A multiplier theorem using the Schechter's method of interpolation

AbstractLet m be a measurable bounded function and let us assume that there exists a bounded functions S so that m(ξ)S(ξ)it−1 is a Fourier multiplier on Lp uniformly in t∈R. Then, using the analytic interpolation theorem of Stein, one can show that necessarily m is a Lp multiplier. The purpose of this work is to show that under the above conditions, it holds that, for every k∈N, m(logS)k∈Mp. The technique is based on the Schechter's interpolation method

Cluster of galaxies Abell 1060

This thesis describes the implementation and results of a prograunue of photometry of the relatively nearby southern cluster of galaxies and X -ray source, Abell 1060 = Hydra I.Photographic photometry in B and V wavebands of galaxies in a ~22 degree² region centred on this cluster have been obtained on the basis of measurements of four U.K. Schmidt telescope plates by the COSMOS measuring machine at the Royal Observatory, Edinburgh. The feasibility of using the COSMOS machine to derive reliable isophotal magnitudes of bright and faint galaxies is demonstrated.Multiaperture photoelectric (U, B, V and R -band) photometry has been secured of sixteen galaxies in the field of the cluster, including seven NGC and IC systems. Besides providing photometric calibration, these observations permit the total magnitudes and colours of several galaxies to be derived.The positions and magnitudes of over 5000 galaxies, supplemented in many cases by morphological classifications, by micrometric measurements of dimensions and alignment, and by redshift data, have been analysed to determine the static and dynamical properties of A1060. The cluster is symmetric, centrally condensed, and rich in disk galaxies; members segregate radially by morphological type, with spiral galaxies generally farthest from the centre.The V -band luminosity function of A1060 has been constructed. It is similar in form but different in detail from those of other clusters. The colour -magnitude relation of early -type galaxies in A1060 is outlined, and is shown to be inconsistent with one prediction of a standard theory of the morphological evolution of galaxies and clusters.Visual searches for low surface brightness members of A1060 are described. It is suggested that fewer such galaxies exist than an extrapolation of the luminosity function for all galaxy types to faint magnitudes would predict.Peculiar properties of four galaxies near the centre of the cluster are interpreted in terms of the influence of their environments. Three are anaemic spirals, for which there is strong evidence that an era of gas depletion has recently begun. The first- ranked member, N3311, is an extended system of low mean surface brightness, and may represent a D galaxy in the early stages of evolution. These observations suggest that the cluster Abell 1060 is relatively young in dynamical terms