Sobolev-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form. (English) Zbl 1511.46019
Summary: We give Sobolev-type inequalities for variable Riesz potentials \(I_{\alpha (\cdot)}f\) of functions in Musielak-Orlicz-Morrey spaces of an integral form \(\mathcal{L}^{\Phi,\omega}(G)\). As a corollary, we give Sobolev-type inequalities on \(\mathcal{L}^{\Phi,\omega}(G)\) for double phase functions \(\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}\).
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
Keywords:
Riesz potentials; maximal functions; Sobolev’s inequality; Musielak-Orlicz-Morrey spaces; double phase functionsReferences:
| [1] | Ahmida, Y.; Chlebicka, I.; Gwiazda, P.; Youssfi, A., Gossez��s approximation theorems in Musielak-Orlicz-Sobolev spaces, J. Funct. Anal., 275, 9, 2538-2571 (2018) · Zbl 1405.42042 · doi:10.1016/j.jfa.2018.05.015 |
| [2] | Almeida, A.; Hasanov, J.; Samko, S., Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J., 15, 195-208 (2008) · Zbl 1263.42002 · doi:10.1515/GMJ.2008.195 |
| [3] | Baroni, P.; Colombo, M.; Mingione, G., Regularity for general functionals with double phase, Calc. Var. Partial Differ. Equs., 57, 2, 62 (2018) · Zbl 1394.49034 · doi:10.1007/s00526-018-1332-z |
| [4] | Baroni, P.; Colombo, M.; Mingione, G., Non-autonomous functionals, borderline cases and related function classes, St Petersburg Math. J., 27, 347-379 (2016) · Zbl 1335.49057 · doi:10.1090/spmj/1392 |
| [5] | Burenkov, VI; Guliyev, HV, Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces, Studia Math., 163, 2, 157-176 (2004) · Zbl 1044.42015 · doi:10.4064/sm163-2-4 |
| [6] | Byun, SS; Lee, HS, Calderón-Zygmund estimates for elliptic double phase problems with variable exponents, J. Math. Anal. Appl., 501 (2021) · Zbl 1467.35064 · doi:10.1016/j.jmaa.2020.124015 |
| [7] | Capone, C.; Cruz-Uribe, D.; Fiorenza, A., The fractional maximal operator and fractional integrals on variable \(L^p\) spaces, Rev. Mat. Iberoamericana, 23, 3, 743-770 (2007) · Zbl 1213.42063 · doi:10.4171/RMI/511 |
| [8] | Colombo, M.; Mingione, G., Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215, 2, 443-496 (2015) · Zbl 1322.49065 · doi:10.1007/s00205-014-0785-2 |
| [9] | Diening, L., Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \(L^{p(\cdot )}\) and \(W^{k, p(\cdot )}\), Math. Nachr., 263, 1, 31-43 (2004) · Zbl 1065.46024 · doi:10.1002/mana.200310157 |
| [10] | De Filippis, C.; Mingione, G., Lipschitz bounds and nonautonomous integrals, Arch. Ration. Mech. Anal., 242, 973-1057 (2021) · Zbl 1483.49050 · doi:10.1007/s00205-021-01698-5 |
| [11] | Futamura, T.; Mizuta, Y.; Shimomura, T., Sobolev embeddings for Riesz potential space of variable exponent, Math. Nachr., 279, 13-14, 1463-1473 (2006) · Zbl 1109.31004 · doi:10.1002/mana.200410432 |
| [12] | Futamura, T.; Mizuta, Y.; Shimomura, T., Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent, J. Math. Anal. Appl., 366, 391-417 (2010) · Zbl 1193.46016 · doi:10.1016/j.jmaa.2010.01.053 |
| [13] | Guliyev, VS; Hasanov, J.; Samko, S., Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces, Math. Scand., 107, 285-304 (2010) · Zbl 1213.42077 · doi:10.7146/math.scand.a-15156 |
| [14] | Guliyev, VS; Hasanov, J.; Samko, S., Boundedness of the maximal potential and Singular integral operators in the generalized variable exponent Morrey type spaces, J. Math. Sci., 170, 4, 423-443 (2010) · Zbl 1307.46018 · doi:10.1007/s10958-010-0095-7 |
| [15] | Harjulehto, P., Hästö, P.: Orlicz spaces and generalized Orlicz spaces. Lecture Notes in Mathematics, vol. 2236. Springer, Cham (2019) · Zbl 1436.46002 |
