IJPAM: Volume 119, No. 4 (2018)

Title

BOUNDEDNESS OF MAXIMAL FUNCTIONS
WITH MIXED HOMOGENEITY

Authors

Hiyam Al-Bataineh$^1$, Mohammed Ali$^2$
$^1$Department of Mathematics and Statistics
Jordan University of Science and Technology
P.O. Box 3030, Irbid, 22110, JORDAN
$^2$Department of Mathematics and Statistics
Jordan University of Science and Technology
P.O. Box 3030, Irbid, 22110, JORDAN

Abstract

In this article, we study a certain class of maximal functions with mixed homogeneity when the kernels in $L^q(\mathbf{S}^{n-1})$ for $1<q\leq2$. We obtain appropriate $L^p$ estimates for such maximal operators. These estimates will allow us to use extrapolation arguments to establish the $L^p$ boundedness of our maximal functions when their kernels belong to $ L(\log L)^{1/\gamma'}(
\mathbf{S}
^{n-1})\cup B^{(0,-1/\gamma)}_q( \mathbf{S}^{n-1})$ with $q>1$ and $1\leq \gamma \leq2$. Our results essentially improve and extend some known results on maximal functions as well as singular integrals.

History

Received: April 5, 2017
Revised: August 2, 2018
Published: August 10, 2018

AMS Classification, Key Words

AMS Subject Classification: 40B20, 40B15, 40B25
Key Words and Phrases: $L^p$ boundedness, maximal operators, rough kernels

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1
M. Al-Dolat, M. Ali, I. Jaradat, K. Al-Zoubi, On the boundedness of a certain class of maximal functionson product spaces and extrapolation, Anal. Math. Phys.,1 (2018), 1-12.

2
H. Al-Qassem, On the boundedness of maximal operators and singular operators with kernels in $L(logL)^\alpha(\mathbf{S}^{n-1})$, J. Ineq. Apll., 1 (2006), 1-16.

3
H. Al-Qassem, Y. Pan, On certain estimatesfor Marcinkiewicz integrals and extrapolation, Collec. Math.,60 (2009), 123-145.

4
A. Al-Salman, A unifying approach for certein classof maximal functions, J. Ineq. Appl., 1 (2006),1-17.

5
A. Al-Salman, On a class of singular integral operators with rough kernels, Can. Math.. Bull., 49(2006), 3-10.

6
A. Al-Salman, On maximal functions with rough kernels in $L(logL)^{1/2}(\mathbf{S}^{n-1})$, Collec. Math., 56(2005), 47-56.

7
A. Al-Salman, Rough oscillatory singular integraloperators of nonconvolution type, J. Math. Anal. Appl.,299 (2004), 72-88.

8
A. Al-Salman, A. Al-Jarrah, Rough oscillatorysingular integral operators, Turk. J. Math., 27(2003), 565-579.

How to Cite?

DOI: 10.12732/ijpam.v119i4.12 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 119
Issue: 4
Pages: 705 - 716


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).