arXiv subjects

Tuğçe Ünver

Publications and source records attributed to Tuğçe Ünver.

15 recordsLinked to original sources

arXiv2026

On Weighted Local Morrey-type and Complementary Local Morrey-Type Spaces

In this paper, we establish necessary and sufficient conditions for continuous embeddings between weighted local Morrey-type spaces and weighted complementary local Morrey-type spaces. Rather than relying on standard multidimensional approaches, our methodology is based on a one-dimensional reduction. Specifically, we reduce the multidimensional embedding problems to corresponding weighted inequalities involving one-dimensional Cesàro and Copson function spaces. Applying this reduction alongside recent results in the characterization of the embeddings between Cesàro and Copson spaces, we provide complete characterizations for the embeddings \[ \LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2)\] and \[\dual\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2).\] Our results cover all possible finite cases of the parameters $0 < p_1, p_2, q_1, q_2 < \infty$.

math.AP↗

arXiv2026

Weighted Hardy inequalities involving supremum for decreasing sequences

In this paper, we provide a complete characterization of the weighted Hardy inequalities involving the supremum operator, restricted to the cone of non-increasing sequences, for all positive parameters. We reduce such inequalities to equivalent ones on the cone of non-negative sequences. The latter setting provides a broader framework for analysis and significantly expands the range of proofs that can be established.

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↗

arXiv2025

On weighted Cesàro function spaces

The main objective of this paper is to provide a comprehensive demonstration of recent results regarding the structures of the weighted Cesàro and Copson function spaces. These spaces' definitions involve local and global weighted Lebesgue norms; in other words, the norms of these spaces are generated by positive sublinear operators and by weighted Lebesgue norms. The weighted Lebesgue spaces are the special cases of these spaces with a specific set of parameters. Our primary method of investigating these spaces will be the so-called discretization technique. Our technique will be the development of the approach initiated by K.G. Grosse-Erdmann, which allows us to obtain the characterization in previously unavailable situations, thereby addressing decades-old open problems. We investigate the relation (embeddings) between weighted Cesàro and Copson function spaces. The characterization of these embeddings can be used to tackle the problems of characterizing pointwise multipliers between weighted Cesàro and Copson function spaces, the characterizations of the associate spaces of Cesàro (Copson) function spaces, as well as the relations between local Morrey-type spaces.

math.FA↗

arXiv2024

New characterization of weighted inequalities involving superposition of Hardy integral operators

Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize the validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}} w(x) dx \bigg)^{\frac{1}{r}} \leq C \bigg(\int_a^b f(x)^p v(x) dx \bigg)^{\frac{1}{p}} \end{equation*} for all non-negative measurable functions $f$ on $(a,b)$, $-\infty \leq a < b \leq \infty$. We construct a more straightforward discretization method than those previously presented in the literature, and we provide some new scales of weight characterizations of this inequality in both discrete and continuous forms and we obtain previous characterizations as the special case of the parameter.

math.AP↗

arXiv2023

Weighted inequalities involving iteration of two Hardy integral operators

Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}} w(x) dx \bigg)^{\frac{1}{r}} \leq C \bigg(\int_a^b f^p(x) v(x) dx \bigg)^{\frac{1}{p}} \end{equation*} for all non-negative measurable functions on $(a,b)$, $-\infty \leq a < b \leq \infty$. We construct a more straightforward discretization method than those previously presented in the literature, and we characterize this inequality in both discrete and continuous forms.

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↗

arXiv2022

Weighted inequalities involving Hardy and Copson operators

We characterize a four-weight inequality involving the Hardy operator and the Copson operator. More precisely, given $p_1, p_2, q_1, q_2 \in (0, \infty)$, we find necessary and sufficient conditions on nonnegative measurable functions $u_1, u_2, v_1, v_2$ on $(0,\infty)$ for which there exists a positive constant $c$ such that the inequality \begin{align*} &\bigg(\int_0^{\infty} \bigg(\int_0^t f(s)^{p_2} v_2(s)^{p_2} ds \bigg)^{\frac{q_2}{p_2}} u_2(t)^{q_2} dt \bigg)^{\frac{1}{q_2}} \notag \\ & \hspace{3cm} \leq c \bigg(\int_0^{\infty} \bigg(\int_t^{\infty} f(s)^{p_1} v_1(s)^{p_1} ds \bigg)^{\frac{q_1}{p_1}} u_1(t)^{q_1} dt \bigg)^{\frac{1}{q_1}} \end{align*} holds for every non-negative measurable function $f$ on $(0, \infty)$. The proof is based on discretizing and antidiscretizing techniques. The principal innovation consists in development of a new method which carefully avoids duality techniques and therefore enables us to obtain the characterization in previously unavailable situations, solving thereby a long-standing open problem. We then apply the characterization of the inequality to the establishing of criteria for embeddings between weighted Copson spaces $\operatorname{Cop}_{p_1,q_1} (u_1, v_1)$ and weighted Cesàro spaces $\operatorname{Ces}_{p_2, q_2} (u_2, v_2)$, and also between spaces $S^q(w)$ equipped with the norm $\|f\|_{S^q(w)}= \bigg(\int_0^\infty [f^{**}(t)-f^*(t)]^q w(t)\,dt\bigg)^{{1}/{q}}$ and classical Lorentz spaces of type $Λ$.

