Related papers: p-Summing Bloch mappings on the complex unit disc

The $p$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil's exponential sum theorem presents an upper bound of the exponential sum over the $p$-set. Based on the result, one shows that the…

Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<\infty$ let…

We study uniform $\epsilon-$BPB approximations of bounded linear operators between Banach spaces from a geometric perspective. We show that for sufficiently small positive values of $\epsilon,$ many geometric properties like smoothness,…

A subset P of N x N is called Schur bounded if every infinite matrix with bounded entries which is zero off of P yields a bounded Schur multiplier on B(H). Such sets are characterized as being the union of a subset with at most k entries in…

We prove modulation invariant embedding bounds from Bochner spaces $L^p(\mathbb{W};X)$ on the Walsh group to outer-$L^p$ spaces on the Walsh extended phase plane. The Banach space $X$ is assumed to be UMD and sufficiently close to a Hilbert…

We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of…

In this paper we extend the scope of three important results of the linear theory of absolutely summing operators. The first one was proved by Bu and Kranz in \cite{BK} and it asserts that a continuous linear operator between Banach spaces…

Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of…

We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak$^*$-compact set. We prove that a weak$^*$-compact set has summable Szlenk index if and only if its weak$^*$-closed, absolutely convex hull does. As a…

Let $f$ be a complex-valued harmonic mapping defined in the unit disk $\mathbb D$. We introduce the following notion: we say that $f$ is a Bloch-type function if its Jacobian satisfies $$ \sup_{z\in\mathbb D}(1-|z|^2)\sqrt{|J_f(z)|}<\infty.…

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric…

The following result was announced in the earlier version(s) of this paper: On weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded,…

We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump…

We study the free Banach lattice $FBL^{(p,\infty)}[E]$ with upper $p$-estimates generated by a Banach space $E$. Using a classical result of Pisier on factorization through $L^{p,\infty}(\mu)$ together with a finite dimensional reduction,…

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the…

This article examines a family of smooth mappings between Banach spaces and establishes conditions for the existence of their zeros. Applications to fixed-point problems and the Implicit Function Theorem are also discussed.

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and…

Given a set $B$ of operators between subspaces of $L_p$ spaces, we characterize the operators between subspaces of $L_p$ spaces that remain bounded on the $X$-valued $L_p$ space for every Banach space on which elements of the original class…

A display of a topological group G on a Banach space X is a topological isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm ||.|| on X, where the latter group is equipped with the strong operator topology.…

We consider two questions on the geometry of Lipschitz free $p$-spaces $\mathcal F_p$, where $0<p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal M, \rho)$ is an infinite…