Riesz Means of Dirichlet Eigenvalues for the Sub-Laplace Operator on the Engel Group ()
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) Noting
for 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
, for
we 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 value
then (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).