Dahlberg’s Bilinear Estimate for Solutions of Divergence Form Complex Elliptic Equations

Ω ∇u · v, where u is harmonic in the domain Ω ≡ {(x, t) ∈ Rn+1 : t > φ(x)}, with φ Lipschitz, and where v ∈ W loc is vector valued. He showed that the bilinear form (1.1) is bounded by the L2 norm of the square function plus the non-tangential maximal function of u, times the same expression for v. In the present note, we generalize Dahlberg’s Theorem to variable coefficient divergence form elliptic operators. To be precise, let