Varopoulos extensions of boundary functions in Lp and BMO in domains with Ahlforsregular boundaries and applications

Abstract

This thesis focuses on the construnction of Varopoulos-type extensions of Lp and BMO boundary functions in rough domains. To be more specific, let Ω ⊂ Rn+1, n ≥ 1, be an open set with s-Ahlfors regular boundary ∂Ω, for some s ∈ (0, n], such that either s = n and Ω is a corkscrew domain with the pointwise John condition, or s < n and Ω = Rn+1 \ E, for some s-Ahlfors regular set E ⊂ Rn+1. In this thesis we provide a unifying method to construct Varopoulos type extensions of Lp and BMO boundary functions. In particu-lar, we show that a) if f ∈ Lp(∂Ω), 1 < p ≤ ∞, there exists F ∈ C∞(Ω) such that the non-tangential maximal functions of F , dist(·, Ωc)|∇F |, as well as the Carleson functional of dist(·, Ωc)s−n∇F are in Lp(∂Ω), with norms controlled by the Lp-norm of f, and F → f in some non-tangential sense Hs|∂Ω-almost everywhere; b) if f¯ ∈ BMO(∂Ω) there exists F¯ ∈ C∞(Ω) such that dist(x, Ωc)|∇F¯(x)| is uniformly bounded in Ω and the Carleson func-tional of dist(x, Ωc)s−n∇F¯(x), as well the the sharp non-tangential maximal function of F¯ are uniformly bounded on ∂Ω with norms controlled by the BMO-norm of f¯, and F¯ → f¯ in a certain non-tangential sense Hs|∂Ω-almost everywhere. If, in addition, the boundary function is Lipschitz with compact support then both F and F¯ can be constructed so that they are also Lipschitz on Ω and converge to the boundary data continuously. The latter results hold without the additional pointwise John condition assumption. Finally, for elliptic systems of equations in divergence form with merely bounded complex-valued coeÿcients, we show some connections between the solvability of Poisson problems with interior data in the appropriate Carleson or tent spaces and the solvability of Dirichlet problem with Lp and BMO boundary data.