In the semi-classical regime we study the resonances of the operator$P$_{$t$}=$h$^{2}Δ+$V$+$t$·δ$V$in some small neighborhood of the first spectral band of$P$_{0}. Here$V$is a periodic potential, δ$V$a compactly supported potential and$t$a small coupling constant. We construct a meromorphic multivalued continuation of the resolvent of$P$_{$t$}, and define the resonances to be the poles of this continuation. We compute these resonances and study the way they turn into eigenvalues when$t$crosses a certain threshold.

A function$G$in a Bergman space$A$^{$p$}, 0<$p$<∞, in the unit disk$D$is called$A$^{$p$}-inner if |$G$|^{$p$}−1 annihilates all bounded harmonic functions in$D$. Extending a recent result by Hedenmalm for$p$=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ_{$D$}denotes the characteristic function of$D$and$G$is an arbitrary$A$^{$p$}-inner function, is continuous in$C$, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman space$A$^{2}to a general$A$^{$p$}-setting.

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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.