arXiv subjects

Anna Kamińska

Publications and source records attributed to Anna Kamińska.

14 recordsLinked to original sources

arXiv2024

On Köthe duals of Orlicz-Lorentz spaces

In this article, we study a number of properties of the Köthe duals $\mathcal{M}_{φ,w}$ of Orlicz-Lorentz spaces. An explicit description of the order-continuous subspace of $\mathcal{M}_{φ,w}$ is provided. Moreover, the separability of these spaces is characterized by the growth condition $Δ_2$. Consequently, the Köthe dual space $\mathcal{M}_{φ,w}$ has the Radon-Nikodým property if and only if the N-function at infinity $φ$ satisfies the appropriate $Δ_2$-condition. The comparison between $\mathcal{M}_{φ,w}$ spaces is characterized via standard orders between Orlicz functions. As applications of these results, we provide sufficient conditions for M-embedded order-continuous subspaces of Orlicz-Lorentz spaces equipped with the Luxemburg norm and prove the existence of a unique norm-preserving extension on Orlicz-Lorentz spaces equipped with the Orlicz norm.

math.FA↗

arXiv2023

Daugavet and diameter two properties in Orlicz-Lorentz spaces

In this article, we study the diameter two properties (D2Ps), the diametral diameter two properties (diametral D2Ps), and the Daugavet property in Orlicz-Lorentz spaces equipped with the Luxemburg norm. First, we characterize the Radon-Nikodým property of Orlicz-Lorentz spaces in full generality by considering all finite real-valued Orlicz functions. To show this, the fundamental functions of their Köthe dual spaces defined by extended real-valued Orlicz functions are computed. We also show that if an Orlicz function does not satisfy the appropriate $Δ_2$-condition, the Orlicz-Lorentz space and its order-continuous subspace have the strong diameter two property. Consequently, given that an Orlicz function is an N-function at infinity, the same condition characterizes the diameter two properties of Orlicz-Lorentz spaces as well as the octahedralities of their Köthe dual spaces. The Orlicz-Lorentz function spaces with the Daugavet property and the diametral D2Ps are isometrically isomorphic to $L_1$ when the weight function is regular. In the process, we observe that every locally uniformly nonsquare point is not a $Δ$-point. This fact provides another class of real Banach spaces without $Δ$-points. As another application, it is shown that for Orlicz-Lorentz spaces equipped with the Luxemburg norm defined by an N-function at infinity, their Köthe dual spaces do not have the local diameter two property, and so as other (diametral) diameter two properties and the Daugavet property.

math.FA↗

arXiv2023

Diameter two properties and the Radon-Nikodým property in Orlicz spaces

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikodým property. Consequently it is shown that an Orlicz space $L_φ$ over a non-atomic $σ$-finite measure space $(Ω, Σ,μ)$, not necessarily separable, has the Radon-Nikodým property if and only if $φ$ is an $N$-function at infinity and satisfies the appropriate $Δ_2$ condition. For an Orlicz sequence space $\ell_φ$, it has the Radon-Nikodým property if and only if $φ$ satisfies condition $Δ_2^0$. In the second part the relationships between uniformly $\ell_1^2$ points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space $L_φ$ has the Daugavet property only if $φ$ is linear, so when $L_φ$ is isometric to $L_1$. The other consequence is that the Orlicz spaces equipped with the Orlicz norm generated by $N$-functions never have local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, the strong diameter two property are equivalent in function and sequence Orlicz spaces with the Luxemburg norm under appropriate conditions on $φ$.

math.FA↗

arXiv2022

Density of smooth functions in Musielak-Orlicz spaces

We provide necessary and sufficient conditions for the space of smooth functions with compact supports $C^\infty_C(Ω)$ to be dense in Musielak-Orlicz spaces $L^Φ(Ω)$ where $Ω$ is an open subset of $\mathbb{R}^d$. In particular we prove that if $Φ$ satisfies condition $Δ_2$, the closure of $C^\infty_C(Ω)\cap L^Φ(Ω)$ is equal to $L^Φ(Ω)$ if and only if the measure of singular points of $Φ$ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of $Φ$, which implies that the measure of the singular points of $Φ$ is zero. As a corollary we obtain analogous results for Musielak-Orlicz spaces generated by double phase functional and we recover the well known result for variable exponent Lebesgue spaces.

