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Potential space estimates for Green potentials in convex domains. (English) Zbl 0789.35047

The author proves some (1,1) weak type bounds for the operators \[ f\mapsto \int\nabla_ x \nabla_ x G(x,y)f(y)dy \qquad\text{and} \qquad f\mapsto\int \nabla_ x \nabla_ y G(x,y)f(y)dy, \] \(G\) being the Green operator associated to the Dirichlet problem for the Poisson equation in a bounded and convex domain in \(\mathbb{R}^ n\). The smoothing properties of the Green operator in the Bessel potential space is then studied, and an application to the restriction of Sobolev spaces is included.
Reviewer: I.Vrabie (Iaşi)

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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