Abstract
We first prove some new weighted refinements of inequalities of Hardy’s type with negative powers. Next, we prove that any \(A_{\lambda }^{1}\) Muckenhoupt class with a weight \(\lambda \) belongs to some weighted Gehring class \(G_{\lambda }^{p}\) for \(p>1\). We also prove that the self-improving property of the weighted Muckenhoupt class \(A_{\lambda }^{q}\) holds. The main results give exact values of the limit exponents and the constants of the new classes. The self-improving property of the weighted Muckenhoupt class will then be applied to prove the self-improving property of the Gehring class with a sharp value on the exponents.
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Agarwal, R.P., O’Regan, D. & Saker, S.H. Self-improving properties of a generalized Muckenhoupt class. Acta Math. Hungar. 164, 113–134 (2021). https://doi.org/10.1007/s10474-021-01136-8
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DOI: https://doi.org/10.1007/s10474-021-01136-8