## Local Reproducing Kernels on Wedge-Like Domains with Type 2 Edges

### Al Boggess and Alexander Nagel

In this paper, we show that holomorphic functions on wedge domains'' can be locally represented via an integral against a measure defined on the edge of the wedge'' where an edge'' is a submanifold M of Cn of real codimension greater than 1. Our edge is required to have type 2'', which means that the image of the Levi form sweeps out the normal cross section of the wedge domain.

Our approach is to first study a certain model case'' where the edge is a nilpotent Lie group of step 2, and then show that the general case can be obtained by a three stage process which is inspired by Stein's work: (i) we pass from the original object of study to a free'' object by adding appropriate variables; (ii) we solve the problem on the freed object by approximating it suitably by the model case; (iii) we return to the orginal object by integrating out the extra variables.

We also apply these local integral representation formulas to the study of Hp functions on wedge domains with a generic type 2 edge. In particular, we prove that Hp functions have admissible limits almost everywhere along the edge, for 0 < p <= infinity, and we obtain a necessary condition for a CR distribution on the edge to be the boundary values of an Hp function.

Full postscript version is available.





## Model Rigid CR Submanifolds of CR Dimension 1

### Al Boggess, Laura Ann Glenn and Alexander Nagel

In this paper, we discuss models that are appropriate for submanifolds of Cn of high codimension that are of higher type. The analysis of the most general submanifold of higher type is complicated. Here, we focus on the simpler class of rigid submanifolds of higher type where the dimension of the holomorphic tangent space at each point is one. For this class, we discuss three models: a tube-like model that only depends on the real part of the holomorphic tangent coordinate; a radial model that depends on the modulus of the holomorphic tangent coordinate; and a free model which is the most general of our models. As we show, the tube-like and free models have a Lie group structure that are generalizations of the Lie group structure for the boundary of a Siegel domain.

We show that the hull of holomorphy of the tube-like model is its convex hull. We also show that the hull of holomorphy of the more general free model is strictly smaller than its convex hull. This leaves open the interesting question as to the exact nature of the hull of holomorphy of this basic model.





## Local Representing Measures for Holomorphic Functions

### Al Boggess and Alexander Nagel

This paper is a proceedings conference version of a longer manuscript entitled Local Reproducing Kernels on Wedge-Like Domains with Type 2 Edges''. This manuscript was delivered at a conference honoring Eli Stein. In this paper, we show that holomorphic functions on wedge domains'' can be locally represented via an integral against a measure defined on the edge of the wedge'' where an edge'' is a submanifold M of Cn of real codimension greater than 1. Our edge is required to have type 2'', which means that the image of the Levi form sweeps out the normal cross section of the wedge domain.

Our approach is to first study a certain model case'' where the edge is a nilpotent Lie group of step 2, and then show that the general case can be obtained by a three stage process which is inspired by Stein's work: (i) we pass from the original object of study to a free'' object by adding appropriate variables; (ii) we solve the problem on the freed object by approximating it suitably by the model case; (iii) we return to the orginal object by integrating out the extra variables.

We also apply these local integral representation formulas to the study of Hp functions on wedge domains with a generic type 2 edge. In particular, we prove that Hp functions have admissible limits almost everywhere along the edge, for 0 < p <= infinity, and we obtain a necessary condition for a CR distribution on the edge to be the boundary values of an Hp function.

Full postscript version is available.





## The Holomorphic Extension of Hp - CR Functions on Tube Submanifolds

### Al Boggess

We consider the set of CR functions on a connected tube submanifold of Cn satisfying a uniform bound on the Lp - norm in the tube direction. We show that all such CR functions holomorphically extend to Hp functions on the convex hull of the tube (1 <= p <= infinity). The Hp - norm of the extension is shown to be the same as the uniform Lp - norm in the tube direction of the CR function.





## The Holomorphic Extension of CR Functions of class Ck from a Tube Submanifold

### Al Boggess

We show that a CR function of class Ck, 0 <= k <= infinity, on a tube submanifold of Cn holomorphically extends to the convex hull of the submanifold. The extension and all its derivatives through order k are shown to have nontangential pointwise boundary values on the original tube submanifold. The Ck - norm of the extension is shown to be no bigger than the Ck - norm of the original CR function.





## CR Extension for Lp CR Functions on a Quadric Submanifold of Cn

### Al Boggess

We consider the space, CRp (M), consisting of CR functions which also lie in Lp (M) on a quadric submanifold M of Cn of codimension at least one. For 1 <= p <= infty, we prove that each element in CRp(M) extends uniquely to an Hp function on the interior of the convex hull of M. As part of the proof, we establish a semi-global version of the CR approximation theorem of Baouendi and Treves for submanifolds which are graphs and whose graphing functions have polynomial growth. AMS Classification Numbers 32, 42 and 43.





## CR Approximation on a Non-rigid Hypersurface Graph in Cn

### Al Boggess and Roman Dwilewicz

Let M be a hypersurface in Cn that is the graph over a (2n-1)-real dimensional linear space. The main result of the paper is that any CR function on M can be uniformly approximated on compact subsets by entire functions on Cn.





