************************************************************************* 73018 The soft X-ray luminosity function of galaxies is dominated by AGN above L_X=10^42 erg/s, below this value by normal galaxies. We have investigated a complete flux-limited sample of ROSAT Bright Survey AGN and galaxies with log L_X<42.5 to discriminate between the nuclear and galaxy flux. Using ROSAT HRI data we calculated the radial wobble- corrected profiles of the sources which were compared with a re- calibrated ROSAT HRI Point Spread function. About half of the studied low-luminosity AGN show significant contribution of stellar emission to the soft X-ray luminosity. Some galaxies are not X-ray extended indicating a possible X-ray flux contribution from a hidden AGN. We propose to perform ASCA SIS and GIS hard X-ray spectroscopy of 3 nearby galaxies in order to search for AGN activity. *************************************************************************