  The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for
$1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman
kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for
$1\le p\le 2$. As applications, we show that for each $1\le p\le 2$,
$K_p(\cdot,z)\in L^q(\Omega)$ for $q< \frac{2pn}{2n-\alpha(\Omega)}$ and
$|K_s(z)-K_p(z)| \lesssim |s-p||\log |s-p||$ whenever the hyperconvexity index
$\alpha(\Omega)$ is positive. Counterexamples for $2<p<\infty$ are given
respectively. An optimal upper bound for the holomorphic sectional curvature of
the $p-$Bergman metric when $2\le p<\infty$ is obtained. For bounded $C^2$
domains, it is shown that the Hardy space and the Bergman space satisfy
$H^p(\Omega)\subset A^q(\Omega)$ where $q=p(1+\frac1n)$. A new concept
so-called the $p-$Schwarz content is introduced. As applications, upper bounds
of the Banach-Mazur distance between $p-$Bergman spaces are given, and
$A^p(\Omega)$ is shown to be non-Chebyshev in $L^p(\Omega)$ for $0<p\le 1$. For
planar domains, we obtain a rigidity theorem for the $p-$Bergman kernel (which
is not valid in high dimensional cases), and a characterization of non-isolated
boundary points through completeness of the Narasimhan-Simha metric.
