## A new hypoelliptic operator on almost CR manifolds.(English)Zbl 1262.32042

Summary: The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $$Q_L$$ on almost CR manifolds equipped with a real structure. The operator acts on all $$(p,q)$$-forms, but when restricted to $$(p,0)$$-forms and $$(p,n)$$-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only a finite type condition is needed to have hypoellipticity on those forms. However, outside these forms $$Q_L$$ may fail to be hypoelliptic, as it is shown in the example of the Heisenberg group $$\mathbb{H}^{5}$$.

### MSC:

 32V99 CR manifolds 32W50 Other partial differential equations of complex analysis in several variables
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### References:

 [1] Ali, S.T. and Engliš, M.: Quantization methods: a guide for physicists and analysts. Rev. Math. Phys. 17 (2005), no. 4, 391-490. · Zbl 1075.81038 [2] Baouendi, M.S., Rothschild, L.P. and Trèves, F.: CR structures with group action and extendability of CR functions. Invent. Math. 82 (1985), 359-396. · Zbl 0598.32019 [3] Beals, R. and Greiner, P.C.: Calculus on Heisenberg manifolds . Annals of Math. Studies 119 . Princeton University Press, Princeton, NJ, 1988. · Zbl 0654.58033 [4] Christ, M.: Regularity properties of the $$\overline\partial_b$$ equation on weakly pseudoconvex CR manifolds of dimension 3. J. Amer. Math. Soc. 1 (1988), 587-646. JSTOR: · Zbl 0671.35017 [5] Christ, M.: On the $$\overline\partial_b$$ equation for three-dimensional CR manifolds. In Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989) , 63-82. Proc. Symposia Pure Math. 52 , Part 3. Amer. Math. Soc., Providence, RI, 1991. · Zbl 0747.32009 [6] Connes, A. and Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5 (1995), no. 2, 174-243. · Zbl 0960.46048 [7] Fefferman, C. and Kohn, J.J.: Hölder estimates on domains of complex dimension two and on three dimensional CR manifolds. Adv. in Math. 69 (1988), 233-303. · Zbl 0649.35068 [8] Fefferman, C., Kohn, J.J. and Machedon, M.: Hölder estimates on CR manifolds with a diagonalizable Levi form. Adv. Math. 84 (1990), 1-90. · Zbl 0763.32004 [9] Folland, G.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13 (1975), 161-207. · Zbl 0312.35026 [10] Folland, G. and Stein, E.M.: Estimates for the $$\bar \partial_b$$ complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974), 429-522. · Zbl 0293.35012 [11] Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147-171. · Zbl 0156.10701 [12] Kaup, W. and Zaitsev, D.: On symmetric Cauchy-Riemann manifolds. Adv. Math. 149 (2000), no. 2, 145-181. · Zbl 0954.32016 [13] Koenig, K.D.: On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian. Amer. J. Math. 124 (2002), no. 1, 129-197. · Zbl 1014.32031 [14] Kohn, J.J.: Boundaries of complex manifolds. In 1965 Proc. Conf. Complex Analysis (Minneapolis, 1964) , 81-94. Springer, Berlin. · Zbl 0166.36003 [15] Kohn, J.J.: Estimates for $$\overline\partial_b$$ on pseudoconvex CR manifolds. In Pseudodifferential operators and applications (Notre Dame, Ind., 1984) , 207-217. Proc. Symposia Pure Math. 43 . Amer. Math. Soc., Providence, RI, 1985. · Zbl 0571.58027 [16] Kohn, J.J. and Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold. Ann. of Math. (2) 81 (1965), 451-472. JSTOR: · Zbl 0166.33802 [17] Nagel, A. and Stein, E.M.: The $$\overline\partial_b$$-complex on decoupled boundaries in $$\mathbb C^n$$. Ann. of Math. (2) 164 (2006), no. 2, 649-713. · Zbl 1126.32031 [18] Ponge, R.: Géométrie spectrale et formules d’indices locales pour les variétés CR et contact. C.R. Acad. Sci. Paris Sér. I Math. 332 (2001), 735-738. · Zbl 0987.58012 [19] Rothschild, L.P. and Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976), 247-320. · Zbl 0346.35030 [20] Rothschild, L.P. and Tartakoff, D.: Parametrices with $$C^\infty$$-error for $$\square_b$$ and operators of Hörmander type. In Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) , 255-271. Lecture Notes in Pure and Appl. Math. 48 . Dekker, New York, 1979. · Zbl 0409.35021 [21] Taylor, M.E.: Noncommutative microlocal analysis. I. Mem. Amer. Math. Soc. 52 (1984), no. 313. · Zbl 0554.35025 [22] Woodhouse, N.: Geometric quantization . Oxford Mathematical Monographs. The Clarendon Press, Oxford Univ. Press, New York, 1992. · Zbl 0747.58004
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