Related papers: Pisier's inequality revisited

J. Elton proved that every normalized weakly null sequence in a Banach space admits a subsequence that is nearly unconditional which is a weak form of unconditionality. The notion of near-unconditionality is quantified by a constant…

In the setting of a complete, doubling metric measure space $(X,d,\mu)$ supporting a $(1,1)$-Poincar\'e inequality, we show that for all $0<\theta<1$, the following fractional Poincar\'e inequality holds for all balls $B$ and locally…

Consider the linear stochastic evolution equation dU(t) = AU(t) + dW_H(t), t\ge 0, where A generates a C_0-semigroup on a Banach space E and W_H is a cylindrical Brownian motion in a continuously embedded Hilbert subspace H of E. Under the…

It is shown that if $1<p<\infty$ and $X$ is a subspace or a quotient of an $\ell_p$-direct sum of finite dimensional Banach spaces, then for any compact operator $T$ on $X$ such that $\|I+T\|>1$, the operator $I+T$ attains its norm. A…

We study the Dirichlet problem for a class of fractional $p$-Laplacian operators of order $s \in (0,1)$ defined through the Riesz fractional gradient, which differs fundamentally from the standard fractional $p$-Laplacian. Our analysis…

We give a derivation for the value of inf-sup constant for the bilinear form (p, div u). We prove that the value of inf-sup constant is equal to 1.0 in all cases and is independent of the size and shape of the domain. Numerical tests for…

Motivated by a problem stated by S.A.Argyros and Th. Raikoftsalis, we introduce a new class of Banach spaces. Namely, for a sequence of separable Banach spaces $(X_n,\|\cdot\|_n)_{n\in\mathbb{N}}$, we define the Bourgain Delbaen…

Let $Z$ and $X$ be Banach spaces. Suppose that $X$ is Asplund. Let $\mathcal{M}$ be a bounded set of operators from $Z$ to $X$ with the following property: a bounded sequence $(z_n)_{n\in \mathbb{N}}$ in $Z$ is weakly null if, for each $M…

We investigate the distance function $\boldsymbol{\delta}_{K}^{\phi}$ from an arbitrary closed subset $ K $ of a~finite-dimensional Banach space $ (\mathbf{R}^{n}, \phi) $, equipped with a uniformly convex $\mathcal{C}^{2}$-norm $ \phi $.…

Let $\{\tau_j\}_{j=1}^n$ be a collection in a finite dimensional Banach space $\mathcal{X}$ of dimension $d$ and $\{f_j\}_{j=1}^n$ be a collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) such that $f_j(\tau_j) =1$, $\forall 1\leq j\leq…

Let $E$ be a Sidon subset of the integers and suppose $X$ is a Banach space. Then Pisier has shown that $E$-spectral polynomials with values in $X$ behave like Rademacher sums with respect to $L_p-$norms. We consider the situation when $X$…

The Lipschitz extension modulus $e(M)$ of a metric space $M$ is the infimum over $L\ge 1$ such that for any Banach space $Z$ and any $C\subset M$, any 1-Lipschitz function $f:C\to Z$ can be extended to an $L$-Lipschitz function $F:M\to Z$.…

We prove uniform parabolic H\"older estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals \[ \mathcal{E}_\varepsilon(W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{…

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in…

The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, defined on Banach spaces…

It was shown by G. Pisier that any finite-dimensional normed space admits an $\alpha$-regular $M$-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball…

In this paper, we obtain a minimax theorem by means of which, in turn, we prove the following result: Let $E$ be an infinite-dimensional reflexive real Banach space, $T:E\to E$ a non-zero compact linear operator, $\varphi:E\to {\bf R}$ a…

Let $\mathcal{P}({\bf N})$ be the power set of $\bf N$. An upper density (on $\bf N$) is a non\-decreasing and subadditive function $\mu^\ast: \mathcal{P}({\bf N})\to\bf R$ such that $\mu^\ast({\bf N}) = 1$ and $\mu^\ast(k \cdot X + h) =…

Let $1<p<\infty$. Let $\{T_t\}_{t>0}$ be a noncommutative symmetric diffusion semigroup on a semifinite von Neumann algebra $\mathcal{M}$, and let $\{P_t\}_{t>0}$ be its associated subordinated Poisson semigroup. The celebrated…

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent,…