  We introduce a new approach to obtaining pointwise estimates for solutions of
elliptic boundary value problems when the operator being considered satisfies a
certain type of weighted integral inequalities. The method is illustrated on
several examples, including a scalar second-order elliptic equation, the 3D
Lam\'{e} system, and a scalar higher-order elliptic equation. The techniques
can be extended to other elliptic boundary value problems provided that the
corresponding weighted integral inequalities are satisfied.
