arXiv subjects

Amiran Gogatishvili

Publications and source records attributed to Amiran Gogatishvili.

At least 19 recordsLinked to original sources

arXiv2026

Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces

In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_γ$, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $γ\in [0,n)$, between the classical Lorentz sequence spaces $λ_p(v)$ and $λ_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_γx$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_γ$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.

math.FA↗

arXiv2026

A solution to a conjecture on the norm of the Copson operator minus the Identity for monotone sequences

We investigate the operator distance between the Identity and the Copson operator on the cone of nonnegative, nonincreasing sequences within the Lebesgue sequence spaces $\ell^p(\mathbb{N})$. By proving the exact value of this distance, we confirm a conjecture originally formulated by A. Ben Said, S. Boza, and J. Soria in ``The norm of Hardy-type oscillation operators in the discrete and continuous settings'' (\emph{Analysis and Applications}, 2025).

math.FA↗

arXiv2026

Compact sets in Banach lattices over spaces of homogeneous type

We prove a Kolmogorov--Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for $L^1$. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov--Riesz theorem in $L^1$, which avoids reflexivity and yields a unified treatment of $L^p$-spaces for $1\le p<\infty$. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.

math.FA↗

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

The Hardy averaging operator between Orlicz spaces and a new gauge functional characterizing a given Orlicz space

We prove that the largest Orlicz space into which the Hardy averaging operator maps coincides with the largest rearrangement-invariant (r.i.) space mapping into that space; in other words, the optimal Orlicz domain is automatically the optimal r.i.\ domain. More generally, we show that every Luxemburg--Orlicz norm is equivalent to a sublinear functional which makes more computable a certain expression for a dual norm arising in the $\K$-theory of interpolation. As applications we obtain Orlicz-space mapping properties for the Hardy--Littlewood maximal function, approximate identities and Calderón--Zygmund singular integral operators. These mapping properties are shown to be optimal for the Hardy-Littlewood maximal function and the approximate identities.

math.AP↗

arXiv2026

Interior a priori estimate for higher order elliptic systems in Orlicz spaces

We investigate singular integral operators with variable Calderón--Zygmund kernels and their commutators with $VMO$ functions on Orlicz spaces. After revisiting the classical $L^p$ theory, we establish boundedness results in $L^Φ$ under the standard $Δ_2$ and $\nabla_2$ conditions on the Young function. The analysis combines decomposition techniques with weak-type estimates to derive the main operator bounds. As an application, these results provide a functional-analytic framework for establishing a priori estimates and proving the interior regularity of solutions to higher-order elliptic equations and systems with discontinuous coefficients.

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↗

arXiv2026

Rubio de Francia Extrapolation Theorem for Quasi non-increasing Sequences

We prove the discrete Rubio de Francia extrapolation theorem for a pair of quasi non-increasing sequences with $\mathcal{QB}_{β, p}$ weight class. Also, a weight characterization of the boundedness of the generalized discrete Hardy averagin19g operator on the class of quasi non-increasing sequences from $l_w^p(\mathbb{Z}^+)$ is proved.

math.FA↗

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

Weighted inequalities involving two Hardy operators

We find necessary and sufficient conditions on weights $u_1, u_2, v_1, v_2$, i.e. measurable, positive, and finite, a.e. on $(a,b)$, for which there exists a positive constant $C$ such that for given $0 < p_1,q_1,p_2,q_2 <\infty$ the inequality \begin{equation*} \begin{split} \bigg(\int_a^b \bigg(\int_a^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}}& \\ & \hspace{-3cm}\le C \bigg(\int_a^b \bigg(\int_a^t 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{split} \end{equation*} holds for every non-negative, measurable function $f$ on $(a,b)$, where $0 \le a <b \le \infty$. The proof is based on a recently developed discretization method that enables us to overcome the restrictions of the earlier results.

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↗

arXiv2025

Weighted inequalities for sub-monotone functionals

We establish a set of relations between several quite diverse types of weighted inequalities involving various integral operators and fairly general quasinorm-like functionals which we call sub-monotone. The main result enables one to solve a specific problem by transferring it to another one for which a solution is known. The main result is formulated in a rather surprising generality, involving previously unknown cases, and it works even for some nonlinear operators such as the geometric or harmonic mean operators. Proofs use only elementary means.

math.CA↗

arXiv2024

Applications of Interpolation theory to the regularity of some equasilinear PDEs

We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla u|^{p-2}\nabla u)+V(x;u)=f, \end{equation*} where $Ω$ is a bounded smooth domain of $\mathbb R^n$, $V$ is a nonlinear potential and $f$ belongs to non-standard spaces like Lorentz-Zygmund spaces. Moreover, we collect some well-known and new results for identifying some interpolation spaces and enrich some contents with details.

math.AP↗

arXiv2024

A generalized version of Holmstedt's formula for the K-functional

Let $(A_0, A_1)$ be a compatible couple of quasi-normed spaces, and let $Φ_0$ and $Φ_1$ be two general parameters of $K$-interpolation method. We compute $K$-functional for the couple $((A_0,A_1)_{Φ_0}, (A_0, A_1)_{Φ_1})$ in terms of $K$-functional for the couple $(A_0, A_1)$.

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↗

arXiv2024

On the embedding between the variable Lebesgue space $L^{p(\cdot)}(Ω)$ and the Orlicz space $L(\log L)^α(Ω)$

We give a sharp sufficient condition on the distribution function, $|\{x\in Ω:\,p(x)\leq 1+λ\}|$, $λ>0$, of the exponent function $p(\cdot): Ω\to [1,\infty)$ that implies the embedding of the variable Lebesgue space $L^{p(\cdot)}(Ω)$ into the Orlicz space $L(\log L)^α(Ω)$, $α>0$, where $Ω$ is an open set with finite Lebesgue measure. As applications of our results, we first give conditions that imply the strong differentiation of integrals of functions in $L^{p(\cdot)}((0,1)^{n})$, $n>1$. We then consider the integrability of the maximal function on variable Lebesgue spaces, where the exponent function $p(\cdot)$ approaches $1$ in value on some part of the domain. This result is an improvement of the result in~\cite{CUF2}.

math.CA↗