Real Analytic Boundary Research Articles

Local analyticity for the $$\overline \partial $$ -Neumann problem in completely decoupled pseudoconvex domains-Neumann problem in completely decoupled pseudoconvex domains

In this paper we prove local analyticity of solutions to the\(\overline \partial \)-Neumann problem up to the boundary of rigid, completely decoupled pseudoconvex domains with real-analytic boundary. These are domains that are locally of the form Imw > Σ ¦hk(zk)¦2 with eachhkholomorphic and vanishing only at 0.

Local real analytic boundary regularity of the $\bar{\partial}$-equation on convex domains

Local real analytic boundary regularity of the $\bar{\partial}$-equation on convex domains

Local real analytic boundary regularity of an integral solution operator of the∂-equation on convex domains

In this paper we show that a well known integral solution operator of the ∂-equation on a convex domain Ω locally preserves the real analyticity of ∂-closed (0, 1) forms at boundary points near which ∂Ω is totally convex in the complex tangential directions

On the Vector Sum of Two Convex Sets in Space

In the paper [KIS2], C. Kiselman studied the boundary smoothness of the vector sum of two smoothly bounded convex sets A and B in . He discovered the startling fact that even when A and B have real analytic boundary the set A + B need not have boundary smoothness exceeding C20/3 (this result is sharp). When A and B have C∞ boundaries, then the smoothness of the sum set breaks down at the level C5 (see [KIS2] for the various pathologies that arise).

Smoothness of sums of convex sets with real analytic boundaries.

Smoothness of sums of convex sets with real analytic boundaries.

Complete Localization of Domains with Noncompact Automorphism Groups

We prove a characterization of the domains in ${{\mathbf {C}}^n}$ with an automorphism orbit accumulating at a boundary point at which the boundary is real analytic and convex up to a biholomorphic change of local coordinates. This result generalizes the well-known Wong-Rosay theorem on strongly pseudoconvex domains to the case of locally convex domains with real analytic boundaries.

Real analytic boundary regularity of the Cauchy kernel on convex domains

It is well-known that in one complex variable the Cauchy integral preserves real analyticity near the boundary. In this paper we show that the same conclusion also holds on convex domains with real analytic boundary in higher dimension, where the Cauchy kernel is given by the Cauchy-Fantappiè form of order zero generated by the (l.0)-form C ( ξ , z ) C\left ( {\xi ,z} \right ) , \[ C ( ξ , z ) = ( ∑ j = 1 n ∂ r ∂ ξ j ( ξ ) d ξ j ) ( ∑ j = 1 n ∂ r ∂ ξ j ( ξ ) ( ξ j − z j ) ) − 1 , C\left ( {\xi ,z} \right ) = \left ( {\sum \limits _{j = 1}^n {\frac {{\partial r}}{{\partial {\xi _j}}}} \left ( \xi \right )d{\xi _j}} \right ){\left ( {\sum \limits _{j = 1}^n {\frac {{\partial r}}{{\partial {\xi _j}}}} \left ( \xi \right )\left ( {{\xi _j} - {z_j}} \right )} \right )^{ - 1}}, \] where r ( ξ ) r\left ( \xi \right ) is the defining function of the domain.

Complete localization of domains with noncompact automorphism groups

We prove a characterization of the domains in C n {{\mathbf {C}}^n} with an automorphism orbit accumulating at a boundary point at which the boundary is real analytic and convex up to a biholomorphic change of local coordinates. This result generalizes the well-known Wong-Rosay theorem on strongly pseudoconvex domains to the case of locally convex domains with real analytic boundaries.

DOMAINS IN $ \mathbf{C}^2$ WITH NONCOMPACT HOLOMORPHIC AUTOMORPHISM GROUPS

It is shown that if a bounded domain with real analytic boundary has a noncompact automorphism group, then it is biholomorphically equivalent to a domain for some .Bibliography: 18 titles.

Extending proper holomorphic mappings of positive codimension

In this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real-analytic boundaries in complex spaces of different dimensions. Equivalently, we obtain results concerning the analyticity of Cauchy-Riemann mappings between real-analytic pseudoconvex hypersurfaces in complex spaces of different dimensions. To begin with, we recall the corresponding results for mappings of equidimensional domains. Let D and D' be bounded pseudoconvex domains with smooth boundaries in C". If the boundaries of D and D' are strictly pseudoconvex or, more generally, of finite type in the sense of D'Angelo [15], then every proper holomorphic map of D onto D' extends smoothly t o / ) according to the results of Bell and Catlin [7, 8] and Diederich and Forn~ess [19]. For mappings between strictly pseudoconvex domains this was proved by Fefferman [26] and Nirenberg, Webster, and Yang [37]. If the boundaries of the pseudoconvex domains D, D ' c C" are real-analytic, then every proper holomorphic mapping of D onto D' extends holomorphically to a neighborhood of /3 according to Baouendi and Rothschild [2] and Diederich and Forn~ess [21]. This 'reflection principle' was first discovered by Lewy [34] and Pin6uk [38] for mappings between strictly pseudoconvex domains. In the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of Baouendi, Jacobowitz, and Treves [5]. Results in this direction were obtained in recent years by several authors; see the papers [4, 6, 18, 22, 33, 47, 49]. In this paper we are treating the case when the domains D and D' have different dimensions. To be specific, we assume that D e C " and D ' c C N are bounded pseudoconvex domains with real-analytic boundaries and N > n > l . In this situation a proper holomorphic map f : D , D ' need not be regular at the boundary. For instance, there exist proper holomorphic maps of balls of different dimensions

