Inequalities of the form $$\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $$ for all$f$∈$H$^{1}, where {$m$_{$k$}} are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterize$B$-convexity. It is also shown that for 1<$q$<∞ and 0<α<∞ the space$X=H$(1,$q$,γa) consisting of analytic functions on the unit disc such that $$\int_0^1 {(1 - r)^{q\alpha - 1} M_1^q (f,r) dr< \infty } $$ satisfies the previous inequality for vector valued functions in$H$^{1}($X$), defined as the space of$X$-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {$m$_{$k$}}={2^{k}} but not for {$m$_{$k$}}={$k$^{$a$}} for any α ∈ N.

Let$B$be the unit ball of$C$^{$n$}, I give necessary conditions on sequence$S$of points in$B$to be$H$^{∞}($B$) interpolating in term of a$C$^{$n$}valued holomorphic function zero on$S$(a substitute for the interpolating Blaschke product).

Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and let$A(Δ)$be the algebra of complex functions continuous on Δ and analytic in int Δ. Let$K$be a compact set in C^{2}such that Π($K$)=Γ, and let$K$_{λ}≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,$K$_{λ}is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))^{2}-S(λ) quadratic in w with$R,S∈A(Δ)$such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ int$K$_{λ}, where$S$has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component of$K$_{λ}. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull of$K$. Furthermore, we show that БК/K is the disjoint union of such disks.

Let$M$be a connected, noncompact, complete Riemannian manifold, consider the operator$L=Δ+∇V$for some$V∈C$^{2}(M) with exp[$V$] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of$L$. As a consequence of the main result, let ϱ be the distance function from a point o, then the spectral gap exists provided lim_{ϱ→∞}sup$L$_{ϱ<0}while the spectral gap does not exist if o is a pole and lim_{ϱ→∞}inf$L$_{ϱ≥0}. Moreover, the elliptic operators on$R$^{$d$}are also studied.

For Ξ∈$R$^{$n$},$t$∈$R$and$f$∈$S$($R$^{$n$}) define $$\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$$ . We determine the optimal regularity$s$_{0}such that $$\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$$ holds where$C$is independent of$f$∈$S$($R$^{$n$}) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11].