This course presents a unified approach to extending boundary data from rough domains into the interior, with a focus on applications to boundary value problems for elliptic operators. We study recent advances in constructing \emph{smooth harmonic-type extensions} of BMO and ( L^p ) functions from the boundary ( \partial \Omega ) of a domain ( \Omega \subset \mathbb{R}^{n+1} ), where the geometry of ( \Omega ) may be highly irregular.
The domains under consideration include:
\begin{itemize}
\item \textbf{Corkscrew domains} when ( \partial \Omega ) is ( n )-dimensional and Ahlfors regular,
\item and \textbf{complements of ( s )-Ahlfors regular sets} when ( s < n ).
\end{itemize}
The core objectives of the course include:
\begin{itemize}
\item Constructing \emph{smooth interior extensions} of boundary functions with optimal control in terms of \emph{Carleson measures} and \emph{non-tangential maximal functions},
\item Establishing \emph{pointwise convergence} of these extensions back to the boundary data in a non-tangential sense,
\item Showing how \emph{Lipschitz boundary data} yields Lipschitz continuous extensions up to the closure of the domain.
\end{itemize}
A significant portion of the course will be dedicated to \textbf{applications in elliptic boundary value problems}, particularly for \emph{divergence-form elliptic systems with rough (e.g., merely bounded, complex-valued) coefficients}. We will explore:
\begin{itemize}
\item The role of these extensions in solving \emph{Dirichlet problems with ( L^p ) and BMO boundary data},
\item Connections between \emph{interior regularity in Carleson or tent spaces} and the \emph{solvability of Poisson problems},
\item How these tools fit into the modern framework of harmonic analysis on non-smooth domains.
\end{itemize}
The course is aimed at graduate students and researchers interested in \emph{elliptic PDEs, harmonic analysis, and geometric measure theory}. It will balance theoretical development with motivation from concrete problems in analysis and PDE.
