It is shown that the punctual quotient scheme Q ${}_{l}^{r}$ parametrizing all zero-dimensional quotients $\mathcal{O}_{A^2 }^{ \oplus ^r } \to T$ of length l and supported at some fixed point O∈A² in the plane is irreducible.

This paper examines the boundary behaviour of superharmonic functions on a half-space in terms of their behaviour along lines normal to the boundary. It is shown that, if the set of lines along which such functions grow quickly is (in a certain sense) metrically dense, then the set of lines along which they are bounded below is topologically small.

We construct singular solutions to equations $div\mathcal{A}(x,\nabla u) = 0,$ similar to the p-Laplacian, that tend to ∞ on a given closed set of p-capacity zero. Moreover, we show that every G_δ-set of vanishing p-capacity is the infinity set of some A-superharmonic function.

We study removable sets for the Sobolev space W^1,p. We show that removability for sets lying in a hyperplane is essentially determined by their thickness measured in terms of a concept of p-porosity.

The equation x⁽ⁿ⁾(t)=(−1)ⁿ│x(t)│^k with k>1 is considered. In the case n≦4 it is proved that solutions defined in a neighbourhood of infinity coincide with C(t−t₀)^−n/(k−1), where C is a constant depending only on n and k. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t₀)^−n/(k−1). It is shown that they do not necessarily coincide with C(t−t₀)^−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics.

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=−1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L_p(w₀), N∩L_p(w₁)), where N is the linear space which consists of all functions with the integral equal to 0. We calculate the K-functional for the couple (N∩L_p(w₀), N∩L_p (w₁)), which turns out to be essentially different from the K-functional for (L_p(w₀), L_p(w₁)), even for the case when N∩L_p(w_i) is dense in L_p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.

A solution to a Dirichlet problem for the complex Monge-Ampère operator is characterized as a minimizer of an energy functional. A mutual energy estimate and a generalization of Hölder's inequality is proved. A comparison is made with corresponding results in classical potential theory.

We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps. If a holomorphic Collet-Eckmann map f is topologically conjugate to a holomorphic map g, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, then g is Collet-Eckmann, too.

Let Ω⊂Rⁿ be an arbitrary open set. In this paper it is shown that if a Sobolev function f∈W^{1, p}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then f is weakly zero on ϖΩ in the sense that f∈W ${}_{0}^{1,p}$ (Ω).

For Ξ∈Rⁿ, t∈R and f∈S(Rⁿ) 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 s₀ 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(Rⁿ) 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].

Our theorems can be generalized to the case where the exp(it|ξ|²) is replaced by exp(it|ξ|^a), a≠2.

The proof uses Parseval's formula on R, orthogonality arguments arising from decomposing L²(Rⁿ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.

LetM be a connected, noncompact, complete Riemannian manifold, consider the operator L=Δ+∇V for some V∈C²(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.

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 C² 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.