Solvability of second-order left-invariant differential operators on the Heisenberg group

Ricci, Fulvio

Journées équations aux dérivées partielles (2000), article no. 15, 10 p.

We present some recent results, obtained jointly with Detlef Müller, on solvability of operators of the form

$$\sum _{j,k=1}^{2n}{a}_{jk}{V}_j{V}_k+i\alpha U$$

where the $V_j$ are left-invariant vector fields on the Heisenberg group, such that $[V_j,V_{j+n}]=U$ ($1\le j\le n$) are the only nontrivial relations, and $A=\left(a_{jk}\right)$ is a complex symmetric matrix with semi-definite real part. The presentation also contains references on the work done in the past few years in this area.

[BG] R. Beals and P.C. Greiner, Calculus on heisenberg manifolds, Annals of Math. Studies, Princeton Univ. Press, 1988. | MR | Zbl

[DPR] F. De Mari, M.M. Peloso, and F. Ricci, Analysis of second order differential operators with complex coefficients on the heisenberg group, J. Reine Angew. Math. 464 (1995), 67-96. | MR | Zbl

[Fo] G. Folland, Harmonic analysis in phase space, Annals of Math. Studies, Princeton Univ. Press, 1989. | MR | Zbl

[FS] G. Folland and E.M. Stein, Estimates for the ∂b complex and analysis on the heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. | MR | Zbl

[G] A. Grigis, Hypoellipticité pour une classe d'opérateurs pseudodifférentiels à caractéristiques doubles et paramétrix associées, C.R. acad. Sci. Paris A280 (1975), 1063-1065. | MR | Zbl

[Hw] R. Howe, The oscillator semigroup, Proc. Symp. Pure Appl. Math. 48 (1988), 61-132. | MR | Zbl

[Hö1] L. Hörmander, A class of hypoelliptic pseudodifferential operators with double characteristics, Math. Ann. 217 (1975), 165-188. | MR | Zbl

[Hö2] L. Hörmander, Symplectic classification of quadratic forms, and general mehler formulas, Math. Zeitschr. 219 (1995), 413-449. | MR | Zbl

[KM] G. Karadzhov and D. Müller, Local solvability for a class of second order complex coefficient differential operators on the heisenberg group h2, preprint Mathematisches Seminar, Kiel (1999).

[MPR1] D. Müller, M.M. Peloso, and F. Ricci, On the solvability of homogeneous left-invariant differential operators on the heisenberg group, J. Funct. Anal. 148 (1997), 368-383. | MR | Zbl

[MPR2] D. Müller, M.M. Peloso, and F. Ricci, On local solvability for complex coefficient differential operators on the heisenberg group, J. Reine Angew. Math. 513 (1999), 181-234. | MR | Zbl

[MR1] D. Müller and F. Ricci, Analysis of second order differential operators on the heisenberg group. i, Invent. Math. 101 (1990), 545-582. | MR | Zbl

[MR2] D. Müller and F. Ricci, Analysis of second order differential operators on the heisenberg group. ii, J. Funct. Anal. 108 (1992), 296-346. | MR | Zbl

[MT] D. Müller and C. Thiele, Normal forms for involutive complex hamiltonian matrices under the real symplectic group, J. Reine Angew. Math. 513 (1999), 97-114. | MR | Zbl

[MZ] D. Müller and Z. Zhang, Local solvability for positive combinations of generalized sub-laplacians on the heisenberg group, preprint Mathematisches Seminar, Kiel (1999). | Zbl

[P] R. Penney, Non-elliptic laplace equations on nilpotent lie groups, Ann. Math. 119 (1984), 309-385. | MR | Zbl

[Sj] Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. för Math. 12 (1974), 85-130. | MR | Zbl