Weighted Inequalities for Discrete Iterated Hardy Operators
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Weighted Inequalities for Discrete Iterated Hardy Operators Amiran Gogatishvili , Martin Kˇrepela , Rastislav Ol’hava and Luboˇs Pick Abstract. We characterize a three-weight inequality for an iterated discrete Hardy-type operator. In the case when the domain space is a weighted space p with p ∈ (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 R, for all cases of involved positive exponents. Mathematics Subject Classification. 46E30, 26D20, 47B38, 46B70. Keywords. Weighted discrete inequality, supremum operator, iterated operator.
1. Introduction In this paper, we focus on a three-weight inequality for the composition of a discrete supremal and integral Hardy operator. Let us denote by RZ+ the space of all double-infinite sequences of positive (nonnegative) real numbers. We are interested in the question under what conditions on given u, v, w ∈ RZ+ , there exist constants C1 , C2 ∈ (0, ∞), such that the inequalities: ⎛ ⎝
n∈Z
⎛ ⎝sup ui i≥n
⎞q
⎞ q1
ak ⎠ wn ⎠ ≤ C1
p1 apn vn
(1.1)
n∈Z
k≤i
This research was supported by the Grants P201-13-14743S and P201-18-00580S of the Czech Science Foundation and by the Grant 8X17028 of the Czech Ministry of Education. The research of A. Gogatishvili was partially supported by Shota Rustaveli National Science Foundation (SRNSF), Grant no: FR17-589.
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⎛ ⎝
n∈Z
A. Gogatishvili et al.
⎛ ⎝sup ui i≥n
⎞q
⎞ q1
ak ⎠ wn ⎠ ≤ C2
MJOM
p1 apn vn
(1.2)
n∈Z
k≥i
hold for every sequence a ∈ RZ+ . We study several aspects of such an inequality including its relationship to an analogous one for integral operators. Before continuing, let us recall that (1.1) being satisfied for all a ∈ RZ+ is equivalent to: ⎛ ⎛ ⎞q ⎞ q1 p1 p ⎝ ⎝sup ui ak ⎠ wn ⎠ ≤ C1 an v n , (1.3) n∈Z
i≤n
n∈Z
k≥i
RZ+ .
also being satisfied for all a ∈ This is obvious by the index change un = u−n , v n = v−n and wn = w−n . Analogously, the inequality: ⎛ ⎞q ⎞ q1 ⎛ p1 p ⎝sup ui ⎝ ak ⎠ wn ⎠ ≤ C1 an v n (1.4) n∈Z
i≤n
k≤i
n∈Z
is equivalent to (1.2). It is common to refer to (1.3) and (1.4) as to the dual versions of (1.1) and (1.2), respectively. In contrast, inequalities (1.1) and (1.2) (hence also (1.3) and (1.4)) are essentially different. The success that the theory of weighted inequalities has seen in last 3 decades can be credited greatly to a clever combination of classical techniques such as symmetrization or interpolation with new methods such as discretization (the blocking technique), antidiscretization, reduction theorems, and the use of supremum operators. The research of problems in mathematical physics often leads to the investigation of certain Sobolev-type embeddings. Under certain circumstances, these can be quite successfully attacked by classical symmetri
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