  Recently, a precise (sub-arcsecond) localization of the repeating fast radio
burst (FRB) 121102 has led to the discovery of persistent radio and optical
counterparts, the identification of a host dwarf galaxy at a redshift of
$z=0.193$, and several campaigns of searches for higher-frequency counterparts,
which gave only upper limits on the emission flux. Although the origin of FRBs
remains unknown, most of the existing theoretical models are associated with
pulsars, or more specifically, magnetars. In this paper, we explore persistent
high-energy emission from a rapidly rotating highly magnetized pulsar
associated with FRB 121102 if internal gradual magnetic dissipation occurs in
the pulsar wind. We find that the efficiency of converting the spin-down
luminosity to the high-energy (e.g., X-ray) luminosity is generally much
smaller than unity, even for a millisecond magnetar. This provides an
explanation for the non-detection of high-energy counterparts to FRB 121102. We
further constrain the spin period and surface magnetic field strength of the
pulsar with the current high-energy observations. In addition, we compare our
results with the constraints given by the other methods in previous works and
would expect to apply our new method to some other open issues in the future.
