Geometry of strictly convex domains and an application to the uniform estimate of the $\overline \partial$-problem

Abstract:

In this paper, we construct a nice defining function $\rho$ for a bounded smooth strictly convex domain $\Omega$ in ${R^n}$ with explicit gradient and Hessian estimates near the boundary $\partial \Omega$ of $\Omega$. From the approach, we deduce that any two normals through $\partial \Omega$ do not intersect in any tubular neighborhood of $\partial \Omega$ with radius which is less than $\frac {1} {K}$, where $K$ is the maximum principal curvature of $\partial \Omega$. Finally, we apply such $\rho$ to obtain an explicit upper bound of the constant ${C_\Omega }$ in the Henkin’s estimate ${\left \| {{H_\Omega }f} \right \|_{{L^\infty }(\Omega )}} \leqslant {C_\Omega }{\left \| f \right \|_{{L^\infty }(\Omega )}}$ of the $\partial$-problem on strictly convex domains $\Omega$ in ${{\mathbf {C}}^n}$.