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Non-isotropic Gevrey classes in domains of finite type in \(\mathbb{C}^ 2\). (Classes de Gevrey non isotropes dans les domaines de type fini de \(\mathbb{C}^ 2\).) (French) Zbl 0820.32005

The paper develops the theory of Grevrey classes for domains of finite type in \(\mathbb{C}^ 2\). The nonisotropic Gevrey class \(G_{NI}^{1 + \alpha}\) consists of those functions in the Gevrey class \(G^{1 + \alpha}\) which have additional Gevrey regularity in the complex tangential directions. In particular, holomorphic functions in \(G^{1 + \alpha}\) are in \(G_{NI}^{1 + \alpha}\). After developing the necessary machinery of adapted coordinates, pseudoballs, and regions of approach, the author proves a Taylor theorem for functions in \(G_{NI}^{1 + \alpha}\). Then he examines certain domains of finite type having a polynomial defining function: for such domains he obtains formulations of the definition of \(G_{NI}^{1 + \alpha}\) and Taylor’s theorem using globally defined vector fields on the boundary. The author also constructs special partitions of unity by functions in \(G_{NI}^{1 + \alpha}\) and uses these to extend a Whitney jet on a compact set in the boundary to a function in \(G_{NI}^{1 + \alpha}\).
The author refers to a second paper [Math. Z. 212, No. 4, 555-580 (1993; Zbl 0790.32008)] in which he uses these results to study Gevrey interpolation in pseudoconvex domains of finite type in \(\mathbb{C}^ 2\).

MSC:

32A40 Boundary behavior of holomorphic functions of several complex variables
32T99 Pseudoconvex domains

Citations:

Zbl 0790.32008
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References:

[1] T. Bloom, C peak functions for pseudoconvex domains of strict type, Duke Math. J. 45 (1978), 133–147. · Zbl 0376.32014 · doi:10.1215/S0012-7094-78-04510-6
[2] J. Bruna and J.M. Ortega Interpolation by holomorphic functions smooth to the boundary in the unit ball of \(\mathbb{C}\)N, Math. Ann. 274 (1986), 527–575. · Zbl 0585.32018 · doi:10.1007/BF01458591
[3] J. Bruna, An extension theorem of Whitney type for non quasi-analytic classes of functions, J. London Math. Soc. 22 (1980), 495–505. · Zbl 0419.26010 · doi:10.1112/jlms/s2-22.3.495
[4] D. Catlin, Estimates for invariant metrics on weakly pseudoconvex domains in \(\mathbb{C}\)2, Math. Z. 200 (1989), 429–466. · Zbl 0661.32030 · doi:10.1007/BF01215657
[5] J. Chaumat and A.-M. Chollet, Classes de Gevrey non isotropes et application à l’interpolation, Ann. Scuola Norm. Sup. Pisa (IV) 15 (1988), 615–676. · Zbl 0689.32009
[6] R. R. Coifman and G. Weiss, Analyse harmonique non commutative sur certains espaces homogènes, Springer Lectures Notes 242 (1972).
[7] M. Gevrey, Sur la nature analytique des solutions des équations aux dérivées partielles, Ann. Ec. Norm. Sup (III 35 (1918), 129–190. · JFM 46.0721.01
[8] J.J. Kohn and L. Nirenberg, A pseudoconvex domain not admitting a holomorphic support function, Math. Ann. 201 (1973), 265–268. · Zbl 0248.32013 · doi:10.1007/BF01428194
[9] A. Nagel, J.-P. Rosay, E.M. Stein and S. Wainger, Estimates for the Bergman and Szegö kernels in \(\mathbb{C}\)2, Ann. of Math. 129 (1989), 113–149. · Zbl 0667.32016 · doi:10.2307/1971487
[10] A. Nagel, E.M. Stein and S. Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. USA 78 (1981), 6596–6599. · Zbl 0517.32002 · doi:10.1073/pnas.78.11.6596
[11] E.M. Stein, Singular integrals and estimates for the Cauchy-Riemann equations, Bull. Amer. Math. Soc. 79 (1973), 440–445. · Zbl 0257.35040 · doi:10.1090/S0002-9904-1973-13205-7
[12] V. Thilliez, Classes de Gevrey non isotropes et interpolation dans les domaines de type fini de \(\mathbb{C}\)2. C. R. Acad. Sci. Paris 313 (série I), 671–674. · Zbl 0755.32007
[13] V. Thilliez, Interpolation Gevrey dans les domaines de type fini de \(\mathbb{C}\)2, A paraître. Math. Z. · Zbl 0755.32007
[14] V. Thilliez, Classes de Gevrey non isotropes et interpolation dans les domaines de type fini de \(\mathbb{C}\)2, Thèse, Univ. Paris-XI Orsay (1991). · Zbl 0755.32007
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