Abstract
In this paper, we study the structure of the discrete Muckenhoupt class mathcal{A}^{p}(mathcal{C}) and the discrete Gehring class mathcal{G}^{q}(mathcal{K}). In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if uin mathcal{A}^{p}(mathcal{C}) then there exists q< p such that uin mathcal{A}^{q}(mathcal{C}_{1}). Next, we prove that the power rule also holds, i.e., we prove that if uin mathcal{A}^{p} then u^{q}in mathcal{A}^{p} for some q>1. The relation between the Muckenhoupt class mathcal{A}^{1}(mathcal{C}) and the Gehring class is also discussed. For illustrations, we give exact values of the norms of Muckenhoupt and Gehring classes for power-low sequences. The results are proved by some algebraic inequalities and some new inequalities designed and proved for this purpose.
Highlights
We fix an interval I ⊂ R and consider subintervals I of I and denote by |I| the Lebesgue measure of I
Sbordone, and Wik [3] improved the Muckenhoupt result by excluding the monotonicity condition on the weight w by using the rearrangement ω∗ of the function ω over the interval I and established the best constant
Remark 1.1 Lemma 1.2 proves that if the weight w belongs to the Muckenhoupt class A1(C), w belongs to the Gehring class Gp(K) with K = [C1–p/(C – p(C – 1))]1/p–1 for p ∈ [1, C/(C – 1)]
Summary
We fix an interval I ⊂ R and consider subintervals I of I and denote by |I| the Lebesgue measure of I. Sbordone, and Wik [3] improved the Muckenhoupt result by excluding the monotonicity condition on the weight w by using the rearrangement ω∗ of the function ω over the interval I and established the best constant. For a given exponent q > 1 and a constant K > 1, a discrete nonnegative weight u belongs to the discrete Gehring class Gq(K) (or satisfies a reverse Hölder inequality) on the interval I if, for every subinterval J ⊆ I, we have. For illustration, we establish the exact values of the Muckenhoupt norm Aq(nα) and the Gehring norm Gp(nα) for power-low sequences {nα}
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