In this treatise, the boundedness of the multilinear fractional Hardy operators is scrutinized within the context of variable exponent Morrey-Herz spaces, denoted as $ {M\dot K^{\alpha(\cdot), \lambda}_{q, p(\cdot)}(\mathbb{R}^n)} $. Analogous estimations are derived for their commutators, contingent upon the symbol functions residing in the space of bounded mean oscillation (BMO) with variable exponents.
Citation: Muhammad Asim, Ghada AlNemer. Boundedness on variable exponent Morrey-Herz space for fractional multilinear Hardy operators[J]. AIMS Mathematics, 2025, 10(1): 117-136. doi: 10.3934/math.2025007
In this treatise, the boundedness of the multilinear fractional Hardy operators is scrutinized within the context of variable exponent Morrey-Herz spaces, denoted as $ {M\dot K^{\alpha(\cdot), \lambda}_{q, p(\cdot)}(\mathbb{R}^n)} $. Analogous estimations are derived for their commutators, contingent upon the symbol functions residing in the space of bounded mean oscillation (BMO) with variable exponents.
| [1] |
G. H. Hardy, Note on a theorem of Hilbert, Math. Z., 6 (1920), 314–317. https://doi.org/10.1007/BF01199965 doi: 10.1007/BF01199965
|
| [2] |
W. G. Faris, Weak Lebesgue spaces and quantum mechanical binding, Duke Math. J., 43 (1976), 365–373. https://doi.org/10.1215/S0012-7094-76-04332-5 doi: 10.1215/S0012-7094-76-04332-5
|
| [3] |
M. Chris, L. Grafakos, Best constant for two nonconvolution inequalities, Math. Z., 123 (1995), 1687–1693. https://doi.org/10.1090/S0002-9939-1995-1239796-6 doi: 10.1090/S0002-9939-1995-1239796-6
|
| [4] | Z. W. Fu, L. Grafakos, S. Z. Lu, F. Y. Zhao, Sharp bounds for $m$-linear Hardy and Hilbert operators, Houston J. Math., 1 (2012), 225–244. |
| [5] |
L. E. Persson, S. G. Samko, A note on the best constants in some hardy inequalities, J. Math. Inequal., 9 (2015), 437–447. https://doi.org/10.7153/jmi-09-37 doi: 10.7153/jmi-09-37
|
| [6] |
F. Y. Zhao, Z. W. Fu, S. Z. Lu, Endpoint estimates for $n$-dimensional Hardy operators and their commutators, Sci. China Math., 55 (2012), 1977–1990. https://doi.org/10.1007/s11425-012-4465-0 doi: 10.1007/s11425-012-4465-0
|
| [7] |
G. Gao, X. Hu, C. Zhong, Sharp weak estimates for Hardy-type operators, Ann. Funct. Anal., 7 (2016), 421–433. https://doi.org/10.1215/20088752-3605447 doi: 10.1215/20088752-3605447
|
| [8] | A. Hussain, N. Sarfraz, F. Gürbüz, Sharp weak bounds for $p$-adic Hardy operators on $p$-adic linear spaces, Commun. Fac. Sci. Univ., 2002. |
| [9] |
Z. W. Fu, Z. G. Liu, S. Z. Lu, H. Wong, Characterization for commutators of $ n $-dimensional fractional Hardy operators, Sci. China Ser. A, 10 (2007), 1418–1426. https://doi.org/10.1007/s11425-007-0094-4 doi: 10.1007/s11425-007-0094-4
|
| [10] |
F. Y. Zhao, S. Z. Lu, The best bound for $n$-dimensional fractional Hardy operator, Math. Inequal. Appl., 18 (2015), 233–240. https://doi.org/10.7153/mia-18-17 doi: 10.7153/mia-18-17
|
| [11] |
S. R. Wang, J. S. Xu, Commutators of the bilinear Hardy operator on Herz type spaces with variable exponent, J. Funct. Space., 2019 (2019). https://doi.org/10.1155/2019/7607893 doi: 10.1155/2019/7607893
|
| [12] |
M. Asim, I. Ayoob, A. Hussain, N. Mlaiki, Weighted estimates for fractional bilinear Hardy operators on variable exponent Morrey Herz space, J. Inequal. Appl., 2024 (2024), 11. https://doi.org/10.1186/s13660-024-03092-7 doi: 10.1186/s13660-024-03092-7
