  We present the Lya luminosity functions (LFs) derived by our deep Subaru
narrowband survey that identifies a total of 3,137 Lya emitters (LAEs) at $z =
2.2$ in five independent blank fields. The sample of these LAEs is the largest,
to date, and covers a very wide Lya luminosity range of $\log L_{Ly\alpha} =
41.7-44.4$ erg s$^{-1}$. We determine the Lya LF at $z = 2.2$ with
unprecedented accuracies, and obtain the best-fit Schechter parameters of
$L^{*}_{Ly\alpha} = 5.29^{+1.67}_{-1.13} \times 10^{42}$ erg s$^{-1}$,
$\phi^{*}_{Ly\alpha} = 6.32^{+3.08}_{-2.31} \times 10^{-4}$ Mpc$^{-3}$, and
$\alpha = -1.75^{+0.10}_{-0.09}$ showing a steep faint-end slope. We identify a
significant hump at the LF bright end ($\log L_{Ly\alpha} > 43.4$ erg
s$^{-1}$). Because all of the LAEs in the bright-end hump have (a) bright
counterpart(s) either in the X-ray, UV, or radio data, this bright-end hump is
not made by gravitational lensing magnification bias but AGNs. These AGNs allow
us to derive the AGN UV LF at $z \sim 2$ down to the faint magnitude limit of
$M_{UV} \simeq -22.5$, and to constrain the faint-end slope of AGN UV LF,
$\alpha_{AGN}=-1.2 \pm 0.1$, that is flatter than those at $z > 4$. Based on
the Lya and UV LFs from our and previous studies, we find the increase of Lya
escape fraction $f^{Ly\alpha}_{esc}$ from $z \sim 0$ to $6$ by two orders of
magnitude. This large $f^{Ly\alpha}_{esc}$ increase can be explained neither by
the evolution of stellar population nor outflow alone, but the evolution of
neutral hydrogen HI density in inter-stellar medium that enhances dust
attenuation for Lya by resonance scattering. Our uniform expanding shell models
suggest that the typical HI column density decreases from $N_{HI} \sim 7 \times
10^{19}$ ($z \sim 0$) to $\sim 1 \times 10^{18}$ cm$^{-2}$ ($z \sim 6$) to
explain the large $f^{Ly\alpha}_{esc}$ increase.
