Weighted $ L^{p} $ norms of Marcinkiewicz functions on product domains along surfaces

Badriya Al-Azri; Ahmad Al-Salman; Badriya Al-Azri; Ahmad Al-Salman

doi:10.3934/math.2024408

AIMS Mathematics

2024, Volume 9, Issue 4: 8386-8405. doi: 10.3934/math.2024408

Previous Article Next Article

Research article

Weighted $ L^{p} $ norms of Marcinkiewicz functions on product domains along surfaces

Badriya Al-Azri ^{1
,
,},
Ahmad Al-Salman ^1,2

1.
Sultan Qaboos University, College of Science, Department of Mathematics, Muscat, Oman
2.
Department of Mathematics, Yarmouk University, Irbid, Jordan

Received: 23 November 2023 Revised: 14 December 2023 Accepted: 25 December 2023 Published: 28 February 2024
MSC : 42B15, 42B20

We prove a weighted $ L^{p} $ boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space $ L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{m}, \, \omega _{1}(x)dx, \, \omega_{2}(y)dy) $, provided that the weights $ \omega_{1} $ and $ \omega_{2} $ are certain radial weights and that the kernels are rough in the optimal space $ L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1}) $. In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted $ L^{p} $ boundedness of the related square and maximal functions. Our weighted $ L^{p} $ inequalities extend as well as generalize previously known $ L^{p} $ boundedness results.
- Marcinkiewicz integral operators on product domains,
- weighted $ L^{p} $ norm,
- maximal functions,
- Hardy Littlewood maximal function,
- convex functions
Citation: Badriya Al-Azri, Ahmad Al-Salman. Weighted $ L^{p} $ norms of Marcinkiewicz functions on product domains along surfaces[J]. AIMS Mathematics, 2024, 9(4): 8386-8405. doi: 10.3934/math.2024408

Related Papers:

Abstract

We prove a weighted $ L^{p} $ boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space $ L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{m}, \, \omega _{1}(x)dx, \, \omega_{2}(y)dy) $, provided that the weights $ \omega_{1} $ and $ \omega_{2} $ are certain radial weights and that the kernels are rough in the optimal space $ L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1}) $. In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted $ L^{p} $ boundedness of the related square and maximal functions. Our weighted $ L^{p} $ inequalities extend as well as generalize previously known $ L^{p} $ boundedness results.

