Related papers: N^p Spaces

We introduce the $\mathcal{L}^p$ spaces of measurable functions whose $p$-th power is summable with respect to the uniform measure over the Levi-Civita field $\mathcal{R}$. These spaces are the counterparts of the real $L^p$ spaces based…

We study norm-estimates for the $\bar\partial$-equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen-Macaulay and whose underlying reduced space is smooth, the…

In this paper, we study {\it operator spaces\/} in the sense of the theory developed recently by Blecher-Paulsen [BP] and Effros-Ruan [ER1]. By an operator space, we mean a closed subspace $E\subset B(H)$, with $H$ Hilbert. We will be…

We show how the $A_\infty$ class of weights can be considered as a metric space. As far as we know this is the first time that a metric d is considered on this set. We use this metric to generalize the results obtained in [9]. Namely, we…

In this paper, we investigate the approximation behavior of both one and multidimensional neural network type operators for functions in $L^p(I^d,\rho)$, where $1\leq p<\infty$, associated with a general measure $\rho$ defined over a…

We prove that $W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\leq p_{1}<p<p_{2}\leq \infty$ on some classes of manifolds and general metric spaces, where $q_{0}$ depends on our…

Let $X$, $Y$, and $Z$ be Banach spaces, and let $\alpha$ be a tensor norm. Let a bounded linear operator $S\in\mathcal{L}(Z,\mathcal{L}(X,Y))$ be given. We obtain (necessary and/or sufficient) conditions for the existence of an operator…

In this paper we show the weak Banach-Saks property of the Banach vector space $(L_\mu^p)^m$ generated by $m$ $L_\mu^p$-spaces for $1\leq p<+\infty,$ where $m$ is any given natural number. When $m=1,$ this is the famous Banach-Saks-Szlenk…

For $N \geq 3$ and $p \in (1,N)$, we look for $g \in L^1_{loc}(\mathbb{R}^N)$ that satisfies the following weighted logarithmic Sobolev inequality: \begin{equation*} \int_{\mathbb{R}^N} g |u|^p \log |u|^p \ dx \leq \gamma \log \left(…

This paper studies the bounded approximation property (BAP) in quasi Banach spaces. In the first part of the paper we show that the kernel of any surjective operator $\ell_p\to X$ has the BAP when $X$ has it and $0<p\leq 1$, which is an…

A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space…

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $\|Id + T^2\|=1 + \|T^2\|$ holds for every bounded linear operator $T:X\longrightarrow X$. This answers in the…

We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ for $0 < p < \infty$, i. e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C…

The main purpose of this note is to show that the question posed in the paper of Sinha D.P. and Karn A.K.("Compact operators which factor through subspaces of $l_p$ Math. Nachr. 281, 2008, 412-423; see the very end of that paper) has a…

We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse…

In order to find an extension of Brezis -- Lieb's lemma to the case of nets, we replace the almost everywhere convergence by the unbounded order convergence and introduce the Brezis -- Lieb property in normed lattices. Then we identify a…

By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq…

We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We demonstrate an important class of…

We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some…

Related to the concept of $p$-compact operator with $p\in [1,\infty]$ introduced by Sinha and Karn, this paper deals with the space $\mathcal{H}^\infty_{\mathcal{K}_p}(U,F)$ of all Banach-valued holomorphic mappings on an open subset $U$ of…