$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation

The paper is concerned with the properties of the solution to a Dirichlet problem for a homogeneous second-order elliptic equation with $L_p$-boundary function, $p>1$. The same conditions are imposed on the coefficients of the equation and the boundary of the bounded domain as were used to establish the solvability of this problem. The $L_p$-norm of the nontangential maximal function is estimated in terms of the $L_p$-norm of the boundary value. This result depends on a new estimate, proved below, for the nontangential maximal function in terms of an analogue of the Lusin area integral.
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