Resolution of open problems via Orlicz-Zygmund spaces and new geometric properties of Morrey spaces in the Besov sense with non-standard growth

Waqar Afzal; Mujahid Abbas; Najla M. Aloraini; Jongsuk Ro; Waqar Afzal; Mujahid Abbas; Najla M. Aloraini; Jongsuk Ro

doi:10.3934/math.2025626

AIMS Mathematics

2025, Volume 10, Issue 6: 13908-13940. doi: 10.3934/math.2025626

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Resolution of open problems via Orlicz-Zygmund spaces and new geometric properties of Morrey spaces in the Besov sense with non-standard growth

1.
Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
2.
Department of Mechanical Engineering Sciences, Faculty of Engineering and the Built Environment, Doornfontein Campus, University of Johannesburg, South Africa
3.
Department of Medical Research, China Medical University, Taichung 406040, Taiwan, China
4.
Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia
5.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
6.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea

Received: 14 February 2025 Revised: 08 May 2025 Accepted: 13 May 2025 Published: 17 June 2025
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

In this paper, we introduced a novel norm structure and corresponding modular for Orlicz-Zygmund spaces, designed to capture summability and integrability properties under non-standard growth conditions. Moreover, we established the Hermite-Hadamard inequalities and gave a positive answer to the open problem 3.11 of Almeida and Hästö in their article (Besov spaces with variable smoothness and integrability, J. Funct. Anal., 258 (2010), 1628–1655), since the Hermite-Hadamard inequalities improve triangular-type inequalities. In addition, we explored some new structural properties of Besov-type Morrey spaces of summability and integrability. Furthermore, we addressed an open problem concerning the compactness of Morrey-type spaces characterized by summability and integrability properties. This problem was originally posed by Peter Hästö during the conference "Nonstandard Growth Phenomena", held in Turku, Finland, from August 29 to 31, 2017.
- Hermite-Hadamard inequalities,
- Orlicz-Zygmund spaces,
- Besov-Morrey spaces,
- mathematical operators,
- completeness,
- separability,
- compactness
Citation: Waqar Afzal, Mujahid Abbas, Najla M. Aloraini, Jongsuk Ro. Resolution of open problems via Orlicz-Zygmund spaces and new geometric properties of Morrey spaces in the Besov sense with non-standard growth[J]. AIMS Mathematics, 2025, 10(6): 13908-13940. doi: 10.3934/math.2025626

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Abstract

In this paper, we introduced a novel norm structure and corresponding modular for Orlicz-Zygmund spaces, designed to capture summability and integrability properties under non-standard growth conditions. Moreover, we established the Hermite-Hadamard inequalities and gave a positive answer to the open problem 3.11 of Almeida and Hästö in their article (Besov spaces with variable smoothness and integrability, J. Funct. Anal., 258 (2010), 1628–1655), since the Hermite-Hadamard inequalities improve triangular-type inequalities. In addition, we explored some new structural properties of Besov-type Morrey spaces of summability and integrability. Furthermore, we addressed an open problem concerning the compactness of Morrey-type spaces characterized by summability and integrability properties. This problem was originally posed by Peter Hästö during the conference "Nonstandard Growth Phenomena", held in Turku, Finland, from August 29 to 31, 2017.

