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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.

To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. In the language of dagger spaces introduced by Grosse-Kl\"onne, the $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of meromorphic differential forms with logarithmic poles for an affinoid toric dagger space over the dagger unit polydisc. It is a complex of ${\mathcal O}^\dagger$-modules with integrable connections and with Frobenius structures defined on the dagger unit polydisc such that traces of Frobenius on fibers at Techm\"uller points define the hypergeometric function over the finite field introduced by Gelfand and Graev.

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In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X(\mu, b)_J$ for a $p$-adic group $G$ and its parahoric subgroup $P_J$. We show that $X(\mu, b)_J \neq \emptyset$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of $\s$-conjugacy classes of a twisted loop group.