|
| [13] | G. H. Hardy, J. E. Littlewood, G. PÓlya, Inequalities, second edition, Cambridge: Univ. Press, 1952. |
| [14] | D. E. Edmunds, W. D. Evans, Hardy operators, function spaces and embedding, Berlin: Springer Verlag, 2004. |
| [15] |
Z. W. Fu, Q. Y. Wu, S. Z. Lu, Sharp estimates of $p$-adic Hardy and Hardy-Littlewood-Pólya operators, Acta Math. Sin., 29 (2013), 137–150. https://doi.org/10.1007/s10114-012-0695-x doi: 10.1007/s10114-012-0695-x
|
| [16] |
G. Gao, F. Y. Zhao, Sharp weak bounds for Hausdorff operators, Anal. Math., 41 (2015), 163–173. https://doi.org/10.1007/s10476-015-0204-4 doi: 10.1007/s10476-015-0204-4
|
| [17] | S. Z. Lu, D. C. Yang, F. Y. Zhao, Sharp bounds for Hardy type operators on higher dimensional product spaces, J. Inequal. Appl., 148 (2013). https://doi.org/10.1186/1029-242X-2013-148 |
| [18] | A. Hussain, M. Asim, Commutators of the fractional Hardy operator on weighted variable Herz-Morrey spaces, J. Funct. Space., 2021. https://doi.org/10.1155/2021/9705250 |
| [19] |
M. Asim, A. Hussain, Weighted variable Morrey-Herz estimates for fractional Hardy operators, J. Inequal. Appl., 2022 (2022). https://doi.org/10.1186/s13660-021-02739-z doi: 10.1186/s13660-021-02739-z
|
| [20] |
T. L. Yee, K. P. Ho, Hardy's inequalities and integral operators on Herz-Morrey spaces, Open Math., 18 (2020), 106–121. https://doi.org/10.1515/math-2020-0008 doi: 10.1515/math-2020-0008
|
| [21] |
W. Orlicz, Uber konjugierete exponentenfolgen, Stud. Math., 3 (1931), 200–212. https://doi.org/10.4064/sm-3-1-200-211 doi: 10.4064/sm-3-1-200-211
|
| [22] |
O. Kováčik, J. Rákosník, On spaces $ L^{p(x)} $ and $ W^{k, p(x)} $, Czechoslovak Math. J., 41 (1991), 592–618. https://doi.org/10.21136/CMJ.1991.102493 doi: 10.21136/CMJ.1991.102493
|
| [23] | D. Cruz-Uribe, A. Fiorenza, J. M. Martell, C. Pérez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn.-M., 31(2006), 239–264. |
| [24] | D. Cruz-Uribe, D. V. Fiorenza, Variable Lebesgue spaces, Foundations and Harmonic Analysis, Heidelberg: Springer, 2013. https://doi.org/10.1007/978-3-0348-0548-3 |
| [25] |
Y. Chen, S. Levine, M, Rao, Variable exponent, linear growth functionals in image restoration, Appl. Math., 6 (2006), 1383–1406. https://doi.org/10.1137/050624522 doi: 10.1137/050624522
|
| [26] | M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory, Lectures Notes in Math., Berlin: Springer, 2000. https://doi.org/10.1007/BFb0104030 |
| [27] |
P. Harjulehto, P. Hasto, U. V. Le, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551–4574. https://doi.org/10.1016/j.na.2010.02.033 doi: 10.1016/j.na.2010.02.033
|
| [28] |
M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math., 13 (2010), 33–50. http://dx.doi.org/10.1007/s10476-010-0102-8 doi: 10.1007/s10476-010-0102-8
|
| [29] |
A. Almeida, D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl., 394 (2012), 781–795. https://doi.org/10.1016/j.jmaa.2012.04.043 doi: 10.1016/j.jmaa.2012.04.043
|
| [30] |
S. Samko, Variable exponent Herz spaces, Mediterr. J. Math., 10 (2013), 2007–2025. https://doi.org/10.1007/s00009-013-0285-x doi: 10.1007/s00009-013-0285-x
|
| [31] | M. Izuki, Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent, Math. Sci. Res. J., 13 (2009), 243–253. |