math.FA↗

arXiv2021

Uniform convexity, reflexivity, supereflexivity and $B$ convexity of generalized Sobolev spaces $W^{1,Φ}$

We investigate Sobolev spaces $W^{1,Φ}$ associated to Musielak-Orlicz spaces $L^Φ$. We first present conditions for the boundedness of the Voltera operator in $L^Φ$. Employing this, we provide necessary and sufficient conditions for $W^{1,Φ}$ to contain isomorphic subspaces to $\ell^\infty$ or $\ell^1$. Further we give necessary and sufficient conditions in terms of the function $Φ$ or its complementary function $Φ^*$ for reflexivity, uniform convexity, $B$-convexity and superreflexivity of $W^{1,Φ}$. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,φ}$ where $φ$ is an Orlicz function, the variable exponent Sobolev spaces $W^{1,p(\cdot)}$ and the Sobolev spaces associated to double phase functionals.

math.FA↗

arXiv2019

Local geometric properties in quasi-normed Orlicz spaces

Several local geometric properties of Orlicz space $L_ϕ$ are presented for an increasing Orlicz function $ϕ$ which is not necessarily convex, and thus $L_ϕ$ does not need to be a Banach space. In addition to monotonicity of $ϕ$ it is supposed that $ϕ(u^{1/p})$ is convex for some $p>0$ which is equivalent to that its lower Matuszewska-Orlicz index $α_ϕ>0$. Such spaces are locally bounded and are equipped with natural quasi-norms. Therefore many local geometric properties typical for Banach spaces can also be studied in those spaces. The techniques however have to be different, since duality theory cannot be applied in this case. In this article we present complete criteria, in terms of growth conditions of $ϕ$, for $L_ϕ$ to have type $0<p\le2$, cotype $q\ge 2$, to be (order) $p$-convex or $q$-concave, to have an upper $p$-estimate or a lower $q$-estimate, for $0<p,q<\infty$. We provide detailed proofs of most results, avoiding appealing to general not necessary theorems.

math.FA↗

arXiv2019

Abstract Lorentz spaces and Köthe duality

Given a fully symmetric Banach function space $E$ and a decreasing positive weight $w$ on $I = (0, a)$, $0 < a \le \infty $, the generalized Lorentz space $Λ_{E,w}$ is defined as the symmetrization of the canonical copy $E_w$ of $E$ on the measure space associated with the weight. If $E$ is an Orlicz space then $Λ_{E,w}$ is an Orlicz-Lorentz space. An investigation of the Köthe duality of these classes is developed that is parallel to preceding works on Orlicz-Lorentz spaces. First a class of functions $M_{E,w}$, which does not need to be even a linear space, is similarly defined as the symmetrization of the space $w.E_w$. Let also $Q_{E,w}$ be the smallest fully symmetric Banach function space containing $M_{E,w}$. Then the Köthe dual of the class $M_{E,w}$ is identified as the Lorentz space $Λ_{E',w}$, while the Köthe dual of $Λ_{E,w}$ is $Q_{E',w}$. The space $Q_{E,w}$ is also characterized in terms of Halperin's level functions. These results are applied to concrete Banach function spaces. In particular the Köthe duality of Orlicz-Lorentz spaces is revisited at the light of the new results.

math.FA↗

arXiv2017

$M$-ideal properties in Orlicz-Lorentz spaces

We provide explicit formulas for the norm of bounded linear functionals on Orlicz-Lorentz function spaces $Λ_{φ,w}$ equipped with two standard Luxemburg and Orlicz norms. Any bounded linear functional is a sum of regular and singular functionals, and we show that the norm of a singular functional is the same regardless of the norm in the space, while the formulas of the norm of general functionals are different for the Luxemburg and Orlicz norm. The relationship between equivalent definitions of the modular $P_{φ,w}$ generating the dual space to Orlicz-Lorentz space is discussed in order to compute the norm of a bounded linear functional on $Λ_{φ,w}$ equipped with Orlicz norm. As a consequence, we show that the order-continuous subspace of Orlicz-Lorentz space equipped with the Luxemburg norm is an $M$-ideal in $Λ_{φ,w}$, while this is not true for the space with the Orlicz norm when $φ$ is an Orlicz $N$-function not satisfying the appropriate $Δ_2$ condition. The analogous results on Orlicz-Lorentz sequence spaces are given.