math.FA↗

arXiv2021

Weighted inequalities for discrete iterated kernel operators

We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a positive constant $C$ such that \begin{equation*} \Bigg(\sum_{n\in\mathbb{Z}}\Bigg(\sum_{i=-\infty}^n {U}(i,n) a_i\Bigg)^{q} {w}_n\Bigg)^{\frac{1}{q}} \le C \Bigg(\sum_{n\in\mathbb{Z}}a_n^p {v}_n\Bigg)^{\frac{1}{p}} \end{equation*} holds for every sequence of nonnegative numbers $\{a_n\}_{n\in\mathbb{Z}}$ where $U$ is a kernel satisfying certain regularity condition, $0 < p,q \leq \infty$ and $\{v_n\}_{n\in\mathbb{Z}}$ and $\{w_n\}_{n\in\mathbb{Z}}$ are fixed weight sequences. We do the same for the inequality \begin{equation*} \Bigg( \sum_{n\in\mathbb{Z}} w_n \Big[ \sup_{-\infty<i\le n} U(i,n) \sum_{j=-\infty}^{i} a_j \Big]^q \Bigg)^{\frac{1}{q}} \le C \Bigg( \sum_{n\in\mathbb{Z}} a_n^p v_n \Bigg)^{\frac{1}{p}}. \end{equation*} We characterize these inequalities by conditions of both discrete and continuous nature.

math.FA↗

arXiv2021

Embeddings between weighted Tandori and Cesàro Function Spaces

We give the characterization of the embeddings between weighted Tandori and Cesàro function spaces using the combination of duality arguments for weighted Lebesgue spaces and weighted Tandori spaces with estimates for the iterated integral operators.

math.FA↗

arXiv2020

Embeddings Between Weighted Cesàro Function Spaces

In this paper, we give the characterization of the embeddings between weighted Cesàro function spaces. The proof is based on the duality technique, which reduces this problem to the characterizations of some direct and reverse Hardy-type inequalities and iterated Hardy-type inequalities.

math.FA↗

arXiv2019

Embeddings Between weighted local Morrey-type spaces

In this paper certain $n$-dimensional inequalities are shown to be equivalent to the inequalities in the one-dimensional setting. By this means, embeddings between weighted local Morrey-type spaces are characterized for some ranges of parameters.

math.AP↗

arXiv2018

Multidimensional bilinear Hardy inequalities

Our goal in this paper is to find a characterization of $n$-dimensional bilinear Hardy inequalities \begin{align*} \bigg\| \,\int_{B(0,\cdot)} f \cdot \int_{B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} & \leq C \, \|f\|_{p_1,v_1,{\mathbb R}^n} \, \|g\|_{p_2,v_2,{\mathbb R}^n}, \quad f,\,g \in {\mathfrak M}^+ ({\mathbb R}^n), \end{align*} and \begin{align*} \bigg\| \,\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)} f \cdot \int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} &\leq C \, \|f\|_{p_1,v_1,{\mathbb R}^n} \, \|g\|_{p_2,v_2,{\mathbb R}^n}, \quad f,\,g \in {\mathfrak M}^+ ({\mathbb R}^n), \end{align*} when $0 < q \le \infty$, $1 \le p_1,\,p_2 \le \infty$ and $u$ and $v_1,\,v_2$ are weight functions on $(0,\infty)$ and ${\mathbb R}^n$, respectively. Since the solution of the first inequality can be obtained from the characterization of the second one by usual change of variables we concentrate our attention on characterization of the latter. The characterization of this inequality is easily obtained for the range of parameters when $p_1 \le q$ using the characterizations of multidimensional weighted Hardy-type inequalites while in the case when $q < p_1$ the problem is reduced to the solution of multidimensional weighted iterated Hardy-type inequality. To achieve the goal, we characterize the validity of multidimensional weighted iterated Hardy-type inequality $$ \left\|\left\|\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,μ,(0,\infty)}\leq c \|h\|_{θ,v,(0,\infty)},~ h \in \mathfrak{M}^+({\mathbb R}^n) $$ where $0 < p,\,q < +\infty$, $1 \leq θ\le \infty$, $u\in {\mathcal W}(0,\infty)$, $v \in {\mathcal W}({\mathbb R}^n)$ and $μ$ is a non-negative Borel measure on $(0,\infty)$.

math.FA↗

arXiv2016

Embeddings between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces

In this paper embeddings between weighted complementary local Morrey-type spaces ${\,^{^{\bf c}}\!}LM_{pθ,ω}({\mathbb R}^n,v)$ and weighted local Morrey-type spaces $LM_{pθ,ω}({\mathbb R}^n,v)$ are characterized. In particular, two-sided estimates of the optimal constant $c$ in the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_{B(0,t)} f(x)^{p_2}v_2(x)\,dx \bigg)^{\frac{q_2}{p_2}} u_2(t)\,dt\bigg)^{\frac{1}{q_2}} \le c \bigg( \int_0^{\infty} \bigg( \int_{\,^{^{\bf c}\!}B(0,t)} f(x)^{p_1} v_1(x)\,dx\bigg)^{\frac{q_1}{p_1}} u_1(t)\,dt\bigg)^{\frac{1}{q_1}} \end{equation*} are obtained, where $p_1,\,p_2,\,q_1,\,q_2 \in (0,\infty)$, $p_2 \le q_2$ and $u_1,\,u_2$ and $v_1,\,v_2$ are weights on $(0,\infty)$ and ${\mathbb R}^n$, respectively. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted local Morrey-type and complementary local Morrey-type spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of the iterated Hardy-type inequalities.

math.FA↗