  We report X-ray fluxes in the 2--10 keV band from LINERs (low-ionization
nuclear emission-line regions) and low-luminosity Seyfert galaxies obtained
with the ASCA satellite. Observed X-ray luminosities are in the range between
4e39 and 5e41 ergs/s, which are significantly smaller than that of the
``classical'' low-luminosity Seyfert 1 galaxy NGC 4051. We found that X-ray
luminosities in 2--10 keV of LINERs with broad Halpha emission in their optical
spectra (LINER 1s) are proportional to their Halpha luminosities. This
correlation strongly supports the hypothesis that the dominant ionizing source
in LINER 1s is photoionization by hard photons from low-luminosity AGNs. On the
other hand, the X-ray luminosities of most LINERs without broad Halpha emission
(LINER 2s) in our sample are lower than LINER 1s at a given Halpha luminosity.
The observed X-ray luminosities in these objects are insufficient to power
their Halpha luminosities, suggesting that their primary ionizing source is
other than an AGN, or that an AGN, if present, is obscured even at energies
above 2 keV.