Weakly pseudoconvex domains with non-compact automorphism groups

The purpose of this paper is to prove a uniform extendibility result for sequences of biholomorphic mappings between weakly pseudoconvex domains. Suppose that f21 and 02 are bounded weakly pseudoconvex domains in {E" with real analytic boundaries and suppose that {fi} is a sequence of biholomorphic mappings J~ : I2t--,O2 which converges to a holomorphic mapping f of ~2~ into the closure of 122. (We remark that a strictly pseudoconvex domain is not a weakly pseudoconvex domain.) Cartan's Theorem states that either f is a biholomorphic map of f2~ onto 02, or f is a mapping of f21 into the boundary of f22. We shall prove that, in either case, there exists a domain D which contains 121 together with all the strictly pseudoconvex boundary points of f2x such that each of the mappings f~ extend to be hoIomorphic on D. Furthermore, the limit map fextends holomorphically to D and the sequence {fi} converges uniformly on compact subsets of D to f. It is interesting to note that the corresponding statement for a sequence of maps between strictly pseudoconvex domains with real analytic boundaries is not true. Indeed, if I21 is strictly pseudoconvex and the limit map f maps into bf22, then both O t and I22 are biholomorphic to the ball by Wong's theorem. In this situation, if the inverse mappings Fi =f~-1 also converge, then there is an open set D which contains the closure of O1 minus a single point in bf21 to which all the f~ extend, and the sequence f~ converges uniformly on compact subsets of D to f. If no assumption is made about the convergence of the inverses, then the maps./~ need not converge uniformly up to the boundary near any boundary point of O1. When f21 and 02 are assumed to be weakly pseudoconvex, and therefore not biholomorphic to the ball, the assumption about the convergence of the inverse maps is not needed.

Global analytic hypoellipticity of the $$\bar \partial$$ -Neumann problem on circular domains

In this paper we show that if $$D \subseteq \mathbb{C}^n ,n \geqq 2$$ , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that $$\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0$$ for allz near the boundary, then the solutionu to the $$\bar \partial$$ -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if $$D \subseteq \mathbb{C}^n ,n \geqq 2$$ , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ▭ is analytic hypoelliptic.

Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings

A classical theorem of Carath6odory [8] states that every biholomorphic map f: D~ ~D2 between domains in the complex plane C bounded by simple closed Jordan curves extends to a homeomorphism of/31 onto/ )2 . There are some well-known generalizations of this result to domains in IE a. If D t and D2 are bounded pseudoconvex domains in 112" with if2 boundary and if the infinitezimal Kobayashi metric on D2 grows sufficiently fast near the boundary of D2(Ko2(g; X) >= [Xl/d(z, bDz) ~ for some e e (0, 1)), then every proper holomorphic map f : Dt--*D2 extends to a H61der-continuous map of/51 onto/32 [2, 3, 10, 25, 29-32]. This holds in particular if/92 is strictly pseudoconvex or if it is pseudoconvex with real-analytic boundary. The same result holds if D2 is only piecewise smooth strongly pseudoconvex [31]. Further results treat exceptional cases such as the balls [1 ], Reinhardt domains [5, 24], domains with many symmetries [4], and analytically bounded Hartogs domains in C 2 [I1]. Biholomorphic maps between certain types of nonpseudoconvex domains with real-analytic boundaries were treated in [13] and [27]. Besides the papers mentioned above there is a vast literature concerning smooth extension of proper holomorphic maps of smoothly bounded pseudoconvex domains ([6, 7, 12, 14], to mention just a few). In the present paper we obtain some results on the continuous extension of proper holomorphic maps f : D1--*/)2 under local assumptions on the boundaries bD1 and bD2. One of the main results is