| [32] |
M. Asim, G. AlNemer, Results for fractional bilinear Hardy operators in central varying exponent Morrey space, AIMS Math., 9 (2024), 29689–29706. https://doi.org/10.3934/math.20241438 doi: 10.3934/math.20241438
|
| [33] |
K. P. Ho, Extrapolation to Herz spaces with variable exponents and applications, Rev. Mate. Complut., 33 (2020), 437–463. https://doi.org/10.1007/s13163-019-00320-3 doi: 10.1007/s13163-019-00320-3
|
| [34] |
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, T. Am. Math. Soc., 165 (1972), 207–226. https://doi.org/10.1090/S0002-9947-1972-0293384-6 doi: 10.1090/S0002-9947-1972-0293384-6
|
| [35] |
D. Cruz-Uribe, L. Diening, P. Hästö, The maximal operator on weighted variable Lebesgue spaces, Fract. Calc. Appl. Anal., 14 (2011), 361–374. https://doi.org/10.2478/s13540-011-0023-7 doi: 10.2478/s13540-011-0023-7
|
| [36] | L. Diening, L. Harjulehto, P. Hästö, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Heidelberg: Springer, 2011. Available from: https://www.mathematik.uni-muenchen.de/~diening/archive/book.pdf. |
| [37] |
C. Capone, D. Cruz-Uribe, SFO, A. Fiorenza, The fractional maximal operator and fractional integrals on variable $L^p$ spaces, Rev. Mat. Iberoam., 23 (2007), 743–770. https://doi.org/10.4171/rmi/511 doi: 10.4171/rmi/511
|
| [38] |
M. Izuki, T. Noi, Boundedness of fractional integrals on weighted Herz spaces with variable exponent, J. Inequal. Appl., 2016 (2016), 199. https://doi.org/10.1186/s13660-016 doi: 10.1186/s13660-016
|
| [39] | L. Grafakos, Modern Fourier analysis, Berlin, Germany: Springer, 2 Eds., 2008. https://doi.org/10.1007/978-1-4939-1194-3 |
| [40] |
B. H. Dong, J. S. Xu, Herz-Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, Banach J. Math. Anal., 9 (2015), 75–101. https://doi.org/10.15352/bjma/09-1-7 doi: 10.15352/bjma/09-1-7
|
| [41] |
A. Asim, F. Gurbaz, Some variable exponent boundedness and commutators estimates for fractional Hardy operators on central Morrey spaces, Commun. Fac. Sci. Univ., 7 (2024), 802–819. https://doi.org/10.31801/cfsuasmas.1463245 doi: 10.31801/cfsuasmas.1463245
|
| [42] | D. Cruz-Uribe, A. Fiorenza, C. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn.-M., 28 (2003), 223–238. Available from: https://acadsci.fi/mathematica/Vol28/cruz.pdf. |
| [43] | C. Bennett, R. C. Sharpley, Interpolation of operators, Boston: Academic Press, 1988. |
| [44] |
Z. Fu, S. Lu, H. Wang, L. Wang, Singular integral operators with rough kernels on central Morrey spaces with variable exponent, Ann. Acad. Sci. Fenn.-M., 44 (2019), 505–522. https://doi.org/10.5186/aasfm.2019.4431 doi: 10.5186/aasfm.2019.4431
|
| [45] |
M. Izuki, Fractional integrals on Herz Morrey spaces with variable exponent, Hiroshima Math. J., 40 (2010), 343–355. https://doi.org/10.32917/hmj/1291818849 doi: 10.32917/hmj/1291818849
|
| [46] |
J. L. Wu, Boundedness of some sublinear operators on Herz Morrey spaces with variable exponent, Georgian Math. J., 21 (2014), 101–111. https://doi.org/10.1515/gmj-2014-0004 doi: 10.1515/gmj-2014-0004
|
Muhammad Asim, Ghada AlNemer. Boundedness on variable exponent Morrey-Herz space for fractional multilinear Hardy operators[J]. AIMS Mathematics, 2025, 10(1): 117-136. doi: 10.3934/math.2025007
DownLoad: