Abstract
In this paper, we will prove some fundamental properties of the discrete power mean operator M p u n = 1 / n ∑ k = 1 n u p k 1 / p , for n ∈ I ⊆ ℤ + , of order p , where u is a nonnegative discrete weight defined on I ⊆ ℤ + the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, we will study the structure of the generalized discrete class B p q B of weights that satisfy the reverse Hölder inequality M q u ≤ B M p u , for positive real numbers p , q , and B such that 0 < p < q and B > 1 . For applications, we will prove some self-improving properties of weights from B p q B and derive the self improving properties of the discrete Gehring weights as a special case. The paper ends by a conjecture with an illustrative sharp example.
Highlights
In [1], Muckenhoupt introduced a full characterization of the Ap − class of weights in connection with the boundedness of the Hardy-Littlewood maximal operator in the space LpwðR+Þ with a weight w
The study of the discrete analogues in harmonic analysis becomes an active field of research
The study of regularity and boundedness of discrete operator on lp analogues for Lp − regularity, higher summability, and structure of discrete Muckenhoupt and Gehring weights has been considered by some authors, and we refer the reader to the papers [24–34] and the references they are cited
Summary
In [1], Muckenhoupt introduced a full characterization of the Ap − class of weights in connection with the boundedness of the Hardy-Littlewood maximal operator in the space LpwðR+Þ with a weight w. The study of regularity and boundedness of discrete operator on lp analogues for Lp − regularity, higher summability, and structure of discrete Muckenhoupt and Gehring weights has been considered by some authors, and we refer the reader to the papers [24–34] and the references they are cited. For a given exponent q > 1 and a constant K > 1, a discrete nonnegative weight u defined on I belongs to the discrete Gehring class GqðKÞ (or satisfies the reverse Hölder inequality) if for every subinterval J ⊆ I, we have. The smallest constant B independent on the interval J and satisfies the inequality (12) is called the Bqp − norm which is given by ÂBqpðuÞÃ. The paper ends by a conjecture with the selfimproving of the Muckenhoupt weights with an illustrative example
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