LetX and Y be closed subspaces of the Lorentz sequence space d(v, p) and the Orlicz sequence space l_M, respectively. It is proved that every bounded linear operator from X to Y is compact whenever $p > \beta _M : = \inf \{ q > 0:\inf \{ M(\lambda t)/M(\lambda )t^q :0< \lambda ,t \leqslant 1\} > 0.$ As an application, the reflexivity of the space of bounded linear operators acting from d(v, p) to l_M is characterized.

Letf be a one-to-one analytic function in the unit disc with f′(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions (f′)^p, where p<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative of f. We use this to improve known estimates for integral means of the functions |f′|^p for integers p⪯−2.

LetD⊂C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. For R>0, let ω_D(R) denote the harmonic measure at 0 of the set {z:|z|≽R}⋔∂D. Then it is shown that there exist β>0and C>0such that for each such D, ω_D(R)≤Ce^−βR, for every R>0. Thus a natural question is: What is the supremum of all β′s , call it β₀, for which the above inequality holds for every such D? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β₀ is found. Upper bounds for β₀ can be obtained by constructing examples of domains D. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.

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.

LetX be a rearrangement-invariant Banach function space on Rⁿ and let V¹X be the Sobolev space of functions whose gradient belongs to X. We give necessary and sufficient conditions on X under which V¹X is continuously embedded into BMO or into L_∞. In particular, we show that L_{n, ∞} is the largest rearrangement-invariant space X such that V¹X is continuously embedded into BMO and, similarly, L_{n, 1} is the largest rearrangement-invariant space X such that V¹X is continuously embedded into L_∞. We further show that V¹X 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 V¹X into C⁰ is characterized as well.

Let Ω be a bounded domain in Cⁿ. 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 w₀ of Ω. We find conditions on Ω which ensure that lim_w→wog_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 w₀ which satisfies a certain growth condition.

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→∞.

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.