A sufficient condition for global regularity of the [formula omitted]-Neumann operator

The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results, such as the Bochner-Martinelli formula, the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations, and the Hartogs theorem, are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular, the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable, several complex variables, and several quaternionic variables with suitable modifications.

Global regularity for the [formula omitted]-Neumann operator and bounded plurisubharmonic exhaustion functions

Let Ω \Omega be a smooth bounded pseudoconvex domain in C n \mathbb {C}^{n} . Let 1 ≤ q 0 ≤ ( n − 1 ) 1\leq q_{0}\leq (n-1) . We show that if q 0 q_{0} –sums of eigenvalues of the Levi form are comparable, then if the Diederich–Fornæss index of Ω \Omega is 1 1 , the ∂ ¯ \overline {\partial } –Neumann operators N q N_{q} and the Bergman projections P q − 1 P_{q-1} are regular in Sobolev norms for q 0 ≤ q ≤ n q_{0}\leq q\leq n . In particular, for domains in C 2 \mathbb {C}^{2} , Diederich–Fornæss index 1 1 implies global regularity in the ∂ ¯ \overline {\partial } –Neumann problem.

Inheritance of noncompactness of the [formula omitted]-Neumann problem

Letbe a pseudoconvex domain with C 2 boundary in CP n , n ≥ 2. We prove that the ¯ ∂-Neumann operator N exists for square-integrable forms on � . Furthermore, there exists a number � 0 > 0 such that the operators N , ¯ ∂ ∗ N , ¯ ∂N and the Bergman projection are regular in the Sobolev space W � (�) for � 0 and n ≥ 2. As a consequence, we prove the nonexistence of C 2 Levi-flat hypersurfaces in CP n .

In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Th\'eor\`eme de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,) and H. Wu, (Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525--548,) for the de Rham cohomology of strictly p-convex domains in R^n in the sense of F. R. Harvey and H. B. Lawson, (The foundations of p-convexity and p-plurisubharmonicity in riemannian geometry, arXiv:1111.3895v1 [math.DG]). Our proof uses the L^2-techniques developed by L. H\"ormander, (An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990,) and A. Andreotti and E. Vesentini, (Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes \'Etudes Sci. Publ. Math. 25 (1965), 81--130).

In this chapter, we consider some elementary properties of Riemann surfaces, as well as a fundamental technique called the L 2 \(\bar{\partial}\)-method, Rado’s theorem on second countability of Riemann surfaces, and analogues of the Mittag-Leffler theorem and the Runge approximation theorem for open Riemann surfaces. Viewing holomorphic functions as solutions of the homogeneous Cauchy–Riemann equation \(\partial f/\partial\bar{z}=0\) in ℂ allows one to very efficiently obtain their basic properties (see Chap. 1). The intrinsic form of the homogeneous Cauchy–Riemann equation on a Riemann surface is given by \(\bar{\partial}f=0\) (see Sect. 2.5). In order to obtain holomorphic functions (and holomorphic 1-forms) on a Riemann surface (even on an open subset of ℂ), it is useful to consider the inhomogeneous Cauchy–Riemann equation \(\bar{\partial}\alpha=\beta\). One well-known approach to solving this differential equation (as well as differential equations in many other contexts) is to consider weak solutions in L 2. This is the approach taken in this book. In order to do so, we must develop suitable versions of an L 2 space of differential forms (see Sect. 2.6) and an (intrinsic) distributional \(\bar{\partial}\) operator (see Sect. 2.7). The relatively simple approaches to the above appearing in this book are, in part, special to Riemann surfaces; but they do contain important elements of the higher-dimensional versions (see, for example, L. Hormander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990 or J.-P. Demailly, Complex Analytic and Differential Geometry, online book, for the higher-dimensional versions).

A contact structure on a closed oriented 3-manifold 3 is a completely nonintegrable plane field on it. The existence of a contact structure on 3 was first proved by Martinet [14] and then Lutz [12], [13] for each homotopy class of plane fields. A co-orientable contact structure is given as the kernel of a 1-form α with α ∧ α > 0 or < 0, which is called a positive or negative contact form respectively. Thurston and Winkelnkemper [17] deduced the existence of a contact form in an elegant way from Alexander’s theorem [1] on open-book decompositions. Eliashberg [2] showed that the most of these contact structures are, however, too flexible for geometrical interest (see §2). On the other hand, he completely characterized more historic examples related closely to the complex analysis in several variables, namely, the strictly pseudo-convex boundary of compact Stein surfaces ([4]). The symplectic fillability is one of its significant generalizations. Recently Loi and Piergallini [11] translated Eliashberg’s characterization of compact Stein surfaces into the language of Lefschetz fibration by using Gompf’s method [8]. They also showed that 3 is realizable as the boundary of a Stein surface if and only if it admits an open-book decomposition whose monodromy map is a composition of right-handed Dehn-twists. In this note, we improve Thurston-Winkelnkemper’s construction so that we obtain a symplectically fillable contact structure in the case where the monodromy map of a given open-book decomposition is a composition of right-handed Dehn-twists along mutually disjoint curves (Theorem 2). An interesting by-product of our construction is Theorem 3 in §4 which gives a deformation of symplectically fillable contact structures into a foliation with a Reeb component. Note that a foliation admitting a Reeb component itself is not symplectically fillable. I would like to thank the referee for noticing the importance of Theorem 3.

