arXiv subjects

Hana Turčinová

Publications and source records attributed to Hana Turčinová.

8 recordsLinked to original sources

arXiv2025

Interpolation of classical Lorentz spaces measuring oscillation

We obtain an explicit characterization of the $K$-functional of a pair of weighted classical Lorentz spaces of type $S$. We develop a method for obtaining such characterization based on a relation between the desired quantity and the $K$-functional of a specific couple of spaces of type $Λ$, which are substantially more manageable than their companions of type $S$. The core of our techniques is a subtle manipulation with respective fundamental functions. We present several applications, in particular we nail down a formula for the $K$-functional of a Lebesgue space and a classical Lorentz space of type $S$ with a power weight, and using this formula we establish an inequality of a reverse Marchaud type.

math.FA↗

arXiv2025

The Persson--Stepanov theorem revisited

We develop a new proof of the result of L.-E.~Persson and V.D.~Stepanov \cite[Theorems 1 and 3]{Per:02}, which provides a characterization of a Hardy integral inequality involving two weights, and which can be applied to an effective treatment of the geometric mean operator. Our approach enables us to extend their result to the full range of parameters, in particular involving the critical case $p=1$, which was excluded in the original work. Our proof avoids all duality steps and discretization techniques and uses solely elementary means.

math.FA↗

arXiv2025

Embeddings between generalized weighted Lorentz spaces

We give a new characterization of a continuous embedding between two function spaces of type $GΓ$. Such spaces are governed by functionals of type \begin{equation*} \|f\|_{GΓ(r,q;w,δ)} := \left(\int_{0}^{L} \left( \frac1{Δ(t)} \int_0^t f^*(s)^r δ(s) ds \right)^{\frac{q}{r}} w(t) dt \right)^\frac1{q}, \end{equation*} in which $f^*$ is the nonincreasing rearrangement of $f$, $L\in(0,\infty]$, $r,q \in (0, \infty)$, $w, δ$ are weights on $(0,L)$ and $Δ(t)=\int_{0}^{t}δ(s)\,ds$ for $t\in(0,L)$. To characterize the embedding of such a space, say $GΓ(r_1,q_1;w_1,δ_1)$, into another, $GΓ(r_2,q_2;w_2,δ_2)$, means to find a balance condition on the four positive real parameters and the four weights in order that an appropriate inequality holds for every admissible function. We develop a new discretization technique which will enable us to get rid of restrictions on parameters imposed in earlier work such as the non-degeneracy conditions or certain relations between the $r$'s and $q$'s. Such restrictions were caused mainly by the use of duality techniques, which we avoid in this paper. On the other hand we consider here only the case when $q_1 \le q_2$, leaving the reverse case to future work.

math.FA↗

arXiv2024

On the properties of rearrangement-invariant quasi-Banach function spaces

This paper explores some important aspects of the theory of rearrangement-invariant quasi-Banach function spaces. We focus on two main topics. Firstly, we prove an analogue of the Luxemburg representation theorem for rearrangement-invariant quasi-Banach function spaces over resonant measure spaces. Secondly, we develop the theory of fundamental functions and endpoint spaces.

math.FA↗

arXiv2022

Characterization of functions with zero traces via the distance function and Lorentz spaces

Consider a regular domain $Ω\subset \mathbb{R}^N$ and let $d(x)=\operatorname{dist}(x,\partialΩ)$. Denote $L^{1,\infty}_a(Ω)$ the space of functions from $L^{1,\infty}(Ω)$ having absolutely continuous quasinorms. This set is essentially smaller than $L^{1,\infty}(Ω)$ but, at the same time, essentially larger than a union of all $L^{1,q}(Ω)$, $q\in[1,\infty)$. A classical result of late 1980's states that for $p\in (1,\infty)$ and $m \in \mathbb{N}$, $u$ belongs to the Sobolev space $W^{m,p}_0(Ω)$ if and only if $u/d^m\in L^p(Ω)$ and $\left|\nabla^m u\right|\in L^p(Ω)$. During the consequent decades, several authors have spent considerable effort in order to relax the characterizing condition. Recently, it was proved that $u\in W^{m,p}_0(Ω)$ if and only if $u/d^m\in L^1(Ω)$ and $\left|\nabla^m u\right|\in L^p(Ω)$. In this paper we show that for $N\geq1$ and $p\in(1,\infty)$ we have $u\in W^{1,p}_0(Ω)$ if and only if $u/d\in L^{1,\infty}_a(Ω)$ and $\left|\nabla u\right|\in L^p(Ω)$. Moreover, we present a counterexample which demonstrates that after relaxing the condition $u/d\in L^{1,\infty}_a(Ω)$ to $u/d\in L^{1,\infty}(Ω)$ the equivalence no longer holds.

