A quasi-commutativity property of the Poisson and composition operators

Let $\Phi$ be a real valued function of one real variable, let $L$ denote an elliptic second order formally self-adjoint differential operator with bounded measurable coefficients, and let $P$ stand for the Poisson operator for $L$. A necessary and sufficient condition on $\Phi ensuring the equivalence of the Dirichlet integrals of$\Phi\circ Ph$and$P(\Phi\circ h)$ is obtained. We illustrate this result by some sharp inequalities for harmonic functions.