References

[1]	E. M. Stein, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, T. Am. Math. Soc., 88 (1958), 430–466. https://doi.org/10.2307/1993226 doi: 10.2307/1993226
[2]	A. Benedek, A. Calderon, R. Panzone, Convolution operators on Banach space valued functions, P. Natl. Acad. Sci. U. S. A., 48 (1962), 356–365. https://doi.org/10.1073/pnas.48.3.356 doi: 10.1073/pnas.48.3.356
[3]	T. Walsh, On the function of Marcinkiewicz, Stud. Math., 44 (1972), 203–217. https://doi.org/10.1090/S0002-9947-1958-0112932-2 doi: 10.1090/S0002-9947-1958-0112932-2
[4]	H. Al-Qassem, A. Al-Salman, L. Cheng, Y. Pan, $ L^{p}$ bounds for the functions of Marcinkiewicz, Math. Res. Lett., 9 (2002), 697–700. https://doi.org/10.4310/MRL.2002.v9.n5.a11 doi: 10.4310/MRL.2002.v9.n5.a11
[5]	A. Al-Salman, Marcinkiewicz functions with Hardy space kernels, Math. Inequal. Appl., 21 (2018), 553–567. https://doi.org/10.7153/mia-2018-21-40 doi: 10.7153/mia-2018-21-40
[6]	A. Al-Salman, On Marcinkiewicz integrals along flat surfaces, Turk. J. Math., 29 (2005), 111–120.
[7]	J. Chen, D. Fan, Y. Ying, Rough Marcinkiewicz integrals with $ L(\log L)^{2}$ kernels, Adv. Math. (China), 30 (2001), 179–181.
[8]	Y. Ding, $L^{2}$ boundedness of Marcinkiewicz integral with rough kernel, Hokkaido Math. J., 27 (1998), 105–115. https://doi.org/10.14492/hokmj/1351001253 doi: 10.14492/hokmj/1351001253
[9]	S. Sato, Estimates for Littlewood-Paley functions and extrapolation, Integr. Equ. Oper. Theory, 62 (2008), 429–440. https://doi.org/10.1007/s00020-008-1631-4 doi: 10.1007/s00020-008-1631-4
[10]	A. Torchinsky, S. Wang, A note on the Marcinkiewicz integral, Colloq. Math., 60 (1990), 235–243. https://doi.org/10.4064/cm-60-61-1-235-243 doi: 10.4064/cm-60-61-1-235-243
[11]	J. Garcia-Cuerva, J. L. R. de Francia, Weighted norm inequalities and related topics, Amsterdam: Elsevier, 2011.
[12]	Y. Ding, D. Fan, Y. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J., 48 (1999), 1037–1056. https://doi.org/10.1512/iumj.1999.48.1696 doi: 10.1512/iumj.1999.48.1696
[13]	M. Y. Lee, C. C. Lin, Weighted $L^{p}$ boundedness of Marcinkiewicz integral, Integr. Equ. Oper. Theory, 49 (2004), 211–220. https://doi.org/10.1007/s00020-002-1204-x doi: 10.1007/s00020-002-1204-x
[14]	J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, T. Am. Math. Soc., 336 (1993), 869–880. https://doi.org/10.2307/2154381 doi: 10.2307/2154381
[15]	A. Al-Salman, Certain weighted $L^{p}$ bounds for the functions of marcinkiewicz, SE Asian B. Math., 30 (2006), 609–620.
[16]	Y. Choi, Marcinkiewicz integrals with rough homogeneous kernels of degree zero in product domains, J. Math. Anal. Appl., 261 (2001), 53–60. https://doi.org/10.1006/jmaa.2001.7465 doi: 10.1006/jmaa.2001.7465
[17]	H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan, Marcinkiewicz integrals on product spaces, Stud. Math., 167 (2005), 227–234. https://doi.org/10.4064/sm167-3-4 doi: 10.4064/sm167-3-4
[18]	B. Al-Azriyah, A. Al-Salman, Singular and marcinkiewicz integral operators on product domains, Commun. Korean Math. S., 38 (2023), 401–430. https://doi.org/10.4134/CKMS.c210421 doi: 10.4134/CKMS.c210421
[19]	B. Al-Azriyah, A. Al-Salman, A note on marcinkiewicz integral operators on product domains, Kyungpook Math. J., 63 (2023), 577–591. https://doi.org/10.5666/KMJ.2023.63.4.577 doi: 10.5666/KMJ.2023.63.4.577
[20]	A. Al-Salman, Rough Marcinkiewicz integrals on product spaces, Int. Math. Forum, 2 (2007), 1119–1128. https://doi.org/10.12988/imf.2007.07097 doi: 10.12988/imf.2007.07097
[21]	A. Al-Salman, Marcinkiewicz integrals along subvarieties on product domains, International Journal of Mathematics and Mathematical Sciences, 2004 (2004), 730308. https://doi.org/10.1155/S0161171204401264 doi: 10.1155/S0161171204401264
[22]	J. Chen, D. Fan, Y. Ying, The method of rotation and Marcinkiewicz integrals on product domains, Stud. Math., 153 (2002), 41–58. https://doi.org/10.4064/sm153-1-4 doi: 10.4064/sm153-1-4
[23]	H. Al-Qassem, Y. Pan, $L^{p}$ boundedness for singular integrals with rough kernels on product domains, Hokkaido Math. J., 31 (2002), 555–613. https://doi.org/10.14492/hokmj/1350911903 doi: 10.14492/hokmj/1350911903
[24]	D. Fan, Y. Pan, D. Yang, A weighted norm inequality for rough singular integrals, Tohoku Math. J., 51 (1999), 141–161. https://doi.org/10.2748/tmj/1178224808 doi: 10.2748/tmj/1178224808
[25]	E. M. Stein, Harmonic analysis: Real-variable mathods, orthogonality and oscillatory integrals, Princeton: Princeton University Press, 1993.
[26]	R. Fefferman, E. M. Stein, Singular integrals on product spaces, Adv. Math., 45 (1982), 117–143. https://doi.org/10.1016/S0001-8708(82)80001-7 doi: 10.1016/S0001-8708(82)80001-7
[27]	A. Al-Salman, Y. Pan, Singular integrals with rough kernels in $Llog^{+}L(\mathbb{S}^{n-1})$, J. Lond. Math. Soc., 66 (2002), 153–174. https://doi.org/10.1112/S0024610702003241 doi: 10.1112/S0024610702003241

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)