References

[1]	Y. Xiao, Y. L. Yang, D. Ye, J. Q. Zhang, Quantitative precision second-order temporal transformation-based pose control for spacecraft proximity operations, IEEE Trans. Aerosp. Electron. Syst., 61 (2025), 1931–1941. https://doi.org/10.1109/TAES.2024.3469167 doi: 10.1109/TAES.2024.3469167
[2]	C. Ma, R. F. Mu, M. M. Li, J. L. He, C. L. Hua, L. Wang, et al., A multi-scale spatial-temporal interaction fusion network for digital twin-based thermal error compensation in precision machine tools, Expert Syst. Appl., 286 (2025), 127812. https://doi.org/10.1016/j.eswa.2025.127812 doi: 10.1016/j.eswa.2025.127812
[3]	S. S. Dragomir, Advances in inequalities of the Schwarz, Grüss and Bessel type in inner product spaces, New York: Nova Science Publishers Inc., 2005.
[4]	A. A. H. Ahmadini, W. Afzal, M. Abbas, E. S. Aly, Weighted Fejér, Hermite-Hadamard, and trapezium-type inequalities for $(h_1, h_2)$-Godunova-Levin preinvex function with applications and two open problems, Mathematics, 12 (2024), 1–28. https://doi.org/10.3390/math12030382 doi: 10.3390/math12030382
[5]	Z. A. Khan, W. Afzal, W. Nazeer, J. K. K. Asamoah, Some new variants of Hermite-Hadamard and Fejér-type inequalities for Godunova-Levin preinvex class of interval‐valued functions, J. Math., 2024 (2024), 8814585. http://dx.doi.org/10.1155/2024/8814585 doi: 10.1155/2024/8814585
[6]	A. Fahad, Z. Ali, S. Furuichi, S. I. Butt, Ayesha, Y. H. Wang, New inequalities for GA-h convex functions via generalized fractional integral operators with applications to entropy and mean inequalities, Fractal Fract., 8 (2024), 1–21. https://doi.org/10.3390/fractalfract8120728 doi: 10.3390/fractalfract8120728
[7]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
[8]	E. Nwaeze, Set inclusions of the Hermite-Hadamard type for $m$ polynomial harmonically convex interval valued functions, Constr. Math. Anal., 4 (2021), 260–273. http://dx.doi.org/10.33205/cma.793456 doi: 10.33205/cma.793456
[9]	T. S. Du, Y. Peng, Hermite-Hadamard type inequalities for multiplicative Riemann-Liouville fractional integrals, J. Comput. Appl. Math., 440 (2024), 115582. https://doi.org/10.1016/j.cam.2023.115582 doi: 10.1016/j.cam.2023.115582
[10]	J. L. Cardoso, E. M. Shehata, Hermite-Hadamard inequalities for quantum integrals: a unified approach, Appl. Math. Comput., 463 (2024), 128345. https://doi.org/10.1016/j.amc.2023.128345 doi: 10.1016/j.amc.2023.128345
[11]	W. Afzal, M. Abbas, W. Hamali, A. M. Mahnashi, M. D. L. Sen, Hermite-Hadamard-type inequalities via Caputo-Fabrizio fractional integral for $h$-Godunova-Levin and $(h_1, h_2)$-convex functions, Fractal Fract., 7 (2023), 1–19. http://dx.doi.org/10.3390/fractalfract7090687 doi: 10.3390/fractalfract7090687
[12]	W. Afzal, E. Y. Prosviryakov, S. M. El-Deeb, Y. Almalki, Some new estimates of Hermite-Hadamard, Ostrowski and Jensen-type inclusions for $h$-convex stochastic process via interval-valued functions, Symmetry, 15 (2023), 1–17. https://doi.org/10.3390/sym15040831 doi: 10.3390/sym15040831
[13]	L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311–1328. https://doi.org/10.1016/j.na.2008.12.005 doi: 10.1016/j.na.2008.12.005
[14]	D. F. Zhao, M. A. Ali, A. Kashuri, H. Budak, M. Z. Sarikaya, Hermite-Hadamard-type inequalities for the interval-valued approximately $h$-convex functions via generalized fractional integrals, J. Inequal. Appl., 2020 (2020), 1–38. https://doi.org/10.1186/s13660-020-02488-5 doi: 10.1186/s13660-020-02488-5
[15]	S. S. Dragomir, Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Proyecciones, 34 (2015), 323–341. https://doi.org/10.4067/S0716-09172015000400002 doi: 10.4067/S0716-09172015000400002
[16]	R. Niu, H. T. Zheng, B. L. Zhang, Navier-Stokes equations with variable viscosity in variable exponent spaces of Clifford-valued functions, Bound. Value Probl., 2015 (2015), 1–17. https://doi.org/10.1186/s13661-015-0291-y doi: 10.1186/s13661-015-0291-y
[17]	F. Li, Z. B. Li, L. Pi, Variable exponent functionals in image restoration, Appl. Math. Comput., 216 (2010), 870–882. https://doi.org/10.1016/j.amc.2010.01.094 doi: 10.1016/j.amc.2010.01.094
[18]	D. Y. Zheng, X. J. Cao, Provably efficient service function chain embedding and protection in edge networks, IEEE Trans. Netw., 33 (2025), 178–193. http://dx.doi.org/10.1109/TNET.2024.3475248 doi: 10.1109/TNET.2024.3475248
[19]	Z. W. Chen, W. H. Zhou, H. L. Kuang, Z. G. Chen, J. Z. Yang, Z. H. Chen, et al., Dynamic model and vibration of rack vehicle on curve line, Veh. Syst. Dyn., 2025, 1–19. https://doi.org/10.1080/00423114.2025.2494835
[20]	W. Orlicz, Über konjugierte exponentenfolgen, Stud. Math., 3 (1931), 200–211.
[21]	M. Izuki, E. Nakai, Y. Sawano, Function spaces with variable exponents–an introduction, Sci. Math. Japon., 77 (2014), 187–315. https://doi.org/10.32219/isms.77.2_187 doi: 10.32219/isms.77.2_187
[22]	V. Rabinovich, S. Samko, Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces, Integral Equ. Operator Theory, 60 (2008), 507–537. https://doi.org/10.1007/s00020-008-1566-9 doi: 10.1007/s00020-008-1566-9
[23]	D. Anick, B. Gray, Small $H$ spaces related to Morre spaces, Topology, 34 (1995), 859–881. http://dx.doi.org/10.1016/0040-9383(95)00001-1 doi: 10.1016/0040-9383(95)00001-1
[24]	Ö. Kulak, The inclusion theorems for variable exponent Lorentz spaces, Turkish J. Math., 40 (2016), 605–619. http://dx.doi.org/10.3906/mat-1502-23 doi: 10.3906/mat-1502-23
[25]	Z. W. Hao, Y. Jiao, Fractional integral on martingale Hardy spaces with variable exponents, Fract. Calc. Appl. Anal., 18 (2015), 1128–1145. https://doi.org/10.1515/fca-2015-0065 doi: 10.1515/fca-2015-0065
[26]	C. J. Wang, On a maximum principle for Bergman spaces with small exponents, Integral Equ. Operator Theory, 59 (2007), 597–601. https://doi.org/10.1007/s00020-007-1539-4 doi: 10.1007/s00020-007-1539-4
[27]	S. Baasandorj, S. S. Byun, H. S. Lee, Gradient estimates for Orlicz double phase problems with variable exponents, Nonlinear Anal., 221 (2022), 112891. https://doi.org/10.1016/j.na.2022.112891 doi: 10.1016/j.na.2022.112891
[28]	V. Kokilashvili, A. Meskhi, Maximal and Calderón-Zygmund operators in grand variable exponent Lebesgue spaces, Georgian Math. J., 21 (2014), 447–461. http://dx.doi.org/10.1515/gmj-2014-0047 doi: 10.1515/gmj-2014-0047
[29]	H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal., 18 (1975), 115–131. https://doi.org/10.1016/0022-1236(75)90020-8 doi: 10.1016/0022-1236(75)90020-8
[30]	R. Denk, M. Hieber, J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, American Mathematical Society, 2003. http://dx.doi.org/10.1090/memo/0788
[31]	X. L. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395–1412. https://doi.org/10.1016/j.jmaa.2007.08.003 doi: 10.1016/j.jmaa.2007.08.003
[32]	M. Mihăilescu, V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929–2937. http://dx.doi.org/10.1090/S0002-9939-07-08815-6 doi: 10.1090/S0002-9939-07-08815-6
[33]	N. Samko, Weighted Hardy and singular operators in Morrey spaces, J. Math. Anal. Appl., 350 (2009), 56–72. http://dx.doi.org/10.1016/j.jmaa.2008.09.021 doi: 10.1016/j.jmaa.2008.09.021
[34]	H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188–211. https://doi.org/10.1137/080732870 doi: 10.1137/080732870
