  We present 4,987 white dwarf (WD) candidates selected from matched stars
between the multi-band imaging datasets of Subaru Hyper Suprime-Cam (HSC)
Survey and SDSS in the Stripe82 region covering about 165 deg$^2$. We first
select WD candidates from the "reduced proper motion" diagram that is obtained
by combining the apparent magnitude in the range $i=19$ -- 24 and the proper
motion measured by comparing the astrometric positions of each object between
the two datasets over about 14 yr time baseline. We refine the WD candidates by
fitting blackbody and template WD atmosphere models to HSC photometries for
each candidate, enabling the estimation of photometric distance and tangential
velocity ($v_{\rm t}$) with respect to the Sun. The deep HSC data allows us to
identify low-temperature ($<4000$ K) and faint WD candidates down to absolute
magnitude, $M_{\rm bol}\simeq 17$. We evaluate the selection function of our WD
candidates using the mock catalogue for spatial and kinematic distributions of
WDs in the (thin and thick) disc and halo regions based on the standard Milky
Way model. We construct the samples of disc and halo WD candidates by selecting
WDs with the cuts of tangential velocity, $40<v_{\rm t}/[{\rm km}~{\rm
s}^{-1}]<80$ and $200<v_{\rm t}/[{\rm km}~{\rm s}^{-1}]<500$, respectively. The
total number densities of the disc and halo WDs are $(9.45 \pm 0.94) \times
10^{-3}$ pc$^{-3}$ and $(4.20 \pm 1.74) \times 10^{-4}$ pc$^{-3}$,
respectively. Our LFs extend down to fainter absolute magnitudes compared with
previous work. The faint WDs could represent the oldest generation of building
blocks in the tens of billions of years of our Milky Way's assembly history.
