The importance of being Orlicz
The K-functional for rearrangement invariant spaces
The Kreps-Yan theorem for Banach ideal spaces.
The Kreps–Yan theorem for .
The mapping problem for well-behaved convolutions
The L1 Structure of Weak L1.
The level function in rearrangement invariant spaces.
An exact expression for the down norm is given in terms of the level function on all rearrangement invariant spaces and a useful approximate expression is given for the down norm on all rearrangement invariant spaces whose upper Boyd index is not one.
The Lizorkin-Freitag formula for several weighted spaces and vector-valued interpolation
A complete description of the real interpolation space is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces (i ∈ I) such that L is an sum of the restrictions of L to , and L on each is a result of interpolation of just two weighted spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.
The Lusin Theorem and Horizontal Graphs in the Heisenberg Group
In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
The martingale Smirnov class.
The maximal, funtion and derivation of integrals in a space of locally integrable functions.
The maximal theorem for weighted grand Lebesgue spaces
We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality holds with some c independent of f iff w belongs to the well known Muckenhoupt class , and therefore iff for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....
The Mazur intersection property for families of closed bounded convex sets in Banach spaces
The minimal operator and the John--Nirenberg theorem for weighted grand Lebesgue spaces
We introduce the minimal operator on weighted grand Lebesgue spaces, discuss some weighted norm inequalities and characterize the conditions under which the inequalities hold. We also prove that the John-Nirenberg inequalities in the framework of weighted grand Lebesgue spaces are valid provided that the weight function belongs to the Muckenhoupt class.
The moduli of smoothness and convexity and the Rademacher averages of the trace classes (1≤p<∞)
The multiplication functions of Kučera
The Navier-Stokes equations in the critical Morrey-Campanato space.
The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of
Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of . Then the Bochner space is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.
The problem and singular integrals on Orlicz spaces
The property of weak type (p,p) for the Hardy-Littlewood maximal operator and derivation of integrals