We investigate possible extensions of the classical Krein-\v{S}mulian theorem to various weak topologies. In particular, we show that if $X$ is a WCG Banach space and $\tau$ is any locally convex topology weaker than the norm-topology, then for every $\tau$-compact norm-bounded set $H$, $\overline{\operatorname{conv}}^{\,\tau}H$ is $\tau$-compact. In arbitrary Banach spaces, the norm-fragmentability assumption on $H$ is shown to be sufficient for the last property to hold.

A new proof to the following result is given: If a Banach space does not contain a copy of $\ell_1[0,1]$, then the Krein-\v{S}mulian theorem holds for every topology $\tau$ induced by a norming set of functionals. We conclude that in such spaces a norm-bounded set is weakly compact if it is merely compact in the topology induced by a boundary. On the other hand, the same statement is obtained for all $C(K)$ and $\ell_1(\Gamma)$ spaces.

In a related paper we showed that Ruelle's property for a Fuchsian group $G$ fails if the group has a condition we called `big deformations near infinity'. In this paper we give geometric conditions on $R = \disk /G$ which imply this condition. In particular, it holds whenever $G$ is divergence type and $R$ has injectivity radius bounded from below. We will also give examples of groups which do not have big deformations near infinity.

Let $T$ be a power-bounded operator on $L_p(\mu)$, $1< p < \infty$. We use a sublinear growth condition on the norms $\{\Vert \sum_{k=1}^n T^k f\Vert _p\}$ to obtain for $f$ the pointwise ergodic theorem with rate, as well as a.e.~convergence of the one-sided ergodic Hilbert transform. For $\mu$ finite and $T$ a positive contraction, we give a sufficient condition for the a.e.~convergence of the ``rotated one-sided Hilbert transform''; the result holds also for $p=1$ when $T$ is ergodic with $T1=1$.

Our methods apply to norm-bounded sequences in $L_p$. Combining them with results of Marcus and Pisier, we show that if $\{g_{n}\}$ is independent with zero expectation and uniformly bounded, then almost surely any realization $\{b_{n}\}$ has the property that for every $\gamma>3/4$, any contraction $T$ on $L_{2}(\mu)$ and $f \in L_{2}(\mu)$, the series $\sum_{k=1}^{\infty} \sfrac{b_{k} T^{k} f(x)}{k^{\gamma}}$ converges $\mu$-almost everywhere. Furthermore, for every Dunford-Schwartz contraction of $L_{1}(\mu)$ of a probability space and $f \in L_{p}(\mu)$, $1<p< \infty$, the series $\sum_{k=1}^{\infty} \sfrac{b_{k} T^{k} f(x)}{k^{\gamma}}$ converges a.e. for $\gamma \in (\max\{\frac{3}{4},\frac{p+1}{2p}\},1]$.

Two subgroups $H$ and $K$ of a finite group $G$ are said to be $\mathcal N$-connected if the subgroup generated by $x$ and $y$ is a nilpotent group, for every pair of elements $x$ in $H$ and $y$ in $K$. This paper is devoted to the study of pairwise $\mathcal N$-connected and permutable products of finitely many groups, in the framework of formation and Fitting class theory.

We study the complexifications of real operator spaces. We show that for every real operator space $V$ there exists a unique complex operator space matrix norm $\{\|\cdot\|_n\}$ on its complexification $V_c = V \+{\rm i} V$ which extends the original matrix norm on $V$ and satisfies the condition $\|x +{\rm i}y\|_n = \|x -{\rm i}y\|_n$ for all $x + {\rm i} y \in M_n(V_c) = M_n(V) \+ {\rm i} M_n(V)$. As a consequence of this result, we characterize complex operator spaces which can be expressed as the complexification of some real operator space. Finally, we show that some properties of real operator spaces are closely related to the corresponding properties of their complexifications.

In this paper, we introduce a certain combinatorial property $Z^\star(k)$, which is defined for every integer $k\ge 2$, and show that every set $E\subset\Z$ with the property $Z^\star(k)$ is necessarily a noncommutative $\Lambda(2k)$ set. In particular, using number theoretic results about the number of solutions to so-called ``$S$-unit equations,'' we show that for any finite set $Q$ of prime numbers the set $E_Q$ of natural numbers whose prime divisors all lie in the set $Q$ is noncommutative $\Lambda(p)$ for every real number $2<p<\infty$.

It is shown that for a complete surface with constant mean curvature $H>1$ in $\mathbb{H}\kern0.5pt ^3$ with boundary and finite index the distance function to the boundary is bounded. We apply this result to establish a sharp height estimate for certain geodesic graphs with noncompact boundary. We also show that a geodesically complete, embedded surface in $\mathbb{H}\kern0.5pt ^3$ with constant mean curvature $H>1$ and bounded Gaussian curvature is proper and has an $\epsilon -$tubular neighborhood on its mean convex side that is embedded. Finally, we use this last result to obtain a monotonicity formula for such a surface.

Let $K$ be a compact subset of an open Riemann surface. We prove that if $L$ is a peak set for $A(K)$, then $A(K)|L=A(L).$ We also prove that if $E$ is a compact subset of $K$ with no interior such that each component of $E^c$ intersects $K^c$, then $A(K)|E$ is dense in $C(E)$. One consequence of the latter result is a characterization of the real-valued continuous functions that when adjoined to $A(K)$ generate $C(K)$.

