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Öğe Dual Spaces Of Local Morrey-Type Spaces(Springer Heidelberg, 2011) Gogatishvili, Amiran; Mustafayev, RzaIn this paper we show that associated spaces and dual spaces of the local Morrey-type spaces are so called complementary local Morrey-type spaces. Our method is based on an application of multidimensional reverse Hardy inequalities.Öğe Embeddings Between Weighted Copson And Cesaro Function Spaces(Springer Heidelberg, 2017) Gogatishvili, Amiran; Mustafayev, Rza; Unver, TugceIn this paper, characterizations of the embeddings between weighted Copson function spaces Cop(p1,q1)(u(1),v(1)) and weighted Cesaro function spaces Ces(p2,q2) (u(2) , v(2)) are given. In particular, two-sided estimates of the optimal constant c in the inequality (integral(infinity)(0) (integral(t)(0) f(tau)(p2)v2(tau)d tau)(q2/p2) u2(t)dt)(1/q2)& para;& para;<= c(integral(infinity)(0) (integral(t)infinity f(tau)(p1)v1(tau)d tau)(q1/p1) u1(t)dt)(1/q1), where p(1), p(2), q(1), q(2) is an element of (0,infinity), p(2) <= q(2) and u(1), u(2), v(1), v(2) are weights on (0,infinity) are obtained. The most innovative part consists of the fact that possibly different parameters p1 and p2 and possibly different inner weights v(1) and v(2) are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesaro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.Öğe Iterated Hardy-type inequalities involving suprema(Element, 2017) Gogatishvili, Amiran; Mustafayev, Rza Ch.In this paper, the boundedness of the composition of the supremal operators defined, for a non-negative measurable functions f on (0,infinity), by S(u)g(t) := ess sup(0Öğe New characterization of weighted inequalities involving superposition of Hardy integral operators(Wiley-V C H Verlag Gmbh, 2024) Gogatishvili, Amiran; Unver, TugceLet 1 <= p < infinity and 0 < q, r < infinity. We characterize the validity of the inequality for the composition of the Hardy operator, (integral(b)(a) (integral(x )(a)(integral(t )(a)f(s)ds)(q )u(t)dt)(r/q )w(x)dx)(1/r)<= C(integral(b )(a)f(x)(p)v(x)dx)(1/p) for all non-negative measurable functions f on (a,b), -infinity <= a < b <=infinity. 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.Öğe New pre-dual space of Morrey space(Academic Press Inc Elsevier Science, 2013) Gogatishvili, Amiran; Mustafayev, Rza Ch.In this paper, we give new characterization of the classical Morrey space. Complementary global Morrey-type spaces are introduced. It is proved that for particular values of parameters these spaces give new pre-dual space of the classical Morrey space. We also show that our new pre-dual space of the Money space coincides with known pre-dual spaces. (C) 2012 Elsevier Inc. All rights reserved.Öğe A note on maximal commutators and commutators of maximal functions(Math Soc Japan, 2015) Agcayazi, Mujdat; Gogatishvili, Amiran; Koca, Kerim; Mustafayev, RzaIn this paper maximal commutators and commutators of maximal functions with functions of bounded mean oscillation are investigated. New pointwise estimates for these operators are proved.Öğe Pointwise multipliers between weighted copson and cesàro function spaces(Walter De Gruyter Gmbh, 2019) Gogatishvili, Amiran; Mustafayev, Rza Ch.; Unver, TugceIn this paper the solution of the pointwise multiplier problem between weighted Copson function spaces Cop(p1), (q1) ((u1,) (v1)) and weighted Cesaro function spaces Ces(p2, q2) (u(2), v(2)) is presented, where p(1), p(2), q(1), q(2) is an element of (0,infinity), p(2) <= q(2) and u(1), u(2), v(1), v(2) are weights on (0, infinity). (C) 2019 Mathematical Institute Slovak Academy of SciencesÖğe Some new iterated Hardy-type inequalities: the case θ=1(Springer International Publishing Ag, 2013) Gogatishvili, Amiran; Mustafayev, Rza; Persson, Lars-ErikIn this paper we characterize the validity of the Hardy-type inequality parallel to parallel to integral(infinity)(s)h(z)dz parallel to(p,u,(0,t))parallel to(q,w,(0,infinity)) <= c parallel to h parallel to(1,v,(0,infinity)), where 0 < p < infinity, 0 < q <= +infinity, u, w and v are weight functions on (0, infinity). It is pointed out that this characterization can be used to obtain new characterizations for the boundedness between weighted Lebesgue spaces for Hardy-type operators restricted to the cone of monotone functions and for the generalized Stieltjes operator.