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    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, Tugce
    We 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.
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    Öğe
    Weighted inequalities for discrete iterated kernel operators
    (Wiley-V C H Verlag Gmbh, 2022) Gogatishvili, Amiran; Pick, Lubos; Unver, Tugce
    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 (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.
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    Öğe
    Weighted inequalities involving Hardy and Copson operators
    (Academic Press Inc Elsevier Science, 2022) Gogatishvili, Amiran; Pick, Lubos; Unver, Tugce
    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) 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.