Bulletin of the
Korean Mathematical Society
BKMS

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

Abstract

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.
\begin{align*}
\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 ),
\end{align*}
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.
\begin{align*}
\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 ).
\end{align*}
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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