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Amiran Gogatishvili

Publications and source records attributed to Amiran Gogatishvili.

At least 37 records · Page 2Linked to original sources
arXiv2024

Cones of monotone functions generated by a generalized fractional maximal function

In this paper, we consider the generalized fractional maximal function and use it to introduce the space of generalized fractional maximal functions and the various cones of monotone functions generated by generalized fractional maximal functions $M_Φf$. We introduced three function classes. We give equivalent descriptions of such cones when the function $Φ$ belongs to some function classes. The conditions for their mutual covering are given. Then, these cones are used to construct a criterion for embedding the space of generalized fractional maximal functions into the rearrangement invariant spaces (RIS). The optimal RIS for such embedding is also described.

math.FA↗
arXiv2024

A unified approach to inequalities for K-functionals and moduli of smoothness

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.

math.FA↗
arXiv2023

Quasilinear P.D.Es, Interpolation spaces and Hölderian mappings

As in the work of Tartar ( Tartar L. Interpolation non linéaire et régularité, 9, Journal of Functional Analysis, (1972), 469-489) we developed here some new results on non linear interpolation of $α$-Hölderian mappings between normed spaces, namely, by studying the action of the mappings on $K$-functionals and between interpolation spaces with logarithm functors. We apply those results to obtain regularity results on the gradient of the solution to quasilinear equations of the form $$-div(\widehat a(\nabla u ))+V(u)=f, $$ whenever $V$ is a nonlinear potential, $f$ belongs to non standard spaces as Lorentz-Zygmund spaces. We show among other that the mapping $T: \ Tf=\nabla u$ is locally or globally $α$-Hölderian under suitable values of $α$ and adequate hypothesis on $V$ and $\widehat a.$

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

Holmstedt's formula for the $K$-functional: the limit case $θ_0=θ_1$

We consider $K$-interpolation spaces involving slowly varying functions, and derive necessary and sufficient conditions for a Holmstedt-type formula to be held in the limiting case $θ_0=θ_1\in\{0,1\}.$ We also study the case $θ_0=θ_1\in (0,1).$ Applications are given to Lorentz-Karamata spaces, generalized gamma spaces and Besov spaces.

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

The two-weight Hardy inequality: a new elementary and universal proof

We give a~new proof of the known criteria for the inequality \begin{equation*} \left(\int_{0}^{\infty}\left(\int_{0}^{t}f\right)^{q}w(t)\,dt\right)^{\frac{1}{q}} \leq C \left(\int_{0}^{\infty}f^{p}v\right)^{\frac{1}{p}}. \end{equation*} The innovation is in the elementary nature of the proof and its versatility.

math.CA↗
arXiv2020

On the stability of the Lions-Peetre method of real interpolation with functional parameter

Let $\vec{X}=(X_0, X_1)$ be a compatible couple of Banach spaces, $ 1\le p \le \infty$ and let $ φ$ be positive quasi-concave function. Denote by $\overline{X}_{φ,p}=(X_0,X_1)_{φ,p}$ the real interpolation spaces defined by S. Janson (1981). We give necessary and sufficient conditions on $ φ_{0}$, $φ_{1}$ and $φ$ for the validity of \begin{equation*} \left(\overline{X}_{φ_{0},1},\overline{X}_{φ_{1},1} \right) _{φ,p}= \left(\overline{X}_{φ_{0},\infty},\overline{X}_{φ_{1},\infty}\right)_{φ,p} \end{equation*} for all $ 1\le p\le \infty$, and all Banach couples $\overline{X}. $

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↗
arXiv2019

Weighted inequalities for discrete iterated Hardy operators

We characterize a three-weight inequality for an iterated discrete Hardy-type operator. In the case when the domain space is a weighted space $\ell^p$ with $p\in(0,1]$, we develop characterizations which enable us to reduce the problem to another one with $p=1$. This, in turn, makes it possible to establish an equivalence of the weighted discrete inequality to an appropriate inequality for iterated Hardy-type operators acting on measurable functions defined on $\mathbb{R}$, for all cases of involved positive exponents.

math.FA↗
arXiv2017

Construction of function spaces close to $L^\infty$ with associate space close to $L^1$

The paper introduces a variable exponent space $X$ which has in common with $L^{\infty}([0,1])$ the property that the space $C([0,1])$ of continuous functions on $[0,1]$ is a closed linear subspace in it. The associate space of $X$ contains both the Kolmogorov and the Marcinkiewicz examples of functions in $L^{1}$ with a.e. divergent Fourier series.

math.FA↗
arXiv2017

Characterization of interpolation between Grand, small or classical Lebesgue spaces

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally $GΓ$-spaces. As a direct consequence of our results any Lorentz-Zygmund space $L^{a,r}({\rm Log}\, L)^β$, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that $ 1<a<\infty, β\not= 0$. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.

math.FA↗
arXiv2017

Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces

There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq \infty$, into Lorentz-type spaces.

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↗