Bulletin of the
Korean Mathematical Society

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online March 10, 2022

Copyright © The Korean Mathematical Society.

Weighted integral inequalities for modified integral Hardy operators

Duranta Chutia and Rajib Haloi

Tezpur University


In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights $\omega, \rho, \phi$ and $\psi$ to hold the following weak type modular inequalities.
\mathcal{U}^{-1} \bigg ( \int \limits_{ \{ | \mathcal{I}f | > \gamma\}} \mathcal{U} \Big(\gamma \omega \Big ) \rho \bigg ) & \leq \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \Big ( C |f| \phi\Big) \psi \bigg ),
where $\mathcal{I}$ is the modified integral Hardy operators . We also obtain a necesary and sufficient condition for the following extra-weak type integral inequalities.
\omega \bigg ( \Big\{ |\mathcal{I}f| > \gamma \Big \} \bigg) &\leq \mathcal{U}\circ \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \bigg ( \dfrac{C |f| \phi}{\gamma} \bigg) \psi \bigg ).
Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operators and its integral version.

Keywords: Modified Hardy operators; Integral operator; Integral inequalities; Weights

MSC numbers: 42B25, 46E30

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