On Riesz Mean Inequalities for Subelliptic Laplacian

Until now, the eigenvalue estimations of Laplacian on the bounded Euclidean domain have been extensively studied (see [1-5]). In recent years, some academics have already started to pay attention to the Heisenberg group n H , such as P. Levy-Bmhl [6], D. Müller [7], P. C. Niu [8], G. Jia [9] and so on. The Heisenberg group plays an important role in several branches of mathematics such as representation theory, harmonic analysis, several complex variables, partial differential equations and quantum mechanics. In the past decades research on Heisenberg sub-Laplacian has achieved considerable progress. But the problem of the invariant differential operator eigenvalue for the Heisenberg group, did not be studied deeply. In this paper, the Riesz mean inequalities of eigenvalues for the subelliptic Laplacian is treated. And some differential inequalities and difference inequalities are established. The outline of the paper is as follows. In Section 2, we first recall some definitions and the lemmas that will be used in the following, and then establish the trace formula of eigenvalues. Main results and their proofs will be given in Section 3.

The Heisenberg group plays an important role in several branches of mathematics such as representation theory, harmonic analysis, several complex variables, partial differential equations and quantum mechanics.In the past decades research on Heisenberg sub-Laplacian has achieved considerable progress.But the problem of the invariant differential operator eigenvalue for the Heisenberg group, did not be studied deeply.
In this paper, the Riesz mean inequalities of eigenvalues for the subelliptic Laplacian is treated.And some differential inequalities and difference inequalities are established.
The outline of the paper is as follows.In Section 2, we first recall some definitions and the lemmas that will be used in the following, and then establish the trace formula of eigenvalues.Main results and their proofs will be given in Section 3.

Preliminaries and Trace Formula
Let n H denote Heisenberg group which is a Lie group that has algebra , with a nonabelian group law x y t x y t x x y y t t y x x y For every , , , , , The Lie algebra is generated by the left invariant vector fields 2 And T t

Remark 2.1
It is easy to see that i X , , are skew symmetric operators, and The subelliptic Laplacian is defined as By the definitions and properties of i X and i Y , it is easy to see that n H  is invariant with respect to lefttranslations.
Let us concern with the eigenvalue problem , in , 0, on .
where  is a bounded domain of the Heisenberg group  n H with smooth boundary.By [8], we see that the Dirichlet problem (2.4) has a discrete spectrum on a Hilbert space with Inner product denoted ,   , and its eigenvalues by and orthonormalize its eigenfunctions Here, denotes the Hilbert space of the functions such that denotes the closure of .
For the sake of simplicity, let be a form There will be a distinguished subset , , , is the complement of j J , and j J P , c j J P will be the corresponding spectral projections.We shall be interested in traces of

If
is an increasing sequence of real numbers, for , the Riesz mean of order where    here I is an identity operator.In fact, we have : , , , : To derive out Theorem 2.1, we need the following lemma.

Proof of Theorem 2.1.
Observe that because   1 According to [10] the formal commutator identity By [2], we obtain And similarly , , Summing of the (2.10) and (2.11), we obtain , we get (2.6), and the proof of the Theorem is completed.

Riesz Means Inequalities
In this section, we derive differential inequalities and difference inequalities for the Riesz means . Here   are ordered eigen- is a nondecreasing function with respect to z .For 2   and is a nondecreasing function with respect to z .
Proof.Let the first term on the right of (2.7) be , , : By Lemma 2.1, the expression can be simplified to By symmetry in j m  , extending the sum to all subtracting the same quantity from the final term in (2.7), we find where , , .

 
Substituting this into (3.8),we have which is equivalent to (3.1), also we can get (3.2).On the other hand is a nondecreasing function with respect to Then we have , which is (3.4). Similarly is a nondecreasing function with respect to z .This completes the proof of the theorem.
. Substituting it into (3.8), and so on.