Maximal operators and boundary values of solutions of elliptic equations (Q1081750)

The author has considered the uniformly elliptic equation \[ (1)\quad \sum^{m}_{i,k=1}\partial /\partial x_ i(a_{ik} \partial u/\partial x_ k)+au=0 \] and proved the following theorems. Theorem 1. Let \(F\in C^ 2(\Omega)\) be the solution of (1) in the domain \(\Omega\) and \(F\in H_ p(\Omega)\), \(1<p<\infty\). Then there exists a function \(f\in L_ p(\partial \Omega)\) such that \[ F(x)=\int_{\partial \Omega}\epsilon '_ x(y)f(y)dy,\quad x\in \Omega \] where \(\epsilon\) 'x is the density of the sweeped-out Dirac measure \(\epsilon_ x'\) on \(\partial \Omega.\) Theorem 2. If \[ F(x)=\int_{\partial \Omega}\epsilon '_ x(y)f(y)dy,\quad x\in \Omega,\quad f\in L_ p(\partial \Omega), \] then the maximal operator \(Mf(y)=\sup_{0<\gamma <1}| F(y)|\), \(y\in \partial \Omega\), is continuous in the space \(L_ p(\partial \Omega)\), \(1<p<\infty\), and has weak type in \(L_ 1(\partial \Omega).\) Moreover, the properties of the Poisson kernel and the boundary values of the solutions of (1) are also studied.