| Abstract: |
| Let $\Omega$ be a domain in $\R^{n}$ ($n \geq 2$)
and $u$ be a nonnegative $p$-superharmonic function in $\Omega$.
In 1994, Kilpel{\a}inen and Mal{\`y} proved that there exists a constant $C > 0$ such that
\begin{equation*}
u(x)
\leq
C \left( \inf_{B(x, R)} u + \mathbf{W}_{p}^{\mu}(x, 2R) \right)
\end{equation*}
whenever $B(x, 2R) \subset \Omega$,
where $\mu$ is the Riesz measure of $u$ and
$\mathbf{W}_{p}^{\mu}(x, 2R)$ is the Wolff potential of $\mu$.
In this talk, we extend this inequality to near the boundary of $\Omega$.
More precisely, we give
a pointwise estimate for $p$-superharmonic functions which vanish on the boundary and
a global integrability estimate of $p$-superharmonic functions.
Combining the two estimates, we give an analog of the Carleson estimate. |
|