Search

This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on M M which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the L p L^p and L p L^p -Sobolev mapping properties of the associated kernels, and (4) provide examples.

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N N and the projection onto the nullspace of ◻ b \Box _b . The main application of our formulas is the critical case of codimension two quadrics in C 4 \mathbb {C}^4 where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.

The purpose of this paper is to find pointwise estimates on the integral kernel for the $$\Box _b$$ Green operator, N, for a particular quadric submanifold of codimension 2 in $${\mathbb {C}}^4$$ . This particular kernel has singularities away from the diagonal. Furthermore, the estimates exhibited by this kernel, both on and off the diagonal, follow no known paradigm.

This paper is the second of a multi-part series in which we explore geometric and analytic properties of the Kohn–Laplacian and its inverse on general quadric submanifolds of $${\mathbb{C}}^n\times {\mathbb{C}}^m$$ . We have two goals in this paper. The first is to give useable sufficient conditions for a map T between quadrics to be a Lie group isomorphism that preserves $$\Box _b$$ , and the second is to establish a framework for which appropriate derivatives of the complex Green operator are continuous in $$L^p$$ and $$L^p$$ -Sobolev spaces (and hence are hypoelliptic). We apply the general results to codimension two quadrics in $${\mathbb{C}}^4$$ .

In this paper we consider the Hartogs-type extension problem for unbounded domains in \(\mathbb C^2\). An easy necessary condition for a domain to be of Hartogs-type is that there is no a closed (in \(\mathbb C^2\)) complex variety of codimension one in the domain which is given by a holomorphic function smooth up to the boundary. The question is, how far this necessary condition is from the sufficient one? To show how complicated this question is, we give a class of tube-like domains which contain a complex line in the boundary which are either of Hartogs-type or not, depending on how the complex line is positioned with respect to the domain.

In 1906, Hartogs proved his famous holomorphic extension theorem for bounded domains in C n , however, not many results of this type exist for unbounded ones. One of such, already a classical result, is the Bochner tube theorem from 1938, and in this paper we consider its generalization to “widening tubes”. Namely, we establish a sufficient condition for holomorphic extension of CR functions from the boundary of a certain class of domains in C n , n ≥ 2 , which possess a growth condition at infinity.

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×ℂm and the group Fourier transform to reduce the □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 possibly zero eigenvalues), and (2) the canonical quadrics in ℂ4 of codimension two.

In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □ b -heat equation on quadric submanifolds M⊂ℂ n ×ℂ m . As a consequence, we can also compute the heat kernel associated with the weighted $\overline{\partial}$ -equation in ℂ n when the weight is given by exp (−φ(z,z)⋅λ) where φ:ℂ n ×ℂ n →ℂ m is a quadratic, sesquilinear form and λ∈ℝ m . Our method involves the representation theory of the Lie group M and the group Fourier transform.

This work contains an improvement 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 ω, an open subset of a real hypersurface in ℂn, can be graphed over a convex subset in ℝ2n−1, then ω is CR-Runge in the sense that continuous CR functions on ω can be approximated by entire functions on ℂn in the compact open topology of ω. Examples are presented to show that this approximation result does not hold for graphed CR submanifolds in higher codimension.

We define a class of generic CR submanifolds of C n \mathbb {C}^n of real codimension d d , 1 ≤ d ≤ n 1\leq d\leq n , called the Bloom-Graham model graphs, whose graphing functions are partially decoupled in their dependence on the variables in the real directions. We prove a global version of the Baouendi-Treves CR approximation theorem for Bloom-Graham model graphs with a polynomial growth assumption on their graphing functions.

Let M be a hypersurface in C n that is the graph over a (2n - 1)-linear real 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 C n .

