  We report on the detection of excess hard X-ray emission from the TeV BL Lac
object Mrk421 during the historical low-flux state of the source in January
2013. NuSTAR observations were conducted four times between MJD56294 and
MJD56312 with a total exposure of 80.9 ksec. The source flux in the 3-40 keV
range was nearly constant except for MJD56307, when the average flux level
increased by a factor of three. Throughout the exposure, the X-ray spectra of
Mrk421 were well represented by a steep power-law model with a photon index of
3.1, although a significant excess was noted above 20 keV in the MJD56302 data
when the source was in its faintest state. Moreover, Mrk421 was detected at
more than the 4-sigma level in the 40-79 keV count maps for both MJD56307 and
MJD56302 but not during the remaining two observations. The detected excess
hard X-ray emissions connect smoothly with the extrapolation of the high-energy
gamma-ray continuum of the blazar constrained by Fermi-LAT during the source
quiescence. These findings indicate that, while the overall X-ray spectrum of
Mrk421 is dominated by the highest-energy tail of the synchrotron continuum,
the variable excess hard X-ray emission above 20 keV (on the timescale of a
week) is related to the inverse Compton emission component. We discuss the
resulting constraints on the variability and spectral properties of the
low-energy segment of the electron energy distribution in the source.
