Al Boggess

A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group

Let L = −1/4 The purpose of this note is to present a simplified calculation of the Fourier transform of fundmental solution of theb-heat equation on the Heisenberg group. The Fourier transform of the fundamental solution has been computed by a number of authors (Gav77, Hul76, CT00, Tie06). We use the approach of (CT00, Tie06) and compute the heat kernel using Hermite functions but differ from the earlier approaches by working on a different, though biholomorphically equivalent, version of the Heisenberg group. The simplification in the computation occurs because the differential operators on this equivalent Heisenberg group take on a simpler form. Moreover, in the proof of Theorem 1.2, we reduce the n-dimensional heat equation to a 1-dimensional heat equation, and this technique would also be useful when analyzing the heat equation on the nonisotropic Heisenberg group (e.g., see (CT00)). We actually use the same version of the Heisenberg group as Hulanicki (Hul76), but he computes the fundamental solution of the heat equation associated to the sub-Laplacian and not the Kohn Laplacian acting on (0,q)-forms.

The holomorphic extension of Hp-CR functions on tube submanifolds

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 ≤ ∞). 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. 1. Definitions and main results Recently, Boivin and Dwilewicz [BD] have generalized Bochner’s Tube Theorem by showing that continuous CR functions on a tube-submanifold of C holomorphically extend to its convex hull. In this manuscript, we show that on a tubesubmanifold, CR functions that satisfy a uniform L-estimate in the tube direction extend to an H function on the tube over the convex hull (here, 1 ≤ p ≤ ∞). In addition, we show the H-norm on the convex hull of the holomorphic extension is bounded by the H-norm of the CR function. We will be working in C = R + iR with coordinates x+ iy, x ∈ R, y ∈ R. Let N be a connected submanifold of R and let M = N + iR be the (connected) tube over N . For 1 ≤ p ≤ ∞, let CR(M) denote the space of CR functions (solutions to the tangential Cauchy-Riemann equations) on M which satisfy ||f ||pp(M) = sup x∈N ∫ |f(x + iy)|p dy ≤ Ap < ∞ if 1 ≤ p < ∞, ||f ||∞(M) = sup x∈N ||f ||L∞(Tx) ≤ A∞ < ∞ if p = ∞, where Tx = {x}+ iR (the tube over x). If 1 ≤ p ≤ ∞ and T is any tube of the form T = U + iR with U an open set in R, then H(T ) will denote the usual space of H-functions on the tube T with the usual H-norm (defined as above with N replaced by U). Our main theorem is the following. Theorem 1 (Extension Theorem). Suppose N is a connected submanifold of R, and let M = N + iR be the tube over N . Let N̂ and M̂ = N̂ + iR denote the interior of the convex hull of N and M , respectively. If M̂ is nonempty and if 1 ≤ Received by the editors August 22, 1997. 1991 Mathematics Subject Classification. Primary 32A35, 42B30, 32D99.

Subaveraging estimates for CR functions

We give conditions on a CR submanifold M in Cn and a compact submanifold N C M such that the average value on N of a CR function on M can be estimated uniformly by the Ll-norm of the CR function on a neighborhood of N in M. The conditions involve the Levi form of M and the transversality of N to the holomorphic tangent bundle of M.

The holomorphic extension of CR functions near a point of higher type

Suppose M is a submanifold of Cn with real codimension at least one. A geometric description is given of the local hull of holomorphy of an open subset of M which contains a point of higher type in which all Hormander numbers are the same. This result is proved as a consequence of examining the relationship between the hypoanalytic wave front sets of CR functions on M and CR extension to a manifold of one higher dimension than M. 0. Introduction. Recently, Baouendi and Rothschild in [BR] have shown that on a semirigid submanifold of higher type that CR functions near a point p holomorphically extend to an open wedge in Cn. In that paper, higher type means that all the Hormander numbers are finite. The purpose of this paper is to more carefully examine the case when all the Hormander numbers are the same finite number 1 and to give a more precise description of this wedge under this additional assumption. We show (Theorem 1.4 below) that this wedge fills in the convex hull of the image of the lth Levi form at p. This result is the natural generalization of the results in [BP and B] which handle the cases I = 2 and I = 3 respectively. There are two key components of the proof of this result. First, we use a CR extension result in [BPi] which states that CR extension to a submanifold of one higher dimension is always possible near a point p of type I (I < xc). Secondly, we examine (in Lemma 1.7 below) the relationship between CR extension to a submanifold of one higher dimension and the hypoanalytic wave front sets of CR functions defined near p. 1. Notation and the precise statement of results. Throughout this paper M will denote a smooth generic real submanifold of Cn of real codimension d, 1 < d < n and p will be a point in M. We let Tp(M) be the real tangent space of M at p and we let Hp(M) be the J-invariant subspace of Tp(M), where J: R2n -R2 is the linear isometry induced by multiplication by V/T. The totally real tangent space of M at p, denoted T?(M), is the orthogonal complement of Hp(M) in Tp(M) under the usual inner product on R2n. Since M is generic, dimR Hp (M) = 2m where m = n d and dimR T?(M) = d. We let Np(M) (the normal space of M) be the orthogonal complement of Tp(M) in R2n. The map J is an isometry from T?(M) to Np(M). Our description of hulls of holomorphy will involve wedges (or conoids) emanating from M in the normal direction. Therefore to simplify this description, we shall Received by the editors November 5, 1986 and, in revised form, July 13, 1987. Results presented at the AMS special session in Several Complex Variables at Indianapolis, Indiana, April, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 32D15; Secondary 32E99. Research partially supported by NSF Grant #MSC-8301369. (?1988 American Mathematical Society 0002-9939/88

CR manifolds and the tangential Cauchy-Riemann complex

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A First Course in Wavelets with Fourier Analysis

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