J. Math. Computer Sci., 29 (2023), 192–202
Online: ISSN 2008-949X
Journal Homepage: www.isr-publications.com/jmcs
Conformable Gehring inequalities and conformable higher
integrability
Samir H. Saker
a,b
, Mohamed Abdalla Darwish
c
, Hamdi Ali Elshamy
c,∗
a
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.
b
Department of Mathematics, Faculty of Science, New Mansoura University, New Mansoura City, Egypt.
c
Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt.
Abstract
In this paper, we prove some reverse conformable inequalities with weights and employ them to prove some conformable
inequalities of Gehring type. Moreover, we prove some interpolation theorems which are powerful tools in the study of operators
in function spaces. Our results develop a technique based on the applications of a refinement of conformable inequalities.
Keywords: Conformable Gehring’s inequality, conformable H¨ older’s inequality, reverse inequality.
2020 MSC: 40D05, 40D25, 42C10 43A55, 46A35, 46B15.
©2023 All rights reserved.
1. Introduction
We fix an interval I
0
⊂ R
+
=[0, ∞), and consider the subinterval I of I
0
of the form [0, s], for
0 <s< ∞ and donate by |I| the Lebesgue measure of I. The nonnegative weight ω is said to belong to
the Muckenhoupt class A
p
(C) on the interval I
0
for p> 1 and C > 1 (independent of p) if the inequality
1
|I|
I
ω(x)dx C
1
|I|
I
ω
1
1-p
(x)dx
1-p
, (1.1)
holds for every subinterval I ⊂ I
0
. For p> 1, we define the A
p
-norm of the weight ω by
[A
p
(ω)] := sup
I⊂I
0
1
|I|
I
ω(x)dx
1
|I|
I
ω
-1
p-1
(x)dx
p-1
.
The weight ω is said to belong to the Muckenhoupt class A
1
(C) on the interval I
0
, if the inequality
1
|I|
I
ω(x)dx Cω(x), for C > 1, (1.2)
∗
Corresponding author
Email addresses: shsaker@mans.edu.eg; samir.saker@nmu.edu.eg (Samir H. Saker), dr.madarwish@gmail.com (Mohamed
Abdalla Darwish), h_elshamy@sci.dmu.edu.eg (Hamdi Ali Elshamy)
doi: 10.22436/jmcs.029.02.08
Received: 2022-06-07 Revised: 2022-07-17 Accepted: 2022-07-21