and

But is always valid, so (5.7) is sharper than (4.4).

REFERENCES