For which regions $G$ is the Hardy space $H^{2}(G)$ contained in the Bergman space $L^{2}_{a}(G)$? This paper relates the above problem to that of finding the multipliers of $H^{2}(\mathbb{D})$ into $L^{2}_{a}(\mathbb{D})$. When $G$ is a simply connected region this leads to a solution of the above problem in terms of Lipschitz conditions on the Riemann map of $\mathbb{D}$ onto $G$. For arbitrary regions $G$, it is shown that if $G$ is the range of a function whose derivative is a multiplier from $H^{2}(\mathbb{D})$ to $L^{2}_{a}(\mathbb{D})$, then $H^{2}(G)$ is contained in $L^{2}_{a}(G)$. Also, if $G$ has a piecewise smooth boundary, then it is shown that $H^{2}(G)$ is contained in $L^{2}_{a}(G)$ if and only if the angles at all the “corner” points are at least $\pi/2$. Examples of multipliers from $H^{2}(\mathbb{D})$ to $L^{2}_{a}(\mathbb{D})$ are given; and in particular, every Bergman inner function is such a multiplier.

We prove, generalizing a well known property of Delaunay surfaces, that if the Gauss image of a cmc surface in the Euclidean space is a compact surface with boundary, then any connected component of sphere minus the image is a strictly convex domain. We also obtain conditions under which the Gauss image has a regular boundary. These results relate to the question, raised by do Carmo, of whether the Gauss image of a complete cmc surface contains an equator of the sphere.

We deal with three different problems of the multidimensional integral geometry of foliations. First, we establish asymptotic formulas for integrals of powers of curvature of foliations obtained by intersecting a foliation by affine planes. Then we prove an integral formula for surfaces of contact of an affine hyperplane with a foliation. Finally, we obtain a conformally invariant integral-geometric formula for a foliation in three-dimensional space.

The global asymptotic stability conjecture in dynamical systems was solved recently and independently by Feller, Glutsiuk and Gutierrez. Crucial to the approach of Gutierrez is the following theorem of his: A local diffeomorphism $f:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ for which the eigenvalues of $Df(x)$ miss $(0,\infty)$ must be injective. The present paper gives a partial generalization of this theorem to local diffeomorphisms between Hadamard surfaces, the spectral condition being replaced by transversality conditions among certain foliations associated to horocycles. The proofs use arguments from global analysis.

Let $X$, $Y$ and $E$ be complex Banach spaces, and let $u:X \times Y \rightarrow E$ be a bounded bilinear map. If $f$, $g$ are analytic functions in the unit disc taking values in $X$ and $Y$ with Taylor coefficients $x_{n}$ and $y_{n}$ respectively, we define the $E$-valued function $f \ast_{u}\, g$ whose Taylor coefficients are given by $u(x_{n},y_{n})$. Given two bounded bilinear maps, $u:X \times Y \rightarrow E$ and $v:Z \times E \rightarrow F$, in our main theorem we prove that Young's Theorem can be improved by showing that the function $f \ast_{v}(g \ast_{u} h)$ is in the Hardy space $H^{p}(F)$ provided that $f$, $g$ and $h$ are in the vector valued Besov spaces corresponding to those that appear in some classical inequalities by Hardy-Littlewood and Littlewood-Paley.

We also investigate the class of Banach spaces for which these inequalities hold in the vector setting, and we give a number of applications of our theorem for these spaces and for certain bilinear maps (such as convolution, tensor products, $\ldots$) obtaining results both in the scalar and the vector valued cases.

We show that under the Proper Forcing Axiom, the Alaoglu-Birkhoff equivalence on separative posets of the first uncountable size roughly coincides with regular embeddability. We also investigate the behavior of the equivalence under the Continuum Hypothesis.

We develope a local theory for frames on finite-dimensional Hilbert spaces. We show that for every frame. $(f_{i})^{m}_{i=1}$ for an $n$-dimensional Hilbert space, and for every $\epsilon \gt 0$, there is a subset $I \subset {1,2,\ldots,m}$ with $|I| \geq (1-\epsilon)n$ so that $(f_{i})_{i \in I}$ is a Riesz basis for its span with Riesz basis constant a function of $\epsilon$, the frame bounds, and $(||f_{i}||)^{m}_{i=1}$, but independent of m and n. We also construct an example of a normalized frame for a Hilbert space $H$ which contains a subset which forms a Schauder basis for $H$, but contains no subset which is a Riesz basis for $H$. We give examples to show that all of our results are best possible, and that all parameters are necessary.

We show that the classification problem for genus $n$ Banach spaces can be reduced to the unconditionally primary case and that the critical case there is $n=2$. It is further shown that a genus $n$ Banach space is unconditionally primary if and only if it contains a complemented subspace of genus $(n-1)$. We begin the process of classifying the genus 2 spaces by showing they have a strong decomposition property.

Suppose that $P$ is a Sylow-$p$-subgroup of a solvable group $G$. If $G$ is a transitive permutation group of degree $n$, then the number of $P$-orbits is at most $2n/(p + 1)$. This is used to prove that if $G$ is a faithful irreducible linear group of degree $n$, then the dimension of the centralizer of $P$ is at most $2n/(p + 1)$. The latter result generalizes results of Isaacs and Navarro and is also used to affirmatively answer a question ofMonasur and Iranzo regarding indices of centralizers in coprime operator groups.

This paper characterizes Hilbert functions and Hilbert polynomials of standard algebras over an Artinian ring $R_{0}$.

Let $X$ be a projective variety with $\mathbb{Q}$-factorial singularities, over an algebraically closed field $k$ of characteristic $0$, $L$ an ample Cartier divisor on $X$, and $x$ a non-singular point of $X$. We prove that if for two general points $y,z \in X$ there exists a rational curve $C$ passing through $x, y, z$ such that $(L.C) = 2$, then $(X.L) \simeq (\mathbb{P}^{n}.\mathcal{O}(1))$ or $(Q^{n}.\mathcal{O}(1))$, a hyperquadric.

Let $\Lambda$ be a ring endowed with an involution $a \mapsto \tilde{a}$. We say that two units $a$ and $b$ of $\Lambda$ fixed under the involution are congruent if there exists an element $u \in \Lambda^{\times}$ such that $a = ub\tilde{u}$. We denote by $\mathcal{H}(\Lambda)$ the set of congruence classes. In this paper we consider the case where $\Lambda$ is an order with involution in a semisimple algebra $A$ over a local field and study the question of whether the natural map $\mathcal{H}(\Lambda) \rightarrow \mathcal{H}(\Lambda)$ induced by inclusion is injective. We give sufficient conditions on the order $\Lambda$ for this map to be injective and give applications to hermitian forms over group rings.

Some subsets of a Banach ${}^{\ast}$-algebra $A$ are shown to be dense. In the special case of the algebra of $L(H)$ of all bounded linear operators on a Hilbert space $H$, the set of all $T$ in $L(H)$ for which $T^{n}$ is quasi-normal for no positive integers $n$ is dense in $L(H)$.

We transpose Delon's analysis of types in valued fields to unramified henselian valued fields of mixed characteristic, by using coefficient maps of order $n$. This yields the Ax-Kochen-Ershov transfer principle for the independence property in this class of valued fields.