On positive almost weak* Dunford–Pettis operators

In this paper, we introduce the class of almost weak* Dunford–Pettis operators and give a characterization of this class of operators. We study its relation with the classes of weak* Dunford–Pettis operators and almost Dunford–Pettis operators, and its relation with the closely related classes of almost limited operators and L-weakly compact operators.     

On the class of positive almost weak Dunford–Pettis operators

We introduce and study the class of almost weak Dunford–Pettis operators and we derive the following interesting consequence: other characterizations of the weak Dunford–Pettis property. After that we characterize pairs of Banach lattices for which the adjoint of almost weak Dunford–Pettis operator is almost Dunford–Pettis. Finally, we establish a necessary and sufficient conditions on the pair of Banach lattices E and F which guarantees that if T : E → F is a positive almost weak Dunford–Pettis then T is almost Dunford–Pettis.     

Positive almost Dunford–Pettis operators and their duality

We study some properties of almost Dunford–Pettis operators and we characterize pairs of Banach lattices for which the adjoint of an almost Dunford–Pettis operator inherits the same property and look at conditions under which an operator is almost Dunford–Pettis whenever its adjoint is.     

Some characterizations of almost Dunford–Pettis operators and applications

This paper attempts to deal with some characterizations of almost Dunford–Pettis operators from a Banach lattice into a Banach space. It also discusses some of the consequences derived from this study. As an application, we generalize some results of Meyer-Nieberg on the duality between semi-compact operators and order weakly compact operators.     

Some results on unbounded absolute weak Dunford–Pettis operators

We characterize Banach lattices for which each Dunford–Pettis operator (or weak Dunford–Pettis) is unbounded absolute weak Dunford–Pettis and conversely.

     

Polynomial versions of weak Dunford–Pettis properties in Banach lattices

Positivity - In this paper, we introduce polynomial versions of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property for Banach lattices. By using Fremlin...     

Quantitative Dunford–Pettis property

We investigate possible quantifications of the Dunford–Pettis property. We show, in particular, that the Dunford–Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford–Pettis property. We prove that L¹$L^1$ spaces and C(K)$C\left(K\right)$ spaces possess both of them. We also show that several natural measures of weak non-compactness are equal in L¹$L^1$ spaces.     

On the Product of Almost Dunford–Pettis and Order Weakly Compact Operators

In this paper, we give some results on the product of positive almost Dunford–Pettis and interval preserving order weakly compact operators. As consequence, we derive some interesting consequences. Also, we look at the dual counterpart.     

Grothendieck spaces with the Dunford–Pettis property

Banach spaces which are Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet spaces occurs in Bonet and Ricker (Positivity 11:77–93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-space are again GDP-spaces. Also, every complete injective space is a GDP-space. For \({p\in \{0\}\cup[1,\infty)}\) it is shown that the classical co-echelon spaces k _p (V) and \({K_p(\overline{V})}\) are GDP-spaces if and only if they are Montel. On the other hand, \({K_\infty(\overline{V})}\) is always a GDP-space and k _∞(V) is a GDP-space whenever its (Fréchet) predual, i.e., the Köthe echelon space λ ₁(A), is distinguished.     

Almost Dunford–Pettis sets in Banach lattices

We introduce and study the class of almost Dunford–Pettis sets in Banach lattices. It also discusses some of the consequences derived from this study. As an application, we characterize Banach lattices whose relatively weakly compact sets are almost Dunford–Pettis sets. Also, we establish some necessary and sufficient conditions on which an almost Dunford–Pettis set is L-weakly compact (respectively, relatively weakly compact). In particular, we characterize Banach lattices under which almost Dunford–Pettis sets in the topological dual of a Banach lattice coincide with that of L-weakly compact (respectively, relatively weakly compact) sets. As a consequences we derive some results.     

On local weak solutions to Nernst–Planck–Poisson system

We study the one-dimensional nonlinear Nernst–Planck–Poisson system of partial differential equations with the class of nonlinear boundary conditions which cover the Chang–Jaffé conditions. The system describes certain physical and biological processes, for example ionic diffusion in porous media, electrochemical and biological membranes, as well as electrons and holes transport in semiconductors. The considered boundary conditions allow the physical system to be not only closed but also open. Theorems on existence, uniqueness, and nonnegativity of local weak solutions are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.     

On a vector-valued Hopf–Dunford–Schwartz lemma

In this paper, we state as a conjecture a vector-valued Hopf–Dunford–Schwartz lemma and give a partial answer to it. As an application of this powerful result, we prove some Fefferman–Stein inequalities in the setting of Dunkl analysis where covering methods are not available.     

Nonlinear Leray–Schauder Alternatives for Decomposable Operators in Dunford–Pettis Spaces and Application to Nonlinear Eigenvalue Problems

We present some new variants of Leray–Schauder type fixed point theorems and eigenvalue results for decomposable single-valued nonlinear weakly compact operators in Dunford-Pettis spaces.     

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Given a general dyadic grid D and a sparse family of cubes S = {Q _j ^k ∈ D, define a dyadic positive operator A _D,S by $${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$ . Given a Banach function space X(? ⁿ ) and the maximal Calderón-Zygmund operator ${T_\natural }$ , we show that $${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$ This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A ₂ conjecture”; (iii) an extension of certain mixed A _p ?A _r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A ₁ estimates (known for T ) to the maximal Calderón-Zygmund operator $\natural $ .