Results are obtained on the scattering theory for the Schrödinger equation $$i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$$ in spaces$L$^{$r$}($R$;$L$^{$q$}($R$^{$d$})) for a certain range of$r, q$, the so-called space-time scattering. In the linear case (i.e.$F$≡)) the relation with usual configuration space scattering is established.

We construct a singular probability measure μ on the complex sphere such that the Poisson integral of μ is a pluriharmonic function in the ball and the Fourier transform of μ is $$\mathcal{O}(1/\sqrt p )$$ as$p$→∞.

Let Ω be a bounded domain in$C$^{$n$}. This paper deals with the study of the behavior of the pluricomplex Green function$g$_{Ω}($z, w$) when the pole$w$tends to a boundary point$w$_{0}of Ω. We find conditions on Ω which ensure that lim_{$w→wo$}$g$_{$Ω$}$(z, w)$=0, uniformly with respect to$z$on compact subsets of $$\bar \Omega \backslash \{ w_0 \} $$ . Our main result is Theorem 5; it gives a sufficient condition for the above property to hold, formulated in terms of the existence of a plurisubharmonic peak function for Ω at$w$_{0}which satisfies a certain growth condition.

Let$X$be a rearrangement-invariant Banach function space on$R$^{$n$}and let$V$^{1}$X$be the Sobolev space of functions whose gradient belongs to$X$. We give necessary and sufficient conditions on$X$under which$V$^{1}$X$is continuously embedded into BMO or into$L$_{∞}. In particular, we show that$L$_{$n, ∞$}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into BMO and, similarly,$L$_{$n$, 1}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into$L$_{∞}. We further show that$V$^{1}$X$is a subset of VMO if and only if every function from$X$has an absolutely continuous norm in$L$_{$n, ∞$}. A compact inclusion of$V$^{1}$X$into$C$^{0}is characterized as well.

In this paper we present a deterministic Cantor-type construction of linear fractal Salem sets with prescribed dimension. The construction rests on a paper of Kaufman [10] where he investigated the Fourier dimension of the set of α-well approximable numbers in$R$.