STRONGLY SINGULAR INTEGRALS ON STRATIFIED GROUPS

We consider a class of spectral multipliers on stratified Lie groups which generalise the class of Hoermander multipliers and include multipliers with an oscillatory factor. Oscillating multipliers have been examined extensively in the euclidean setting where sharp, endpoint L^p estimates are well known. In the Lie group setting, corresponding L^p bounds for oscillating spectral multipliers have been established by several authors but only in the open range of exponents. In this paper we establish the endpoint L^p(G) bound when G is a stratified Lie group. More importantly we begin to address whether these estimates are sharp

From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

We prove a sharp multiplier theorem of Mihlin–Hörmander type for the Grushin operator on the unit sphere in R 3 , and a corresponding boundedness result for the associated Bochner–Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics

On the maximal operator of a general Ornstein--Uhlenbeck semigroup

If Q is a real, symmetric and positive definite nxn matrix, and B a real nxn matrix whose eigenvalues have negative real parts, we consider the Ornstein--Uhlenbeck semigroup on R^n with covariance Q and drift matrix B. Our main result is that the associated maximal operator is of weak type (1,1) with respect to the invariant measure. The proof has a geometric gist and hinges on the ``forbidden zones method'' previously introduced by the third author. For large values of the time parameter, we also prove a refinement of this result, in the spirit of a conjecture due to Talagrand

Palomar 13: a velocity dispersion inflated by binaries ?

Recently, combining radial velocities from Keck/HIRES echelle spectra with published proper motion membership probabilities, Cote et al (2002) observed a sample of 21 stars, probable members of Palomar 13, a globular cluster in the Galactic halo. Their projected velocity dispersion sigma_p = 2.2 +/-0.4 km/s gives a mass-to-light ratio M/L_V = 40 +24/-17, about one order of magnitude larger than the usual estimate for globular clusters. We present here radial velocities measured from three different CCD frames of commissioning observations obtained with the new ESO/VLT instrument FLAMES (Fibre Large Array Multi Element Spectrograph). From these data, now publicly available, we measure the homogeneous radial velocities of eight probable members of this globular cluster. A new projected velocity dispersion sigma_p = 0.6-0.9 +/-0.3 km/s implies Palomar 13 mass-to-light ratio M/L_V = 3-7, similar to the usual value for globular clusters. We discuss briefly the two most obvious reasons for the previous unusual mass-to-light ratio finding: binaries, now clearly detected, and more homogeneous data from the multi-fibre FLAMES spectrograph.Comment: 9 pages, 2 Postscript figure

Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.Comment: 17 page

Constraining Cosmic Evolution of Type Ia Supernovae

We present the first large-scale effort of creating composite spectra of high-redshift type Ia supernovae (SNe Ia) and comparing them to low-redshift counterparts. Through the ESSENCE project, we have obtained 107 spectra of 88 high-redshift SNe Ia with excellent light-curve information. In addition, we have obtained 397 spectra of low-redshift SNe through a multiple-decade effort at Lick and Keck Observatories, and we have used 45 UV spectra obtained by HST/IUE. The low-redshift spectra act as a control sample when comparing to the ESSENCE spectra. In all instances, the ESSENCE and Lick composite spectra appear very similar. The addition of galaxy light to the Lick composite spectra allows a nearly perfect match of the overall spectral-energy distribution with the ESSENCE composite spectra, indicating that the high-redshift SNe are more contaminated with host-galaxy light than their low-redshift counterparts. This is caused by observing objects at all redshifts with the same slit width, which corresponds to different projected distances. After correcting for the galaxy-light contamination, subtle differences in the spectra remain. We have estimated the systematic errors when using current spectral templates for K-corrections to be ~0.02 mag. The variance in the composite spectra give an estimate of the intrinsic variance in low-redshift maximum-light SN spectra of ~3% in the optical and growing toward the UV. The difference between the maximum light low and high-redshift spectra constrain SN evolution between our samples to be < 10% in the rest-frame optical.Comment: 22 pages, 22 figures, submitted to ApJ. Composite spectra can be downloaded from http://astro.berkeley.edu/~rfoley/composite

A compact dust shell in the symbiotic system HM Sge

We present high spatial resolution observations of the mid-infrared core of the dusty symbiotic system HM Sge. The MIDI interferometer was used with the VLT UTs and ATs providing baselines oriented from PA=42&deg; to 105&deg;. The MIDI visibilities are compared with the ones predicted in the frame of various spherical dust shells published in the literature involving single or double dusty shells. The mid-IR environment is unresolved by a 8m telescope and the MIDI spectrum exhibits a level similar to the ISO spectra recorded 10 yr ago. The estimated Gaussian HWHM of the shell of 12AU in the 8-9$\mu$m range, and 18AU in the 11-12$\mu$m range, are much smaller than the angular separation between the Mira and the White Dwarf of 60AU. The discrepancies between the HWHM at different angle orientations suggest an increasing level of asymmetry from 13 to 8$\mu$m. The observations are well fitted by the densest and smallest model published in the literature based on the ISO data, although such a model does not account for the variations of near-IR photometry due to the Mira pulsation cycle suggesting a much smaller optical thickness. These observations also discard the two shells models, developed to take into account the effect of the WD illumination onto the dusty wind of the Mira. These observations show that a high rate of dust formation is occurring in the vicinity of the Mira which seems to be not highly perturbed by the hot companion

Palomar 13's Last Stand

We present a proper motion and CCD photometric study of stars in the distant halo globular cluster Palomar 13. The absolute proper motion of Pal 13 with respect to the background galaxies, derived from moderate scale photographic plates separated by a 40-year baseline, is $(\mu_{\alpha cos \delta}, \mu_{\delta}) = (+2.30, +0.27) \pm (0.26, 0.25)$ milliarc-seconds per year. The resultant total space velocity (315 km s$^{-1}$) implies that Pal 13 is in the inner part of its orbit near perigalacticon. Orbital integration reveals the cluster to possess an inclined, very eccentric, retrograde orbit. These data confirm that Pal 13 is a paradigm "young halo" globular cluster. The derived proper motions for cluster stars are used to produce membership probabilities and a cleaned CCD UBV catalogue for Pal 13. With this data set we have made small revisions to Pal 13's distance, metallicity, position and light profile. The membership of four previously reported RR Lyrae variables and a proportionally large group of blue straggler stars are confirmed. As expected, the blue stragglers are centrally concentrated. The small size of this cluster, combined with the shape of its light profile, which shows a clear departure from a classical King function beyond the tidal radius, suggests that Pal 13 is in the final throes of destruction. This could explain the large blue straggler specific frequency, as destructive processes would preferentially strip less massive stars.Comment: 54 pages, 9 figures, 7 tables, accapted for publication in February 2001 A

Generalized uncertainty inequalities

In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are obtained for a pair of positive-self adjoint operators on a Hilbert space, whose spectral projectors satisfy a ``balance condition'' involving certain operator norms. This result is then applied to obtain uncertainty inequalities on Riemannian manifolds, Riemannian symmetric spaces of non-compact type, homogeneous graphs and unimodular Lie groups with sublaplacians.Comment: 19 page