math.FA↗

arXiv2017

Diameter of weak neighborhoods and the Radon-Nikodym property in Orlicz-Lorentz spaces

Given an Orlicz $N$-function $φ$ and a positive decreasing weight $w$, we present criteria of the diameter two property and of the Radon-Nikodým property in Orlicz-Lorentz function and sequence spaces $Λ_{φ,w}$ and $λ_{φ,w}$. We show that in the spaces $Λ_{φ,w}$ or $λ_{φ,w}$ equipped with the Luxemburg norm, the diameter of any relatively weakly subset of the unit ball in these spaces is two if and only if $φ$ does not satisfy the appropriate $Δ_2$ condition, while they have the Radon-Nikodým property if and only if $φ$ satisfies the appropriate $Δ_2$ condition.

math.FA↗

arXiv2015

The Daugavet property in the Musielak-Orlicz spaces

We show that among all Musielak-Orlicz function spaces on a $σ$-finite non-atomic complete measure space equipped with either the Luxemburg norm or the Orlicz norm the only spaces with the Daugavet property are $L_1$, $L_{\infty}$, $L_1\oplus_1 L_{\infty}$ and $L_1\oplus_{\infty} L_{\infty}$. We obtain in particular complete characterizations of the Daugavet property in the weighted interpolation spaces, the variable exponent Lebesgue spaces (Nakano spaces) and the Orlicz spaces.

math.FA↗

arXiv2015

k-Extreme Points in Symmetric Spaces of Measurable Operators

Let $\mathcal{M}$ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace $τ$ and $E$ be a strongly symmetric Banach function space on $[0,τ(1))$. We show that an operator $x$ in the unit sphere of $E\left(\mathcal{M},τ\right)$ is $k$-extreme, $k\in\mathbb N$, whenever its singular value function $μ(x)$ is $k$-extreme and one of the following conditions hold (i) $μ(\infty,x)=\lim_{t\to\infty}μ(t,x)=0$ or (ii) $n(x)\mathcal{M} n(x^*)=0$ and $|x|\geq μ(\infty,x)s(x)$, where $n(x)$ and $s(x)$ are null and support projections of $x$, respectively. The converse is true whenever $\mathcal{M}$ is non-atomic. The global $k$-rotundity property follows, that is if $\mathcal{M}$ is non-atomic then $E$ is $k$-rotund if and only if $E\left(\mathcal{M},τ\right)$ is $k$-rotund. As a consequence of the noncommutive results we obtain that $f$ is a $k$-extreme point of the unit ball of the strongly symmetric function space $E$ if and only if its decreasing rearrangement $μ(f)$ is $k$-extreme and $|f|\geq μ(\infty,f)$. We conclude with the corollary on orbits $Ω(g)$ and $Ω'(g)$. We get that $f$ is a $k$-extreme point of the orbit $Ω(g)$, $g\in L_1+L_{\infty}$, or $Ω'(g)$, $g\in L_1[0,α)$, $α<\infty$, if and only if $μ(f)=μ(g)$ and $|f|\geq μ(\infty,f)$. From this we obtain a characterization of $k$-extreme points in Marcinkiewicz spaces.

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↗

arXiv2013

New formulas for decreasing rearrangements and a class of Orlicz-Lorentz spaces

Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{φ,w}$ and $m_{φ,w}$ respectively, introduced earlier in \cite{HKM} for describing the Köthe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($φ$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{φ,w}$ classes in the most general setting, where they may even not be linear, and identify their Köthe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces $\mathcal{M}_{φ,w}$ which proves to be the Köthe biduals of the $M_{φ,w}$ classes. In the case when the class $M_{φ,w}$ is a separable quasi-Banach space, $\mathcal{M}_{φ,w}$ is its Banach envelope.

math.FA↗

arXiv2011

On the dual of Cesàro function space

The goal of this paper is to present an isometric representation of the dual space to Cesàro function space $C_{p,w}$, $1<p<\infty$, induced by arbitrary positive weight function $w$ on interval $(0,l)$ where $0<l\leqslant\infty$. For this purpose given a strictly decreasing nonnegative function $Ψ$ on $(0,l)$, the notion of essential $Ψ$-concave majorant $\hat f$ of a measurable function $f$ is introduced and investigated. As applications it is shown that every slice of the unit ball of the Cesàro function space has diameter 2. Consequently Cesàro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.

math.FA↗