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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On Hardy's inequality in weighted rearrangement invariant spaces and applications. I


Author: Lech Maligranda
Journal: Proc. Amer. Math. Soc. 88 (1983), 67-74
MSC: Primary 46E30; Secondary 26D15, 46M35
DOI: https://doi.org/10.1090/S0002-9939-1983-0691280-6
MathSciNet review: 691280
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Abstract: We give inequalities relating the norm of a function and the norm of its average operators $ {P_\psi },{Q_\psi }$ and $ {S_\psi },{T_\psi }$ in weighted rearrangement invariant spaces $ {E_{\kappa ,\delta }}$ and $ E(\mu ),d\mu (t) = \tau '(t)dt$. These average operators include, for example, the integral mean, the $ {P_p},{Q_p}$ operators of Boyd [4] and Butzer and Fehér [6], the average operators $ {P_\varphi },{Q_\varphi }$ and $ {S_E},{T_E}$ from [14,15,16]. In the particular case, for some $ \psi ,\kappa ,\delta ,\tau $ and $ E$ these inequalities were obtained by many authors and applied to a study of interpolation operators and imbedding theorems for Sobolev weight spaces.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0691280-6
Keywords: Average operator, Hardy inequality, rearrangement invariant spaces, indices
Article copyright: © Copyright 1983 American Mathematical Society