math.FA↗

arXiv2022

Weighted inequalities for a superposition of the Copson operator and the Hardy operator

We study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely \begin{equation*} \bigg(\int_a^b \bigg(\int_t^b \bigg(\int_a^s f(τ)^p v(τ) \,dτ\bigg)^\frac{q}{p} u(s) \,ds \bigg)^{\frac{r}{q}} w(t) \,dt \bigg)^{\frac{1}{r}} \leq C \int_a^b f(t)\,dt, \end{equation*} in which $(a,b)$ is any nontrivial interval, $q,r$ are positive real parameters and $p\in(0,1]$. A simple change of variables can be used to obtain any weighted $L^p$-norm with $p\ge1$ on the right-hand side. Another simple change of variables can be used to equivalently turn this inequality into the one in which the Hardy and Copson operators swap their positions. We focus on characterizing those triples of weight functions $(u,v,w)$ for which this inequality holds for all nonnegative measurable functions $f$ with a constant independent of $f$. We use a new type of approach based on an innovative method of discretization which enables us to avoid duality techniques and therefore to remove various restrictions that appear in earlier work. This paper is dedicated to Professor Stefan Samko on the occasion of his 80th birthday.

math.FA↗

arXiv2021

Discretization and antidiscretization of Lorentz norms with no restrictions on weights

We improve the discretization technique for weighted Lorentz norms by eliminating all "non-degeneracy" restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant $C$ such that the inequality $$\left( \int_0^L (f^*(t))^{p_2} w(t)\,\mathrm{d}t \right)^\frac 1{p_2} \le C \left( \int_0^L \left( \int_0^t u(s)\,\mathrm{d}s \right)^{-\frac {p_1}α} \left( \int_0^t (f^*(s))^αu(s) \,\mathrm{d}s \right)^\frac {p_1}αv(t) \,\mathrm{d}t \right)^\frac 1{p_1}$$ holds for all relevant measurable functions, where $L\in(0,\infty]$, $α, p_1, p_2 \in (0,\infty)$ and $u$, $v$, $w$ are locally integrable weights, $u$ being strictly positive. It the case of weights that would be otherwise excluded by the restrictions, it is shown that additional limit terms naturally appear in the characterizations of the optimal $C$. A weak analogue for $p_1=\infty$ is also presented.

math.FA↗

arXiv2020

Basic functional properties of certain scale of rearrangement-invariant spaces

Let $X$ be a rearrangement-invariant space over a non-atomic $σ$-finite measure space $(\mathscr{R},μ)$ and let $α\in(0,\infty)$. We define the functional \begin{equation*} \|f\|_{X^{\langle α\rangle}} = \|((|f|^α)^{**})^{\frac{1}α}\|_{\overline{X}(0,μ(\mathscr{R}))}, \end{equation*} in which $f$ is a $μ$-measurable scalar function defined on $(\mathscr{R},μ)$ and $\overline{X}(0,μ(\mathscr{R}))$ is the representation space of $X$. We denote by $X^{\langle α\rangle}$ the collection of all almost everywhere finite functions $f$ such that $\|f\|_{X^{\langle α\rangle}}$ is finite. These spaces recently surfaced in connection of optimality of target function spaces in general Sobolev embeddings involving upper Ahlfors regular measures. We present a variety of results on these spaces including their basic functional properties, their relations to customary function spaces and mutual embeddings and, in a particular situation, a characterization of their associate structures. We discover a new one-parameter path of function spaces leading from a Lebesgue space to a Zygmund class and we compare it to the classical one.

math.FA↗