Let $E$ be a subset of a domain $\Omega $ in Euclidean space. This paper deals with the representation, or approximation, of functions on the boundary of $\Omega $ by sums of Poisson, Green or Martin kernels associated with the set $E$, and with the related issue of whether $E$ can be used to determine the suprema of certain harmonic functions on $\Omega $. The results address several questions raised by Hayman.

We prove a general vanishing exponential integrability result for Besov functions. In a basic case, this allows us to improve the known $\mathcal{O}(1)$ estimate to a $o(1)$ estimate. It also leads to improvements of differentiability results for Besov functions.

We prove upper bound estimates for Euler-Zagier multiple zeta-functions. First, by shifting the paths of the relevant Mellin-Barnes type integrals to the right, we prove an estimate for general $r$-fold zeta-functions. Then, in the cases $r=2$ and $r=3$, we give further improvements by shifting the path suitably to the left.

We consider simply connected complete Riemannian manifolds with sectional curvature bounded above by $-C^2 < 0$, and curves on such manifolds with geodesic curvature at most $C>0$ in absolute value. We give an estimate for the rate at which such curves approach the boundary of the manifold.

We give a partial answer to the question in the title by showing that the McShane and Pettis integrals coincide for functions with values in super-reflexive spaces as well as for functions with values in $c_0(\Gamma)$. We also improve an example of Fremlin and Mendoza, according to which these integrals do not coincide in general, by showing that, at least under the Continuum Hypothesis, there is a scalarly negligible function which is not McShane integrable.

Optimal control systems governed by second-order ODEs with boundary data and variable parameters are considered. Using variational methods a theorem on existence of optimal processes is proven and a sufficient condition for continuous (or semicontinuous) dependence of optimal trajectories and controls on parameters is given.

We provide a direct proof that the Haagerup estimate on the completely bounded norm of elementary operators is best possible in the case of $\mathcal{B}(H)$ via a generalisation of a theorem of Stampfli. We show that for an elementary operator $T$ of length $\ell$, the completely bounded norm is equal to the $k$-norm for $k = \ell$. A $C$*-algebra $A$ has the property that the completely bounded norm of every elementary operator is the $k$-norm, if and only if $A$ is either $k$-subhomogeneous or a $k$-subhomogeneous extension of an antiliminal $C$*-algebra.

For an invariant subspace $I$ of the Bergman space we study an integral operator on $I$ defined in terms of the reproducing kernel of $I$. Such an operator will be called the root operator of $I$ and its associated integral kernel will be called the root function of $I$. We obtain fundamental spectral properties of the root operator when the invariant subspace $I$ has finite index.

We determine the correct dimension gauge for measuring generalized quasicircles (the images of a circle under so-called $\mu$-homeomorphisms). We establish a sharp modulus of continuity estimate for the inverse of a homeomorphism with finite exponentially integrable distortion. We exhibit several illustrative examples.

We prove that, for generic systems of polynomial differential equations, the dependence of the solution on the initial conditions is not differentially algebraic. This answers, in the negative, a question posed by L.A. Rubel.

We present new examples of harmonic and minimal unit vector fields on Riemannian symmetric spaces. These examples are constructed from cohomogeneity one actions with a reflective singular orbit. The radial unit vector field associated to such a reflective submanifold is harmonic and minimal.

It is shown that any generalized Cohen-Macaulay module $M$ can be approximated by a maximal generalized Cohen-Macaulay module $X$ up to a module of finite projective dimension, and such that the local cohomology modules of $M$ and $X$ coincide for all cohomological degrees different from the dimensions of the two modules. By a theorem of Migliore there exist graded generalized Cohen-Macaulay rings which, up to a shift, have predescribed local cohomology modules. Bounds for this shift are given in terms of homological data.

Let $X$ and $Y$ be Banach spaces such that $X$ has a boundedly complete basis. Then $X\hat{\otimes}Y$, the projective tensor product of $X$ and $Y$, has the Radon-Nikodym property (resp.~the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of $c_0$) if and only if $Y$ has the same property.

Let $m(\xi,\eta)$ be a bounded continuous function in $\mathbb{R}\times\mathbb{R}$, let $0< p_i,q_i<\infty$ for $i=1,2$, and let $0<p_3,q_3\le\infty$, be such that $1/p_1+1/p_2=1/p_3$. It is shown that

\[ C_m (f,g)(x)=\int_{\mathbb{R}} \int_{\mathbb{R}} \hat f(\xi) \hat g(\eta) m(\xi,\eta) e^{2\pi i x(\xi +\eta )}d\xi d\eta \]

is a bounded bilinear operator from $L^{p_1,q_1}(\mathbb{R})\times L^{p_2,q_2}(\mathbb{R})$ into $L^{p_3,q_3}(\mathbb{R})$ if and only if

\[ P_{D_{\varepsilon^{-1}}m} (f,g)(\theta)=\sum_{k\in \mathbb{Z}} \sum_{k'\in \mathbb{Z}} \hat f(k) \hat g(k') m(\varepsilon k, \varepsilon k') e^{2\pi i \theta(k +k' )} \]

are bounded bilinear operators from $L^{p_1,q_1}(\T)\times L^{p_2,q_2}(\T)$ into $L^{p_3,q_3}(\T)$ with norm bounded by a uniform constant for all $\epsilon >0$.

We study the invertibility of $\lambda I+K$ in $L^p(\partial\Omega\times\mathbf{R})$, for $p$ near $2$ and $\lambda\in\mathbf{R}$, $|\lambda|\geq\sfrac12$, where $K$ is the caloric double layer potential operator and $\Omega$ is a Lipschitz domain. Applications to transmission boundary value problems are also presented.