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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Let $\U_q(\fg)$ be the quantum supergroup of $\gl_{m|n}$ or the modified quantum supergroup of $\osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $\U_q(\fg)$. There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to the category of finite dimensional representations of $\U_q(\fg)$, which preserves ribbon category structures. We show that this functor is full in the cases $\fg=\gl_{m|n}$ or $\osp_{2\ell+1|2n}$. For $\fg=\osp_{2\ell|2n}$, we show that the space $\Hom_{\U_q(\fg)}(V_q^{\otimes r}, V_q^{\otimes s})$ is spanned by images of ribbon graphs if $r+s< 2\ell(2n+1)$. The proofs involve an equivalence of module categories for two versions of the quantisation of $\U(\fg)$.

Three dimensional Einstein gravity with negative cosmological constant -1/\ell^2 deformed by a gravitational Chern-Simons action with coefficient 1/\mu is studied in an asymptotically AdS_3 spacetime. It is argued to violate unitary or positivity for generic \mu due to negative-energy massive gravitons. However at the critical value \mu\ell=1, the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by \ell M=J. The corresponding chiral quantum theory of gravity is conjectured to exist and be dual to a purely right-moving boundary CFT with central charges (c_L,c_R)=(0,3\ell /G).