[35]	V. S. Guliyev, J. J. Hasanov, S. G. Samko, Boundedness of maximal, potential type, and singular integral operators in the generalized variable exponent Morrey type spaces, J. Math. Sci., 170 (2010), 423–443. https://doi.org/10.1007/s10958-010-0095-7 doi: 10.1007/s10958-010-0095-7
[36]	C. Unal, I. Aydın, Compact embeddings of weighted variable exponent Sobolev spaces and existence of solutions for weighted $p(\cdot)$-Laplacian, Complex Var. Elliptic Equ., 66 (2021), 1755–1773. http://dx.doi.org/10.1080/17476933.2020.1781831 doi: 10.1080/17476933.2020.1781831
[37]	A. Dorantes-Aldama, D. Shakhmatov, Completeness and compactness properties in metric spaces, topological groups and function spaces, Topol. Appl., 226 (2017), 134–164. https://doi.org/10.1016/j.topol.2017.04.012 doi: 10.1016/j.topol.2017.04.012
[38]	L. Holá, L. Zsilinszky, Completeness and related properties of the graph topology on function spaces, 2013, arXiv: 1304.6628.
[39]	S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and compactifications, In: Mathematics of fuzzy sets, Boston: Springer, 1999,481–552. https://doi.org/10.1007/978-1-4615-5079-2_9
[40]	J. M. Almira, U. Luther, Compactness and generalized approximation spaces, Numer. Funct. Anal. Optim., 23 (2002), 1–38. https://doi.org/10.1081/NFA-120003668 doi: 10.1081/NFA-120003668
[41]	M. J. Carro, J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal., 112 (1993), 480–494. https://doi.org/10.1006/jfan.1993.1042 doi: 10.1006/jfan.1993.1042
[42]	R. A. Bandaliyev, P. Górka, V. S. Guliyev, Y. Sawano, Relatively compact sets in variable exponent Morrey spaces on metric spaces, Mediterr. J. Math., 18 (2021), 1–23. https://doi.org/10.1007/s00009-021-01828-z doi: 10.1007/s00009-021-01828-z
[43]	J. Xu, Precompact sets in Bochner-Lebesgue spaces with variable exponent, Math. Notes, 110 (2021), 932–941. http://dx.doi.org/10.1134/S0001434621110298 doi: 10.1134/S0001434621110298
[44]	D. D. Haroske, S. D. Moura, L. Skrzypczak, Some embeddings of Morrey spaces with critical smoothness, J. Fourier Anal. Appl., 26 (2020), 1–31. https://doi.org/10.1007/s00041-020-09758-2 doi: 10.1007/s00041-020-09758-2
[45]	K. Stempak, X. X. Tao, Local Morrey and Campanato spaces on quasimetric measure spaces, J. Funct. Spaces, 2014 (2014), 172486. https://doi.org/10.1155/2014/172486 doi: 10.1155/2014/172486
[46]	R. S. Puri, D. Morrey, A Krylov-Arnoldi reduced order modelling framework for efficient, fully coupled, structural-acoustic optimization, Struct. Multidiscip. Optim., 43 (2011), 495–517. https://doi.org/10.1007/s00158-010-0588-5 doi: 10.1007/s00158-010-0588-5
[47]	A. L. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355 (2003), 1297–1364. https://doi.org/10.1090/S0002-9947-02-03214-2 doi: 10.1090/S0002-9947-02-03214-2
[48]	G. D. Fazio, D. K. Palagachev, M. A. Ragusa, Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal., 166 (1999), 179–196. https://doi.org/10.1006/jfan.1999.3425 doi: 10.1006/jfan.1999.3425
[49]	J. Lang, V. Musil, M. Olšák, L. Pick, Maximal non-compactness of Sobolev embeddings, J. Geom. Anal., 31 (2021), 9406–9431. http://dx.doi.org/10.1007/s12220-020-00522-y doi: 10.1007/s12220-020-00522-y
[50]	Y. Komori, S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr., 282 (2009), 219–231. https://doi.org/10.1002/mana.200610733 doi: 10.1002/mana.200610733
[51]	X. X. Xia, J. Zhou, Homogeneous grand mixed Herz-Morrey spaces and their applications, Axioms, 13 (2024), 1–14. https://doi.org/10.3390/axioms13100713 doi: 10.3390/axioms13100713
[52]	M. J. Beltrán-Meneu, J. Bonet, E. Jordá, Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms, Banach J. Math. Anal., 19 (2025), 10. https://doi.org/10.1007/s43037-024-00395-1 doi: 10.1007/s43037-024-00395-1
