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

Maligranda, Lech

doi:10.1090/S0002-9939-1983-0691280-6

On Hardy’s inequality in weighted rearrangement invariant spaces and applications. I
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Proc. Amer. Math. Soc. 88 (1983), 67-74 Request permission

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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On Hardy’s operator in weighted

spaces