Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin Spaces and Applications to Problems in Partial Differential Equations

In their ground-breaking work [42], D. Jerison and C. Kenig have studied the well-posedness of the Poisson problem for the Dirichlet Laplacian on Besov and Bessel potential spaces, ∆u = f ∈ B α (Ω), u ∈ B α+2(Ω), Tru = 0 on ∂Ω, (1.1) ∆u = f ∈ Lα(Ω), u ∈ Lpα+2(Ω), Tru = 0 on ∂Ω, (1.2) in a bounded Lipschitz domain Ω ⊂ R. Let GD be the Green operator associated with the Dirichlet Laplacian in Ω ⊂ R. That is, for f ∈ C∞(Ω̄), the function u := GD f ∈W (Ω) is the unique solution (given by Lax-Milgram’s lemma) of the variational problem