×

On Riesz means of eigenfunction expansions for the Kohn-Laplacian. (English) Zbl 0697.35102

In the recent article of C. D. Sogge [Ann. Math. (2) 126, 439–447 (1987; Zbl 0653.35068)] some classical results by L. Hörmander about the \(L_p\)-convergence of the Riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds were improved. In the reviewed paper analogous problem for the model non-elliptic but subelliptic differential operator, the Kohn-Laplacian operator on the Heisenberg group is treated.
Let \(L=-\sum^n_{j=1}(Z_j\bar Z_j+\bar Z_jZ_j)\), where \(Z_j=\partial /\partial z_j+(i\bar z_j)(\partial /\partial t)\), be the so called Kohn-Laplacian on the Heisenberg group \(H_n\) of dimension \(2n+1\).
\(H_n\) can be realised as \(H_n=\mathbb C^n\times\mathbb R\) with multiplication given by \((z,t)(w,u)=(z+w, t+u+2\, \text{Im}(z)\cdot\bar w)\). Let \(\overline L=\int^\infty_0\lambda \,dE(\lambda)\) be the spectral decomposition of the closure \(\overline L\) of \(L\). The Riesz means of index \(\alpha >0\) of a Schwartz class function \(f\in\mathcal S(H_n)\) is defined by
\[ S^\alpha_N(f)=\int^\infty_0 (1-(\lambda /N))^\alpha_+ \,dE(\lambda)f,\quad N>0. \] Define \(\| f\|_{p,r}=\| f\|_{L_p (\mathbb C^n,L^r(\mathbb R))}\) \((1\leq p,r\leq \infty)\) for functions \(f\) on \(H_n\) and write \(L^{(p,r)}(H_n)\) for the Banach space \(L^p(\mathbb C^n,L^r(\mathbb R))\). The main result is the following:
Theorem. Let \(\alpha (p,r):=| 1/r-1/2| +2n\cdot | 1/p-1/2|\). If \(\alpha >\alpha (p,r)\), and if either \(1\leq p<2\) and \(1\leq r\leq p\), or \(2<p\leq \infty\) and \(p\leq r\leq \infty\), or if \(r=p=2\), then \(S^\alpha_N\) extends to a bounded operator on \(L^{(p,r)}(H_n)\) for every \(N>0\), and
\[ \| S^\alpha_N(f)\|_{(p,r)}\leq C\| f\|_{(p,r)}\quad (f\in L^{(p,r)}(H_n)) \tag{i} \] for some constant \(C>0\) which is independent of \(N\), and
\[ S^\alpha_N(f)\to f\quad\text{in}\; L^{(p,r)}(H_n)\quad\text{as}\;N\to \infty \tag{ii} \] for every \(f\in L^{(p,r)}(H_n)\).

MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35J70 Degenerate elliptic equations
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 0653.35068
PDF BibTeX XML Cite
Full Text: DOI Crelle EuDML