Let \(\Omega\) be a bound domain in \(\mathbb{R}^n\) with smooth boundary. Consider the following elliptic boundary valued problem:

\[\begin{equation} \begin{cases} \Delta u = f \quad \text{in \(\Omega\)}
Xu = g\quad \text{on the boundary} \end{cases} \end{equation} \] Here \(X\) is a transversal vector field to the boundary. This includes the regular Dirichlet and Neumann problem. In this talk, we first introduce suitable Hardy spaces \(H_p(\Omega)\) on \(\Omega\). Then we shall show the inequality \[

\text{norm of second partial differential of $f$} \leq \text{const * norm of $f$}

\]

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