Embedding strictly pseudoconvex domains into balls

Every relatively compact strictly pseudocc)nvex domain D with C2 boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each n > 2 there exist bounded strictly pseudoconvex domains D in Cn with real-analytic boundary such that no proper holomorphic map from D into any finite dimensional ball extends smoothly to D. 0. Introduction. In this paper we study the representations of bounded strictly pseudoconvex domains D c cn. If the boundary of D is of class Ck, k E {2, 3, . . ., oo}, then, by a theorem of Fornaess [8] and Khenkin [13], D can be mapped biholomorphically onto the intersection xn Q of a bounded strictly convex domain Q c CN with Ck boundary and a closed complex submanifold X defirXed in a neighborhood of Q in CN, X intersecting the boundary of Q transversally. Moreover, the map f: D X n Q extends to a holomorphic map on a neighborhood of D. The convex domain Q depends on D; hence a natural question is whether a similar result holds with Q replaced by the unit ball B = (z = (Z1L, * , ZN) E CN ||Z||2 = E |Zjl2 n that intersects bBN transversally such that D is biholomorphically equivalent to xn E3ff? This question has been mentioned by Lempert [14], Pinduk [20], Bedford [2] and others. Our main result is that the answer to this question is negative in general. If D is as above, then every biholomorphism of D onto XnE3N extends smoothly to D according to [3]. However, we will show that not all such domains D admit a proper holomorphic map into a finite dimensional ball that is smooth on D (Theorem 1.1). A similar local result was obtained independently by Faran [7]. We shall show that the answer to the question (Q) is positive if we allow the intersections of complex submanifolds with strictly convex domains Q c CN with real-analytic boundaries (Theorem 1.2). ThetheoremsofFornaess [8] andKhenkin [13] onlygiveanQwith smooth boundary. Received by the editors April 4t 1985. 1980 Mathematics Subject Classification. Primary 32H05. 1Research supported by a fellowship from the Alfred P. Sloan Foundation. (t)1986 American Mathematical Society 0002-9947/86 $1 .00 + $ .25 per page

Extension of proper holomorphic mappings past the boundary

It is proved that proper holomorphic mappings between n-dimensional bounded pseudoconvex domains with real analytic boundaries extend holomorphically past the boundary whenever the target domain is strictly pseudoconvex.

Hölder andL p -estimates for the ∂ equation in some convex domains with real-analytic boundary

Hölder andL p -estimates for the ∂ equation in some convex domains with real-analytic boundary

Proper Holomorphic Mappings from Domains with Real Analytic Boundary

Proper Holomorphic Mappings from Domains with Real Analytic Boundary

Extendability of proper holomorphic mappings and global analytic hypoellipticity of the ∂̄-Neumann problem

IN THIS COMMUNICATION, THE PROOF OF THE FOLLOWING THEOREM IS SKETCHED: If D(1) is a bounded domain in C(n) with real analytic boundary whose partial differential-Neumann problem is globally real analytic hypoelliptic and f is a proper holomorphic mapping of D(1) onto a second bounded domain D(2) in C(n) with real analytic boundary, then the mapping f extends to be holomorphic in a neighborhood of D(1). The proof relies on a transformation formula for the Bergman projection under proper holomorphic mappings.

Non-vanishing of the Bergman kernel function at boundary points of certain domains in ? n

Let f2 be a smooth bounded domain in (t TM whose associated Bergman kernel function K(w, z) extends continuously to O as a function ofw for each z ~ f~. We shall prove that for any fixed we bf2, K(w, z) cannot vanish identically in z for a wide class of domains O. Our theorem applies to any domain for which the L2(O) orthogonal projection P onto holomorphic functions is bounded in Sobolev norms ; in particular, if f2 is 1) smooth bounded strictly pseudoconvex, or 2) bounded pseudoconvex with smooth real analytic boundary (Kohn [7], Diederich and Fornaess [,2]), or 3) any smooth bounded domain for which subelliptic estimates hold for the ~-Neumann opcrator (Kohn [7]). Webster [8] has pointed out that the non-vanishing of K(w, z) for strongly pseudoconvex points of domains for which the c~-Neumann operator is pseudo-local follows easily from the asymptotic expansion for K(w, z) given by Fefferman [-3]. Our proof has no recourse to the asymptotic expansion, is elementary in spirit, and extends Webster's result to include certain weakly pseudoconvex domains. The non-vanishing question arises in connection with the problem of boundary behavior of biholomorphic mappings (see Webster [-8]). As a corollary to the non-vanishing theorem, we prove that if (2 is strictly pseudoconvex and Dis a set of determinacy for holomorphic functions in fL then the linear span of {K(., z) :ze ID} is dense in the subspace of holomorphic functions in w+(Q).

Domains of existence for plurisubharmonic functions

Theorem. Let 12 C ~" be a domain with C 2 boundary, and let the Levi form of OQ be nondeyenerate on an open dense subset of Og2. Then there is a plurisubharmonic function on £2 that cannot be extended to be plurisubharmonic on a set f2: CO" with f21c~0:# 0. We note that the hypotheses of the theorem are "generic" and are satisfied, for instance, if £2 is bounded and has real analytic boundary. The methods here are local, but the situation where a large portion of ~£2 is Levi flat seems to pose a global problem. Let P(f2) denote the plurisubharmonic functions on O. It is well known that if g2 is pseudo-convex, then there is a plurisubharmonic exhaustion function for f2. However, if u~ Lipl(O)c~P(f2) and if f2 is strongly pseudoconvex with C 2 boundary, then u extends to a plurisubharmonic function in a neighborhood of O. On the other hand, if