Potential space estimates for Green potentials in convex domains

Abstract:

Weak type $(1,1)$ bounds are demonstrated for the operators \[ f \mapsto \int {{\nabla _x}{\nabla _x}G(x,y)f(y)dy} \quad {\text {and}}\quad f \mapsto \int {{\nabla _x}{\nabla _y}G(x,y)f(y)dy,} \] where $G$ is the Green operator for the Dirichlet problem for the Poisson equation on a bounded convex domain in ${\mathbb {R}^n}$. These results are used to investigate smoothing properties of the Green operator in potential spaces. An application is given to the restriction of the potential space to the boundary of the domain.