Complex Tangential Directions Research Articles

Tangential Lipschitz gain for holomorphic functions on domains of finite type

Functions that are holomorphic and Lipschitz in a smoothly bounded domain enjoy a gain in the order of Lipschitz regularity in the complex tangential directions near the boundary. We describe this gain explicitly in terms of the defining function near points of finite type in the boundary.

Geometric Sufficient Conditions for Compactness of the Complex Green Operator

We establish compactness estimates for \(\overline{\partial}_{M}\) on a compact pseudoconvex CR-submanifold M of ℂn of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the \(\overline{\partial}\)-Neumann operator given in (Straube in Ann. Inst. Fourier, 54(3):699–710, 2004; Munasinghe and Straube in Pac. J. Math., 232(2):343–354 2007). These conditions are formulated in terms of certain short time flows in complex tangential directions.

Lipschitz and BMO extensions of holomorphic functions from subvarieties to a convex domain

Let be a bounded convex domain with C∞ boundary. Let M be a sub variety in D of dimension one which has no singular points on ∂M and intersects ∂D transversally. Assume that D is weakly totally convex in the complex tangential directions at any point in ∂M We get Lipschitz and BMO extensions of holomorphic functions from M to D

Complex tangential characterizations of Hardy-Sobolev Spaces of holomorphic functions

Let O be a C8-domain in Cn. It is well known that a holomorphic function on O behaves twice as well in complex tangential directions (see [GS] and [Kr] for instance). It follows from well known results (see [H], [RS]) that some converse is true for any kind of regular functions when O satisfies (P) The real tangent space is generated by the Lie brackets of real and imaginary parts of complex tangent vectors In this paper we are interested in the behavior of holomorphic Hardy-Sobolev functions in complex tangential directions. Our aim is to give a characterization of these spaces, defined in a domain which satisfies the property (P), involving only complex tangential derivatives. Our method, which is elementary, is to prove pointwise estimates between gradients and tangential gradients of holomorphic functions and, next, to use them to obtain the characterization of Hardy-Sobolev spaces for 1 = p < 8.

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

Behavior of holomorphic functions in complex tangential directions in a domain of finite type in $\Bbb C^n$

Let O be a domain in Cn. It is known that a holomorphic function on O behaves better in complex tangential directions. When O is of finite type, the best possible improvement is quantified at each point by the distance to the boundary in the complex tangential directions (see the papers on the geometry of finite type domains of Catlin, Nagel-Stein and Wainger for precise definition). We show that this improvement is characteristic: for a holomorphic function, a regularity in complex tangential directions implies the corresponding regularity in all directions. We give a pointwise inequality in both directions between the gradients and the complex tangential gradients. We characterize Besov, Sobolev and Lipschitz spaces of holomorphic functions defined on O by the behavior of complex tangential derivatives.

Tangential boundary limits and exceptional sets for holomorphic functions in Dirichlet-type spaces

According to the well-known theorem of Fatou, the harmonic extension u to the upper half-space R~_ +x of a function f in LP(Rn), 1 1. In this paper we study the analogous question in the unit ball B of C'. Recall that the correct analogue of Fatou's theorem for B is the theorem of Kor/myi [K] stating that invariant harmonic extensions (Poisson-Szeg6 integrals) of functions in LP(S), 1 < p < oo, have "admissible" limits at almost every point of the unit sphere S. In particular, this is true for holomorphic functions in the Hardy space HP(B). As is well known, Korfinyi's admissible regions allow parabolic tangential approach along the complex-tangential directions. These admissible regions are best possible as far as the order of tangency to the boundary is concerned, even if only bounded holomorphic functions are considered, as shown in [H-S] (however, they are not best possible in a stronger sense [Su 1]). By analogy with the real-variable situation, it seems reasonable to expect Kor/myi's regions not to be optimal for holomorphic functions whose (admissible) boundary values are, in some sense, Bessel-type potentials of LP(S) functions. Thus we consider holomorphic functions in the unit ball which are given by "fractional Cauchy integrals" of LP(S) functions. These are obtained by modifying the exponent in the denominator of the Cauchy kernel of B, a procedure suggested by the

Estimates for the Bergman and Szego Kernels in C 2

The purpose of this paper (some of whose conclusions were announced in [NRSW]) is to study the Bergman and Szegd projection operators on pseudoconvex domains Q of finite type in C2. The results we obtain are of three kinds: (i) The size estimates of the Bergman and Szeg6 kernels, and their derivatives. (ii) The cancellation properties of those kernels, expressed in terms of the actions of these operators on suitable bump functions; these properties have an interest in their own right but are also crucial in (iii) below. (iii) The sharp mapping properties of these operators on functions spaces, such as LP, the nonisotropic Sobolev spaces, and the nonisotropic Lipschitz spaces, which are naturally attached to the geometry of the boundary. This also leads ultimately to sharp results in the regularity properties of solutions of the equation du = f in Q, and 9bU = f on the boundary. Let us describe these matters in more detail. Our starting points are certain basic geometric constructs associated to Q: a metric p defined in terms of the vector fields X1 and X2 which are the real and imaginary parts of the (tangential) Levi vector field on d 2; and a function A(p, 8), (a polynomial in 8 for p E d 2), which represents the higher Levi-invariant attached to d 9 . The significance of these to the geometry of d Q can be understood from the following facts: the ball B(p, 8) C d Q (centered at p, of radius 8 in the metric p) when viewed in the appropriate coordinates has width - 8 in the complex tangential directions, but its width is - A(p, 8) in the complementary direction. Thus the volume of the ball is - 82A(p, 8). Among other crucial properties

THE THEOREMS OF LINDELÖF AND FATOU IN Cn

The author proves a generalization of the theorems of Lindelof and Fatou in which approach to the boundary along complex tangential directions is allowed.