## Global Approximation of CR Functions on Bloom-Graham Model Graphs in Cn

### Al Boggess and Daniel Jupiter

We define a class of submanifolds of Cn, of higher codimension, which are graphs, and where the graphing function is independent of some of the real coordinates. This class is said to be of Bloom-Graham type since the leading order terms of the Bloom-Graham normal form have a similar description. This class contains the set of all rigid graphs. We show that if the graphing function has polynomial growth, then continuous CR functions on a graph of Bloom-Graham type can be globally approximated, uniformly on compact sets by entire functions.




## CR Runge Sets on Hypersurface Graphs in Cn

### Al Boggess, Roman Dwilewicz, and Daniel Jupiter

This work contains a refinement of earlier results of Boggess and Dwilewicz regarding global approximation of CR functions by entire functions in the case of hypersurface graphs. In this work, we show that if \omega, an open subset of a real hypersurface in Cn, can be graphed over a convex subset in R2n-1, then \omega is CR-Runge in the sense that continuous CR functions on \omega can be approximated by entire functions on Cn in the compact open topology of \omega.




## Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group

### Al Boggess and Andy Raich

Let Lg = -1/4 Sum_{j=1}^n (X_j^2+Y_j^2)+i\gamma T where \gamma belongs to C, and X_j, \ Y_j and T are the left invariant vector fields of the Heisenberg group structure for Rn \times Rn \times R. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation Ds\rho = - Lg\rho. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the \Boxb-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted \dbar-operator in Cn with weight \exp(-\tau P(z_1,\dots,z_n)) where P(z_1,\dots,z_n) = (1/2)(|\Imm z_1|^2 + \cdots |\Imm z_n|^2) and \tau\in\R.

## The Boxb-heat equation on quadric manifolds

### Al Boggess and Andy Raich

The purpose of this article is to present an explicit calculation of the Fourier transform of the fundamental solution of the Boxb-heat equation on quadric submanifolds in complex Euclidean space. A quadric submanifold can be thought of as a generalization of the Heisenberg group -- it is a Lie group with a known representation theory, and the technique of using Hermite functions to compute the heat kernel, as done in and elsewhere, can be extended to work in this situation as well.

## Heat Kernels, Smoothness Estimates and Exponential Decay

### Al Boggess and Andy Raich

In this article, we characterize functions whose Fourier transforms have exponential decay. We characterize such functions by showing that they satisfy a family of estimates that we call quantitative smoothness estimates (QSE). Using the QSE, we establish Gaussian decay for the Boxb-heat kernel on polynomial models in C2. On the transform side, the problem becomes establishing QSE on a heat kernel associated to the weighted D-bar-operator on L2(C). The bounds are established with Duhamel's formula and careful estimation. We then use our result in C2 to prove QSE estimates for decoupled polynomial models in Cn+1 and establish Gaussian decay of the Boxb heat kernel in the bad direction.

## Fundamental Solutions to Boxb on Certain Quadrics

### Al Boggess and Andy Raich

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of CN and the group Fourier transform to reduce the $\Box_b$ equation to ones that can be solved using modified Hermite functions. We use Mehler's formula and investigate 1) quadric hypersurfaces, where the eigenvalues of the Levi form are not identical (including zero eigenvalues), and 2) the canonical quadrics in C4 of codimension two.

## Hartogs Phenomenon on Unbounded Domains - Conjectures and Examples

### Al Boggess, Roman J. Dwilewicz, and Zbigniew Slodkowski

In this paper we consider the Hartogs type extension problem for unbounded domains Omega in C2. The conjecture is that if the closure of Omega does not contain a complex curve given by the zero set of an entire function f, then any function that satisfies the tangential Cauchy-Riemann equations on the boundary of Omega can be holomorphically extended to Omega. The conjecture is established for Reinhardt tube-like domains and several related examples are given.




## Advanced Placement Calculus Booklet for High School Teachers

### Al Boggess

This booklet is an outgrowth of summer workshops for Advanced Placement Calculus high school teachers that I taught in the summers of 1994 and 1995. I have two goals for this booklet. The first is to explain some of the highlights of calculus, using the graphing calculator, where appropriate, as a teaching aid. The second goal is to provide a set of problems, some of which use the graphing calculator as a tool to help with graphics or computations that may be cumbersome to do by hand.

This booklet is not intended to replace a standard calculus book, but rather to enhance it. My goal is to state the central ideas that are at the heart of some of the more important concepts of calculus at the level of the AB advanced placement calculus curriculum. Standard calculus books also contain the ideas in this booklet. However a published book includes additional material for completeness which sometimes muddles the central ideas at hand. In addition, some of the problems involving the calculator are somewhat novel and may not appear in many of the standard texts. The problems in the booklet vary from the routine to the quite difficult. Most of the more difficult problems are at the very end in the {\em Challenge Problems} section. In addition, there are some ideas for extended project ideas in section VIII for your students to work over a longer period of time than what would be required for routine homework problems.