In the classical complex analysis of several variables Lars Hoermander had been proven for Hartog's extension theorem. His proof is based on solving the inhomogeneous Cauchy–Riemann system. However, it is difficult when generalizing this diagram to prove this problem in Clifford analysis because of the non-commutative property. In this paper we will propose a method to overcome this difficultly and we will prove the Hartog's extension theorem for multi-regular function taking valued in Clifford algebra

In this paper, we define an appropriate version of Morrey spaces for spaces of homogeneous type. As our main result, we give a sufficient condition for an operator to be bounded on the version of Morrey spaces (cf. Theorem 3.1 and Corollary 3.5). Moreover, using thin result, we prove Morrey space estimates for various operators arising in harmonic analysis and complex analysis of several variables.

We give a necessary and sufficient condition for the existence of global solutions of some partial differential equation which is locally solvable and give some applications in complex analysis of several variables.

On a smooth, bounded pseudoconvex domain Ω \Omega in C n \mathbb {C}^n , to verify that Catlin’s Property ( P P ) holds for b Ω b\Omega , it suffices to check that it holds on the set of D’Angelo infinite type boundary points. In this note, we consider the support of the Levi core, S C ( N ) S_{\mathfrak {C}(\mathcal {N})} , a subset of the infinite type points, and show that Property ( P P ) holds for b Ω b\Omega if and only if it holds for S C ( N ) S_{\mathfrak {C}(\mathcal {N})} . Consequently, if Property ( P P ) holds on S C ( N ) S_{\mathfrak {C}(\mathcal {N})} , then the ∂ ¯ \overline {\partial } -Neumann operator N 1 N_1 is compact on Ω \Omega .

&lt;abstract&gt;&lt;p&gt;In this paper, we study a sufficient condition for subelliptic estimates in the weak $ \operatorname{Z}(k) $ domain with $ \operatorname{\operatorname{C}}^{3} $ boundary in an $ n $-dimentionsl $ \operatorname{Stein\, manifold} $ $ \operatorname{X} $. Consequently, the compactness of the $ \overline\partial $-Neumann operator $ \operatorname{N} $ on $ { M } $ is obtained and the closedness ranges of $ \overline\partial $ and $ \overline\partial^* $ are presented. The $ \operatorname{L}^2 $-setting and the Sobolev estimates of $ \operatorname{N} $ on $ { M } $ are proved. We study the $ \overline\partial $ problem with support conditions in $ { M } $ for $ \Xi $-valued $ (\operatorname{p}, \operatorname{k}) $ forms, where $ \Xi $ is the $ m $-times tensor product of holomorphic line bundle $ \Xi^{\otimes \operatorname{m}} $ for positive integer $ m $. Moreover, the compactness of the weighted $ \overline\partial $-Neumann operator is studied on an annular domain in a $ \operatorname{Stein\, manifold} $ $ { M } = { M }_{1}\backslash\overline{ M }_{2} $, between two smooth bounded domains $ { M }_{1} $ and $ { M }_{2} $ satisfy $ \overline{ M }_{2}\Subset{ M }_{1} $, $ { M }_{1} $ is weak $ Z(k) $, $ { M }_{2} $ is weak $ Z(n-1-k) $, $ 1 \leqslant k\leqslant n -2 $ with $ n \geqslant 3 $. In all cases, the closedness of $ \overline\partial $ and $ \overline\partial^* $, global boundary regularity for $ \overline\partial $ and $ \overline\partial_b $ are studied.&lt;/p&gt;&lt;/abstract&gt;

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

In this paper, we solve the O-Neumann problem on (0,q) forms, q > 0, in the strictly pseudoconvex Siegel domain. The metric we use is invariant under the action of the Heisenberg group on the domain. We give a formula for a kernel of the Neumann operator, the fundamental solution of the problem, in (1.6) and Theorem 4.2, and we prove uniqueness of the Neumann operator. As far as we know, this is the first example of an explicit solution of the ~-Neumann problem. Greiner and Stein [GS] constructed a parametrix for the problem on (0, 1) forms in bounded strictly pseudoconvex domains. Phong [P] has announced an explicit construction of a parametric on (03) forms in the Siegel domain in C "+ 1, n > 1. Harvey and Polking [HP] have since constructed a Neumann kernel for the ~-Neumann problem on (p, q) forms on the ball. We obtain the kernel of the Neumann operator by integrating the fundamental solution of the corresponding heat equation ([S1], [$2]) with respect to time. To carry out the integration, we require estimates on the fundamental solution for large time. We did not need such estimates in [$1] and [$2]. Also, to prove that the resulting kernel is locally integrable, we require more precise estimates for small time. In Sect. 1, after some preliminaries, we state our main theorem. We then summarize the results of [S1 ] and [$2]. We solve the corresponding elliptic problem on C" + 1 in Sect. 2. In Sect. 3 we study the operator which corrects the boundary values of the solution we obtained in Sect. 2. We solve the O-Neumann problem in Sect. 4. Most of the work for this paper was done while I was a member of the Institute for Advanced Study. I wish to thank the Institute for its hospitality.