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).