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On the generalization of Hörmander’s inequality. (English) Zbl 1215.35177

Let \(P=P^*\) be a transversally elliptic operator in the class \(OPN^{m,k}(X,\Sigma)\) introduced by J. Sjöstrand [Ark. Mat. 12, 85–130 (1974; Zbl 0317.35076)], where \(m\in\mathbb R\), \(k\in\mathbb N\), \(X\) is an open subset of \(\mathbb R^n\), \(\Sigma\) a smooth conic and connected submanifold of \(T^*X\setminus O\). The authors suppose that the canonical \(z\)-form \(\sum d\xi_j\wedge dx_j\) has constant rank on \(\Sigma\) and the canonical 1-form \(\sum \xi_jdx_j\) does not vanish on \(T_\rho\Sigma\) for any \(\rho\in\Sigma\). Under some conditions on the “ground energy” of \(P\), they prove that for any compact subset \(K\subset X\), there exists \(C_K>0\) such that \[ (P_{u,u})\geq -C_k\| u\|^2_{m/2-(k+2)/4}\quad \forall u\in{\mathcal C}^\infty_0(K). \] When \(k=2\), this result reduces to a well-known inequality of L. Hörmander [J. Anal. Math. 32, 118–196 (1977; Zbl 0367.35054)]. For general (even) \(k\geq 2\) and \(\Sigma\) an involutive or a symplectic manifold, this is a result of C. Parenti and A. Parmeggiani [Commun. Partial Differ. Equations 25, No. 3–4, 457–506 (2000; Zbl 0986.35146)].

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
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[1] Bony J. M., Bull. Soc. Math. France 122 pp 77– (1994)
[2] DOI: 10.1002/cpa.3160270502 · Zbl 0294.35020 · doi:10.1002/cpa.3160270502
[3] Boutet de Monvel L., Astérisque 34 pp 93– (1976)
[4] Brummelhuis R. G. M., C. R. Acad. Sci. Paris, Sér. I Math. 310 pp 95– (1990)
[5] DOI: 10.1081/PDE-100107452 · Zbl 1007.35111 · doi:10.1081/PDE-100107452
[6] DOI: 10.1007/BF02788080 · Zbl 1007.35112 · doi:10.1007/BF02788080
[7] DOI: 10.1080/03605307908820127 · Zbl 0432.35082 · doi:10.1080/03605307908820127
[8] Helffer B., Mémoires de la SMF 51 pp 13– (1977)
[9] Helffer B., Astérisque 112 (1984)
[10] DOI: 10.1081/PDE-120005851 · Zbl 1028.35176 · doi:10.1081/PDE-120005851
[11] DOI: 10.1007/BF02803578 · Zbl 0367.35054 · doi:10.1007/BF02803578
[12] Hörmander L., The Analysis of Linear Partial Differential Operators (1983)
[13] Maniccia L., Annali dell’Università di Ferrara pp 263– (2003)
[14] DOI: 10.1007/BF02383640 · Zbl 0211.17102 · doi:10.1007/BF02383640
[15] DOI: 10.1080/03605308308820269 · Zbl 0522.35069 · doi:10.1080/03605308308820269
[16] DOI: 10.1090/S0002-9939-04-07486-6 · Zbl 1057.35108 · doi:10.1090/S0002-9939-04-07486-6
[17] Parenti C., Boll. Un. Mat. It. 1 pp 187– (1998)
[18] Parenti C., Comm. Part. Diff. Eqs. 25 pp 457– · Zbl 0986.35146 · doi:10.1080/03605300008821521
[19] DOI: 10.1007/BF02786644 · Zbl 1055.35153 · doi:10.1007/BF02786644
[20] DOI: 10.1081/PDE-200037706 · Zbl 1072.35213 · doi:10.1081/PDE-200037706
[21] DOI: 10.1007/BF02384749 · Zbl 0317.35076 · doi:10.1007/BF02384749
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