## 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

Zbl 0653.35068
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