Poisson semigroups and singular integrals

Abstract

Let D ⊂ R n D \subset {{\mathbf {R}}^n} be a Lipschitz domain and consider the bilinear form ∫ D u ( ∂ v / ∂ y ) d P \int _D {u\left ( {\partial v/\partial y} \right )dP} . We show that the form is bounded if v v is harmonic with boundary values in L 2 {L^2} , if u u is smooth with its nontangential maximal function in L 2 {L^2} and ∫ D dist { P , ∂ D } | grad  u | 2 d P > ∞ \int _D {{\text {dist}}\left \{ {P,\partial D} \right \}{{\left | {{\text {grad }}u} \right |}^2}dP > \infty } .