Revisiting the Rectangular Constant in Banach Spaces

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009

Let X be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of X-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents p ≥ 2 if and only if the space X is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.

Mathematical Inequalities & Applications, 2008

For any Banach space X the n -th James constants J n (X) and the n -th Khintchine constants K n p,q (X) are investigated and discussed. Some new properties of these constants are presented. The main result is an estimate of the n -th Khintchine constants K n p,q (X) by the n -th James constants J n (X) . In the case of n = 2 and p = q = 2 this estimate is even stronger and improvs an earlier estimate proved by . : 46B20, 46E30, 46A45, 46B25.

Bulletin of the Australian Mathematical Society, 2007

Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.

2022

For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. We also introduce the following geometric notion of $p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \in \mathbb{N}$. For $p\in [2,\infty)$, a complex Banach space $X$ is called $p$-uniformly $\mathbb{C}$-convex of order $N$ if there exists a constant $\lambda > 0$ such that \begin{align}\label{e-0.1} \left(\norm{x_0}^p + \lambda \norm{x_1}^p + {\lambda}^2 \norm{x_2}^p + \cdots + {\lambda}^N \norm{x_N}^p \right)^{1/p} \leq \max_{\theta \in [0,2\pi)} \norm{x_0 + \sum_{k=1}^{N}e^{i \theta}x_k} \end{align} for all $x_0$, $x_1$,$\dots$, $x_N$ $\in X$. We denote $A_{p,N}(X)$,...

2002

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from L p. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov-Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that L p is the only space where it is possible to change this order.

2016

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Revista Matemática Complutense, 2011

We study the reflexivity, the uniform convexity, the Daugavet property and the Radon-Nikodym property of the generalized Lebesgue spaces L p(x) . in , and continued later by J. Musie lak . The theory of variable exponent spaces of functions defined on the real line was also developed by I. Tsenov [33], I. Sharapudinov [30] and V. V. Zhikov [36] and [37]. In the late 1980's these spaces were investigated in connection with certain specific applications in the continuum mechanics (see, for example, V.V. Zhikov [36]). Some of their fundamental properties were established in [18] by O. Kováčik and J. Rákosník. A deeper study of the norm in L p(x) can be found in [10] by D.E. Edmunds, J. Lang and A. Nekvinda. Yet other authors considered inequalities of Sobolev type and related questions.

As in 2-normed spaces, we can also give two definitions for Cauchy sequences in n-normed spaces. It is known that in some cases, especially in the finite dimensional case and the standard case, two definitions are equivalent. What is not clear is in the infinite dimensional case. In this paper, we will prove that these two definitions are still equivalent in p space.

Journal of Functional Analysis, 2008