We analyze Hubble Space Telescope/Goddard High Resolution Spectrograph spectra of AB Doradus, the prototypical, ultrarapidly rotating K dwarf. We observed chromospheric (Mg II) and transition-region (C II, Si IV, C IV, and N V) lines periodically throughout the stellar rotation period and provide a low-dispersion stellar atlas of 78 emission lines. The quiescent line profiles of the chromospheric and transition-region lines show narrow cores superposed on very broad wings. The broad wings of the Mg II k and h lines and of the transition-region lines can be explained by emission from gas corotating with the star and extending out to near the Keplerian corotation radius (2.8 stellar radii). While this is not a unique solution, it is consistent with previous studies of Hα emission, which are naturally explained by large corotating prominences. We find no evidence for rotational modulation of the emission-line fluxes. The density diagnostics suggest that the transition region is formed at constant pressure, with an electron density of 2–3 × 1012 cm-3 at a temperature of 3 × 104 K. The electron pressure is about 100 times larger than that for the quiet Sun. The emission-measure distribution shows a minimum between log T = 5 and 5.5. The Mg II line exhibits three interstellar absorption components along the 15 pc line of sight. We identify the lowest velocity component with the G Cloud, but the other components are not identified with any interstellar clouds previously detected from other lines of sight.

Observations of the ultra-sharp-lined, chemically peculiar star chi Lupi taken by the Goddard High Resolution Spectrograph in echelle mode are presented. Thirty-six intervals of the spectral region between 1249 and 2688 Angstrom are covered with resolving powers in the range 75,000-93,000. Line identifications are provided, and the observed spectra are compared with synthetic spectra calculated using the SYNTHE program and associated line lists with changes to the line lists. The significance of these spectra for the chi Lupi Pathfinder Project and the closely related atomic physics effort is discussed in a companion paper.

Goddard High Resolution Spectrograph echelle-mode observations of the interstellar absorption lines of Mg II, Si IV, C IV, and N V toward μ Columbae (HD 38666) are presented. The observations have a spectral resolution of 3.5 km s-1 and signal-to-noise ratios (S/Ns) of 20–200. The μ Col sight line (l = 2373, b = -271, d = 0.40 kpc, z = -0.18 kpc) extends though the Local Bubble and the warm neutral, warm ionized, and hot ionized phases of the interstellar medium (ISM). The high-ionization column densities toward μ Col are log N(Si IV) = 12.16 ± 0.05, log N(C IV) = 12.88 ± 0.02, and logN(N V) = 11.8–12.3. Profile fits to Copernicus satellite measures of O VI absorption toward μ Col yield log N(O VI) = 13.82±0.01 and b = 38.7 km s-1. This implies N(C IV)/N(O VI) = 0.11 ± 0.01, which is typical of the values found for the hot ISM of the Galactic disk. The O VI profile is twice as broad as the C IV and N V profiles, even though these species have roughly similar average velocities. Some of the C IV, N V, and O VI absorption toward μ Col may occur at the interface of the Local Cloud and Local Bubble, although additional contributions to these ions probably also occur in more distant gas along the sight line. A substantial part of the Si IV absorption likely arises in warm photoionized gas in an H II region surrounding μ Col. The profile width differences among the high-ionization lines of C IV, N V, and O VI could be produced if the line of sight passes through a highly evolved supernova remnant. The observations for μ Col and for other stars observed at high resolution with the GHRS reveal that multiple gas types (warm and hot) contribute to the absorption by the highly ionized atoms along both nearby and distant sight lines. Disentangling the relative contributions from the different gas types requires high-resolution and high-S/N observations. The Mg II observations, combined with a solar Mg reference abundance, imply that the Mg depletion toward μ Col is -0.31 dex. As observed for other sight lines through the warm neutral medium, the gas-phase observations of Mg, when combined with results for Fe and Si, suggest that Mg and Fe are more deficient from the gas phase than one would expect if these elements are only contained in silicate dust grains.

The Space Telescope Imaging Spectrograph (STIS) instrument was installed on the Hubble Space Telescope (HST) during the second servicing mission, in 1997 February. Four bands cover the wavelength range of 115–1000 nm, with spectral resolving powers between 26 and 200,000. Camera modes are used for target acquisition and deep imaging. Correction for HST's spherical aberration and astigmatism is included. The 115–170 nm range is covered by a CsI MAMA (Multianode Microchannel Array) detector and the 165–310 nm range by a Cs2Te MAMA, each with a format of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cy...

Observations of the narrow-lined O-type star 10 Lacertae taken with the Goddard High Resolution Spectrograph in 1992 November are presented. The spectra cover the wavelength range 1181–1777 A with a resolution of 15 km s-1 and signal-to-noise ratio greater than 100:1. Absorption lines arising in the interstellar medium, the photosphere, and the stellar wind are identified and discussed.