Öğe Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function(Tokyo Journal Mathematics Editorial Office Acad Center, 2018) Gogatishvili, Amiran; Mustafayev, Rza; Agcayazi, MujdatIn this paper it is shown that the Hardy-Littlewood maximal operator M is not bounded on Zygmund-Money space M-L(log L),M-lambda,M- O < lambda < n, but M is still bounded on M-L(logL),M-lambda for radially decreasing functions. The boundedness of the iterated maximal operator M-2 from M-L(log L),M-lambda to weak Zygmund-Morrey space WML(log L),lambda is proved. The class of functions for which the maximal commutator C-b is bounded from ML((log L),lambda)to WM(L(log L),lambda)are characterized. It is proved that the commutator of theHIardy-Littlewood maximal operator M with function b is an element of BMO(R-n)A such that b(-)is an element of L-infinity(R-n)A is bounded fom M(L(log L),lambda)to WM(L(log L),lambda. )New pointwise characterizations of M alpha M by means of norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.Öğe Weighted Inequalities for a Superposition of the Copson Operator and the Hardy Operator(Springer Birkhauser, 2022) Gogatishvili, Amiran; Mihula, Zdenek; Pick, Lubos; Turcinova, Hana; Unver, TugceWe study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely (integral(b)(a)(integral(b)(t)integral(s)(a) f(tau)p upsilon(tau)d tau)(q/p) u(s) ds)(r/q)w(t)dt)(1/r) <= C integral(b)(a) f(t) dt, in which (a, b) is any nontrivial interval, q, r are positive real parameters and p is an element of (0, 1]. A simple change of variables can be used to obtain any weighted L-p-norm with p >= 1 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 newtype 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.Öğe Weighted inequalities for discrete iterated kernel operators(Wiley-V C H Verlag Gmbh, 2022) Gogatishvili, Amiran; Pick, Lubos; Unver, TugceWe 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 (Sigma(n is an element of z)(Sigma(n)(t = -infinity) U(i, n)a(i))(q)w(n))(1/q) <= C(Sigma(n is an element of Z)a(n)(p)v(n))(1/p) holds for every sequence of nonnegative numbers where {a(n)}(nzZ) where U is a kernel satisfying certain regularity condition, 0 < p,q <= infinity and (u(n))(nzZ) and {w(n)}(nzZ) are fixed weight sequences. We do the same for the inequality (Sigma(n is an element of z)w(n)(sup-infinity<= n U(i, n) Sigma(i)(j=-infinity) a(j)](q))(1/q) <= C(Sigma(n is an element of Z)a(n)(p)v(n))(1/p) . We characterize these inequalities by conditions of both discrete and continuous nature.Öğe Weighted inequalities involving Hardy and Copson operators(Academic Press Inc Elsevier Science, 2022) Gogatishvili, Amiran; Pick, Lubos; Unver, TugceWe characterize a four-weight inequality involving the Hardy operator and the Copson operator. More precisely, given p(1), p(2), q(1), q(2) is an element of (0, infinity), we find necessary and sufficient conditions on non-negative measurable functions u(1), u(2), v(1), v(2) on (0, infinity) for which there exists a positive constant c such that the inequality (integral(infinity)(0)(integral(t)(0)f(s)(p2) v(2)(s)(p2)ds)(q2/p2) u(2)(t)(q2)dt)(1/q2) <= c(integral(infinity)(0)(integral(infinity)(t)f(s)(p1) v(1) (s)(p1) ds)(q1/p1) u(1)(t)(q1)dt)(1/q1) holds for every non-negative measurable function f on (0, infinity). 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 Cop(p1,q1)(u(1), v(1)) and weighted Cesaro spaces Ces(p2,q2)(u(2), v(2)), and also between spaces S-q(w) equipped with the norm parallel to f parallel to(Sq(w)) = (integral(infinity)(0)[f**(t) - f*(t)](q)w(t) dt)(1/q) and classical Lorentz spaces of type Lambda. (C) 2022 Published by Elsevier Inc.Öğe Weighted Iterated Hardy-Type Inequalities(Element, 2017) Gogatishvili, Amiran; Mustafayev, Rza Ch.In this paper reduction and equivalence theorems for the boundedness of the composition of a quasilinear operator T with the Hardy and Copson operators in weighted Lebesgue spaces are proved. New equivalence theorems are obtained for the operator T to be bounded in weighted Lebesgue spaces restricted to the cones of monotone functions, which allow to change the cone of non-decreasing functions to the cone of non-increasing functions and vice versa not changing the operator T. New characterizations of the weighted Hardy-type inequalities on the cones of monotone functions are given. The validity of so-called weighted iterated Hardy-type inequalities are characterized.
