arXiv subjects

Karol Leśnik

Publications and source records attributed to Karol Leśnik.

8 recordsLinked to original sources

arXiv2025

Looking for a continuous version of Bennett--Carl theorem

We study absolute summability of inclusions of r.i. function spaces. It appears that such properties are closely related, or even determined by absolute summability of inclusions of subspaces spanned by the Rademacher system in respective r.i. spaces. Our main result states that for $1<p<2$ the inclusion $X_p\subset L^p$ is $(q,1)$-absolutely summing for each $p<q<2$, where $X_p$ is the unique r.i. Banach function space in which the Rademacher system spans copy of $l^p$. This result may be regarded as a continuous version of the well-known Carl--Bennett theorem. Two different approaches to the problem and extensive discussion on them are presented. We also conclude summability type of a kind of Sobolev embedding in the critical case.

math.FA↗

arXiv2020

Pointwise multipliers of Musielak--Orlicz spaces and factorization

We prove that the space of pointwise multipliers between two distinct Musielak--Orlicz spaces is another Musielak-Orlicz space and the function defining it is given by an appropriately generalized Legendre transform. In particular, we obtain characterization of pointwise multipliers between Nakano spaces. We also discuss factorization problem for Musielak-Orlicz spaces and exhibit some differences between Orlicz and Musielak-Orlicz cases.

math.FA↗

arXiv2016

Pointwise multipliers of Orlicz function spaces and factorization

In the paper we find representation of the space of pointwise multipliers between two Orlicz function spaces, which appears to be another Orlicz space and the formula for the Young function generating this space is given. Further, we apply this result to find necessary and suffcient conditions for factorization of Orlicz function spaces.

math.FA↗

arXiv2015

Isomorphic structure of Cesàro and Tandori spaces

We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers to the question posted in \cite{AM09}. This is rather surprising since $Ces_{\infty}$ has no natural lattice predual similarly as the known Talagrand's example \cite{Ta81}. We prove that neither $ces_{\infty}$ is isomorphic to $l_{\infty}$ nor $Ces_{\infty}$ is isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}= \|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t):= \esssup_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Cesàro and Tandori spaces. In particular, using Bourgain's results we show that a wide class of Ces{à}ro-Marcinkiewicz and Ces{à}ro-Lorentz spaces have the latter property.

math.FA↗

arXiv2014

Abstract Cesàro spaces. II. Optimal range

Cesàro spaces are investigated from the optimal domain and optimal range point of view. There is a big difference between the cases on $[0, \infty)$ and on $[0, 1]$, as we can see in Theorem 1. Moreover, we present an improvement of Hardy inequality on $[0, 1]$ which plays an important role in these considerations.

math.FA↗

arXiv2014

Abstract Cesàro Spaces. I. Duality

We study abstract Cesàro spaces $CX$, which may be regarded as generalizations of Cesàro sequence spaces $ces_p$ and Cesàro function spaces $Ces_p(I)$ on $I = [0,1]$ or $I = [0,\infty)$, and also as the description of optimal domain from which Cesàro operator acts to $X$. We find the dual of such spaces in a very general situation. What is however even more important, we do it in the simplest possible way. Our proofs are more elementary than the known ones for $ces_p$ and $Ces_p(I)$. This is the point how our paper should be seen, i.e. not as generalization of known results, but rather like grasping and exhibiting the general nature of the problem, which is not so easy visible in the previous publications. Our results show also an interesting phenomenon that there is a big difference between duality in the cases of finite and infinite interval.

math.FA↗

arXiv2014

Dual spaces to Orlicz - Lorentz spaces

For an Orlicz function $φ$ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the Köthe dual of an Orlicz-Lorentz function space $Λ_{φ,w}$ or a sequence space $λ_{φ,w}$, equipped with either Luxemburg or Amemiya norms. The first description of the dual norm is given via the modular $\inf\{\intφ_*(f^*/|g|)|g|: g\prec w\}$, where $f^*$ is the decreasing rearrangement of $f$, $g\prec w$ denotes the submajorization of $g$ by $w$ and $φ_*$ is the complementary function to $φ$. The second one is stated in terms of the modular $\int_I φ_*((f^*)^0/w)w$, where $(f^*)^0$ is Halperin's level function of $f^*$ with respect to $w$. That these two descriptions are equivalent results from the identity $\inf\{\intψ(f^*/|g|)|g|: g\prec w\}=\int_I ψ((f^*)^0/w)w$ valid for any measurable function $f$ and Orlicz function $ψ$. Analogous identity and dual representations are also presented for sequence spaces.

math.FA↗

arXiv2012

Pointwise products of some Banach function spaces and factorization

The well-known factorization theorem of Lozanovski{\u ı} may be written in the form $L^{1}\equiv E\odot E^{\prime}$, where $\odot $ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize $F$ through $E$, i.e., when $F\equiv E\odot M(E, F) \,$, where $M(E, F) $ is the space of pointwise multipliers from $E$ to $F$. Properties of $M(E, F) $ were investigated in our earlier paper [KLM12] and here we collect and prove some properties of the construction $E\odot F$. The formulas for pointwise product of Calderón-Lozanovski{\u ı} $E_φ$ spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for these spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through Marcinkiewicz space.

math.FA↗