Let$D$be a strictly pseudoconvex domain in$C$^{$n$}. We prove that $$\bar \partial u = \phi , \phi $$ , ϕ a $$\bar \partial $$ (0,1)-form, admits solutions in$L$^{$p$}(∂$D$), 1≤$p$<∞ and in BMO, under certain Wolff type conditions of ϕ. Some such results (for 1<$p$<∞) have previously been obtained by Amar in the ball, but under slightly stronger hypotheses. As a corollary we obtain a$H$^{$p$}-corona result for two generators.

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It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. We discuss its recent extensions given in \cite{TiDu16} in some heterogeneously localized nonlocal function spaces. The new trace theorems are stronger and more general than the classical result. They can be established essentially for all functions having only square integrability away from the boundary or in any compact subset of interior domain. Yet, the heterogeneous localization offers the necessary regularity precisely at the boundary to have well-defined traces. A consequence is that we may study associated Dirichlet type boundary value problems, as well as the coupling of local and nonlocal equations through co-dimension-1 interfaces. %Their derivations not only involve various extensions of classical inequalities but also %require new techniques absent from traditional approaches.