| [16] | Hästö, P.; Ok, J., Maximal regularity for local minimizers of non-autonomous functionals, J. Eur. Math. Soc., 24, 4, 1285-1334 (2022) · Zbl 1485.49044 · doi:10.4171/JEMS/1118 |
| [17] | Hedberg, LI, On certain convolution inequalities, Proc. Amer. Math. Soc., 36, 505-510 (1972) · Zbl 0283.26003 · doi:10.1090/S0002-9939-1972-0312232-4 |
| [18] | Krasnoesl’skii, MA; Rutickii, YaB, Convex Functions and Orlicz Spaces (1961), Groningen: P. Noordhoff Ltd., Groningen · Zbl 0095.09103 |
| [19] | Maeda, F-Y; Mizuta, Y.; Ohno, T.; Shimomura, T., Boundedness of maximal operators and Sobolev’s inequality on Musielak-Orlicz-Morrey spaces, Bull. Sci. Math., 137, 76-96 (2013) · Zbl 1267.46045 · doi:10.1016/j.bulsci.2012.03.008 |
| [20] | Maeda, F.-Y., Mizuta Y., Ohno T., Shimomura T.: Sobolev’s inequality for double phase functionals with variable exponents, Forum Math. 31 517-527 (2019) · Zbl 1423.46049 |
| [21] | Maeda, F-Y; Ohno, T.; Shimomura, T., Boundedness of the maximal operator on Musielak-Orlicz-Morrey spaces, Tohoku Math. J., 69, 483-495 (2017) · Zbl 1387.42017 · doi:10.2748/tmj/1512183626 |
| [22] | Mizuta, Y.; Nakai, E.; Ohno, T.; Shimomura, T., Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent, Complex Vari. Elliptic Equ., 56, 7-9, 671-695 (2011) · Zbl 1228.31004 · doi:10.1080/17476933.2010.504837 |
| [23] | Mizuta, Y.; Nakai, E.; Ohno, T.; Shimomura, T., Maximal functions, Riesz potentials and Sobolev embeddings on Musielak-Orlicz-Morrey spaces of variable exponent in \(\textbf{R}^n\), Rev. Mat. Complut., 25, 2, 413-434 (2012) · Zbl 1273.31005 · doi:10.1007/s13163-011-0074-7 |
| [24] | Mizuta Y., Nakai E., Ohno T., Shimomura T.: Campanato-Morrey spaces for the double phase functionals with variable exponents, Nonlinear Anal. 197, 111827, 19 (2020) · Zbl 1441.31004 |
| [25] | Mizuta, Y.; Ohno, T.; Shimomura, T., Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space \(L^{p(\cdot )}(\log L)^{q(\cdot )}\), J. Math. Anal. Appl., 345, 70-85 (2008) · Zbl 1153.31002 · doi:10.1016/j.jmaa.2008.03.067 |
| [26] | Mizuta, Y.; Ohno, T.; Shimomura, T., Sobolev inequalities for Musielak-Orlicz spaces, Manuscripta Math., 155, 209-227 (2018) · Zbl 1392.46037 · doi:10.1007/s00229-017-0944-5 |
| [27] | Mizuta, Y.; Ohno, T.; Shimomura, T., Sobolev’s theorem for double phase functionals, Math. Ineq. Appl., 23, 17-33 (2020) · Zbl 1453.46021 |
| [28] | Mizuta, Y.; Shimomura, T., Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent, J. Math. Soc. Japan, 60, 583-602 (2008) · Zbl 1161.46305 · doi:10.2969/jmsj/06020583 |
| [29] | Mizuta, Y.; Shimomura, T., Sobolev’s inequality for Riesz potentials of functions in Morrey spaces of integral form, Math. Nachr., 283, 9, 1336-1352 (2010) · Zbl 1211.46026 · doi:10.1002/mana.200710122 |
| [30] | Mizuta, Y., Shimomura, T.: Hardy-Sobolev inequalities in the unit ball for double phase functionals, J. Math. Anal. Appl. 501 124133, 17 (2021) · Zbl 1478.46037 |
| [31] | Ohno, T.; Shimomura, T., Sobolev’s inequality for Musielak-Orlicz-Morrey spaces over metric measure spaces, J. Aust. Math. Soc., 110, 371-385 (2021) · Zbl 1477.46045 · doi:10.1017/S144678871900034X |
| [32] | Ohno, T.; Shimomura, T., Sobolev-type inequalities on variable exponent Morrey spaces of an integral form, Ricerche Mat., 71, 189-204 (2022) · Zbl 1501.46029 · doi:10.1007/s11587-021-00635-8 |
| [33] | Ragusa, MA; Tachikawa, A., Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9, 1, 710-728 (2020) · Zbl 1420.35145 · doi:10.1515/anona-2020-0022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
