We study order convexity and concavity of quasi-Banach Lorentz spaces ${\Lambda }_{p,w}$, where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that ${\Lambda }_{p,w}$ contains an order isomorphic copy of $l^p$. We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for ${\Lambda }_{p,w}$. We conclude with a characterization of the type and cotype of ${\Lambda }_{p,w}$ in the case when ${\Lambda }_{p,w}$ is a normable space.

Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ to a quasi-Banach space ℬ if and only if sup${\left|\right|T\left(a\right)\left|\right|}_{ℬ}$: a is an infinitely differentiable (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ < ∞, where the (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ is as defined by Han, Paluszyński and Weiss.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.

Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into ${ℝ}^d$. The paper deals with Y-weak cluster points ϕ̅ of the sequence $\varphi (·,z_j\left(·\right))$ in X, where $z_j:\Omega \to {ℝ}^m$ is measurable for j ∈ ℕ and $\varphi :\Omega \times {ℝ}^m\to {ℝ}^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{\varphi }$, the integral $I(\varphi ,{\nu }_x):={\int }_{{ℝ}^m}\varphi (x,\lambda )d{\nu }_x\left(\lambda \right)$ exists for $x\in \Omega \setminus A_{\varphi }$ and $\varphi ̅\left(x\right)=I(\varphi ,{\nu }_x)$ on $\Omega \setminus A_{\varphi }$, where $\nu ={{\nu }_x}_{x\in \Omega }$ is a measurable-dependent family of Radon probability measures on ${ℝ}^m$.

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain...

Geometric structure of Cesàro function spaces $Ce{s}_p\left(I\right)$, where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $Ce{s}_p[0,1]$ contains isomorphic and complemented copies of $l_q$-spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $Ce{s}_p[0,1]$.

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We show that $E\left(\mu \right)$ is order bounded if and only if it is contained in a principal ideal in $ba\left(\Re \right)$ if and only if it is weakly compact and $extrE\left(\mu \right)$ is contained in a principal ideal in $ba\left(\Re \right)$. We also establish some criteria for the coincidence of the ideals, in $ba\left(\Re \right)$, generated by $E\left(\mu \right)$ and $extrE\left(\mu \right)$.

We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{\epsilon }\left(x\right):={\sum }_{j\in D_{\epsilon }\left(x\right)}e{*}_j\left(x\right)e_j$, where $D_{\epsilon }\left(x\right):=j:|e{*}_j\left(x\right)|\ge \epsilon$. We study a generalized version of $T_{\epsilon }$ that we call the weak thresholding approximation. We modify the $T_{\epsilon }\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,\epsilon }\left(x\right):=j:t\epsilon \le |e{*}_j\left(x\right)|<\epsilon$ and consider...

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...

Let ${\beta \left(n\right)}_{n=0}^{\infty }$ be a sequence of positive numbers and 1 ≤ p < ∞. We consider the space ${\ell }^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }f̂\left(n\right)zⁿ$ such that ${\sum }_{n=0}^{\infty }{\left|f̂\left(n\right)\right|}^p{\left|\beta \left(n\right)\right|}^p<\infty$. We give some sufficient conditions for the multiplication operator, $M_z$, to be unicellular on the Banach space ${\ell }^p\left(\beta \right)$. This generalizes the main results obtained by Lu Fang [1].

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

It is known that $ℬ\left({\ell }^p\right)$ is not amenable for p = 1,2,∞, but whether or not $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞), then so are ${\ell }^{\infty }\left(ℬ\left({\ell }^p\right)\right)$ and ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$. Moreover, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable so is ${\ell }^{\infty }(,\left(E\right))$ for any index set and for any infinite-dimensional ${ℒ}^p$-space E; in particular, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$. We show that ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over...

The Rademacher sums are investigated in the Cesàro spaces $Ce{s}_p$ (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces $K_{p,w}$ on [0,1]. They span l₂ space in $Ce{s}_p$ for any 1 ≤ p < ∞ and in $K_{p,w}$ if and only if the weight w is larger than $tlog{₂}^{p/2}(2/t)$ on (0,1). Moreover, the span of the Rademachers is not complemented in $Ce{s}_p$ for any 1 ≤ p < ∞ or in $K_{1,w}$ for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.