Bulletin of the
Korean Mathematical Society
BKMS

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

Article

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Bull. Korean Math. Soc. 2022; 59(3): 757-780

Online first article March 10, 2022      Printed May 31, 2022

https://doi.org/10.4134/BKMS.b210469

Copyright © The Korean Mathematical Society.

Weighted integral inequalities for modified integral Hardy operators

Duranta Chutia, Rajib Haloi

Tezpur University; 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 inequality \begin{align*} \mathcal{U}^{-1} \bigg ( \int_{ \{ | \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 inequality \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 operator and its integral version.

Keywords: Modified Hardy operators, integral operator, integral inequalities, weights

MSC numbers: Primary 42B25, 46E30

Supported by: Duranta Chutia is thankful to DST INSPIRE, Govt. of India for the financial support DST/INSPIRE Fellowship/2017/IF170509. Rajib Haloi is thankful to Department of Science and Technology, Govt. of India for the financial support DST MATRICS(SERB/F/12082/2018-2019).