  We study the Cauchy problem of the damped wave equation \begin{align*}
  \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp
$L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$
with derivative loss. This is an improvement of the so-called Matsumura
estimates. Moreover, as its application, we consider the nonlinear problem with
initial data in $(H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r)$ with $r \in
(1,2]$, $s\ge 0$, and $\beta = (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the
local and global existence of solutions. In particular, we prove the existence
of the global solution with small initial data for the critical nonlinearity
with the power $1+\frac{2r}{n}$, while it is known that the critical power
$1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the
asymptotic behavior of the global solution in supercritical cases. Moreover, we
present blow-up results in subcritical cases. We give estimates of lifespan and
blow-up rates by an ODE argument.
