  Nuclear hard X-ray luminosities (Lx,nuc) for a sample of 112 early type
galaxies within a distance of 67 Mpc are used to investigate their relationship
with the central galactic black hole mass Mbh, the inner galactic structure
(using the parameters describing its cuspiness), the age of the stellar
population in the central galactic region, the hot gas content and the core
radio luminosity. Lx,nuc ranges from 10^{38} to 10^{42} erg/s, and the
Eddington ratio Lx,nuc/Ledd from 10^{-9} to 10^{-4}. Lx,nuc increases on
average with the galactic luminosity Lb and Mbh, with a wide variation by up to
4 orders of magnitude at any fixed Lb>6x10^9 Lb,sun or Mbh>10^7 Msun. This
large range should reflect a large variation of the mass accretion rate
\dotMbh. On the circumnuclear scale, \dotMbh at fixed Lb (or Mbh) could vary
due to differences in the fuel production rate from the stellar mass return
linked to the inner galactic structure; however, \dotMbh should vary with
cuspiness by a factor exceeding a few only in hot gas poor galaxies and for
large differences in the core radius. Lx,nuc does not depend on age, but less
luminous nuclei are found among galaxies with a younger stellar component.
Lx,nuc is detected both in gas poor and gas rich galaxies, on average increases
with the total galactic hot gas cooling rate L_{X,ISM}, but again with a large
variation. The lack of a tight relationship between Lx,nuc and the
circumnuclear and total gas content can be explained if the gas is heated by
black hole feedback, and/or the mass effectively accreted can be largely
reduced with respect to that entering the circumnuclear region. Differently
from Lx,nuc, the 5 GHz VLA luminosity shows a trend with the inner galactic
structure similar to that of the total soft X-ray emission; therefore they
could both be produced by the hot gas.
