Let$L$=−Δ+$V$be a Schrödinger operator on ℝ^{$d$},$d$≥3. We assume that$V$is a nonnegative, compactly supported potential that belongs to$L$^{$p$}(ℝ^{$d$}), for some$p$>$d$$/$2. Let$K$_{$t$}be the semigroup generated by −$L$. We say that an$L$^{1}(ℝ^{$d$})-function$f$belongs to the Hardy space $H^{1}_{L}$ associated with$L$if sup_{$t$>0}|$K$_{$t$}$f$| belongs to$L$^{1}(ℝ^{$d$}). We prove that $f\in H^{1}_{L}$ if and only if$R$_{$j$}$f$∈$L$^{1}(ℝ^{$d$}) for$j$=1,…,$d$, where$R$_{$j$}=($∂$/$∂$$x$_{$j$})$L$^{−1$/$2}are the Riesz transforms associated with$L$.

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Let X and Y be two complex manifolds, let$D$⊂$X$and$G$⊂$Y$be two nonempty open sets, let$A$(resp.$B$) be an open subset of ∂$D$(resp. ∂$G$), and let$W$be the 2-fold cross (($D$∪$A$)×$B$)∪($A$×($B$∪$G$)). Under a geometric condition on the boundary sets$A$and$B$, we show that every function locally bounded, separately continuous on$W$, continuous on$A$×$B$, and separately holomorphic on ($A$×$G$)∪($D$×$B$) “extends” to a function continuous on a “domain of holomorphy” $\widehat{W}$ and holomorphic on the interior of $\widehat{W}$ .