[53]	C. Bellavita, V. Daskalogiannis, S. Miihkinen, D. Norrbo, G. Stylogiannis, J. Virtanen, Generalized Hilbert matrix operators acting on Bergman spaces, J. Funct. Anal., 288 (2025), 110856. https://doi.org/10.1016/j.jfa.2025.110856 doi: 10.1016/j.jfa.2025.110856
[54]	N. Apollonio, Cantelli's bounds for generalized tail inequalities, Axioms, 14 (2025), 1–18. https://doi.org/10.3390/axioms14010043 doi: 10.3390/axioms14010043
[55]	A. Almeida, P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal., 258 (2010), 1628–1655. https://doi.org/10.1016/j.jfa.2009.09.012 doi: 10.1016/j.jfa.2009.09.012
[56]	A. Almeida, A. Caetano, Variable exponent Besov-Morrey spaces, J. Fourier Anal. Appl., 26 (2020), 5. https://doi.org/10.1007/s00041-019-09711-y doi: 10.1007/s00041-019-09711-y
[57]	G. A. Chacón, G. R. Chacón, Variable exponent spaces of analytic functions, In: Advances in complex analysis and applications, IntechOpen, 2020. https://doi.org/10.5772/intechopen.92617
[58]	K. Saibi, Variable Besov-Morrey spaces associated with operators, Mathematics, 11 (2023), 1–22. https://doi.org/10.3390/math11092038 doi: 10.3390/math11092038
[59]	A. Almeida, J. Hasanov, S. Samko, Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J., 15 (2008), 195–208. http://dx.doi.org/10.1515/GMJ.2008.195 doi: 10.1515/GMJ.2008.195
[60]	V. Kokilashvili, A. Meskhi, Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure, Complex Var. Elliptic Equ., 55 (2010), 923–936. https://doi.org/10.1080/17476930903276068 doi: 10.1080/17476930903276068
[61]	T. Ohno, Continuity properties for logarithmic potentials of functions in Morrey spaces of variable exponent, Hiroshima Math. J., 38 (2008), 363–383. https://doi.org/10.32917/hmj/1233152775 doi: 10.32917/hmj/1233152775
[62]	V. S. Guliyev, J. J. Hasanov, S. G. Samko, Boundedness of maximal, potential type, and singular integral operators in the generalized variable exponent Morrey type spaces, J. Math. Sci., 170 (2010), 423–443. http://dx.doi.org/10.1007/s10958-010-0095-7 doi: 10.1007/s10958-010-0095-7
[63]	P. Hästö, Local-to-global results in variable exponent spaces, Math. Res. Lett., 16 (2009), 263–278.
[64]	V. S. Guliyev, S. G. Samko, Maximal, potential and singular operators in the generalized variable exponent Morrey spaces on unbounded sets, J. Math. Sci., 193 (2013), 228–248. https://doi.org/10.1007/s10958-013-1449-8 doi: 10.1007/s10958-013-1449-8
[65]	W. Afzal, M. Abbas, O. M. Alsalami, Bounds of different integral operators in tensorial Hilbert and variable exponent function spaces, Mathematics, 12 (2024), 1–33. https://doi.org/10.3390/math12162464 doi: 10.3390/math12162464
[66]	P. Harjulehto, P. Hästö, R. Klén, Generalized Orlicz spaces and related PDE, Nonlinear Anal., 143 (2016), 155–173. http://dx.doi.org/10.1016/j.na.2016.05.002 doi: 10.1016/j.na.2016.05.002
[67]	T. Iwaniec, A. Verde, On the operator $L(f) = f \log \|f\|$, J. Funct. Anal., 169 (1999), 391–420. https://doi.org/10.1006/jfan.1999.3443 doi: 10.1006/jfan.1999.3443
[68]	T. Iwaniec, G. Martin, Geometric function theory and non-linear analysis, Clarendon Press, 2001. https://doi.org/10.1093/oso/9780198509295.001.0001
[69]	D. R. Sahu, D. O'Regan, R. P. Agarwal, Fixed point theory for Lipschitzian-type mappings with applications, New York: Springer, 2009. https://doi.org/10.1007/978-0-387-75818-3
[70]	D. V. Cruz-Uribe, A. Fiorenza, Variable Lebesgue spaces, Basel: Birkhäuser, 2013. https://doi.org/10.1007/978-3-0348-0548-3
[71]	R. Bandaliyev, P. Górka, Relatively compact sets in variable-exponent Lebesgue spaces, Banach J. Math. Anal., 12 (2018), 331–346. http://dx.doi.org/10.1215/17358787-2017-0039 doi: 10.1215/17358787-2017-0039

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