Article
| Title: | Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces (English) |
| Author: | Mizuta, Yoshihiro |
| Author: | Shimomura, Tetsu |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 4 |
| Year: | 2023 |
| Pages: | 1201-1217 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_{\mathbb H,\nu }f$ and Riesz potentials $I_{\mathbb H,\alpha }f$ in weighted Morrey spaces of variable exponent on the half space $\mathbb H$. We also obtain Sobolev type inequalities for a $C^1$ function on $\mathbb H$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $\Phi (x,t) = t^{p(x)} + (b(x) t)^{q(x)}$, where $p(\cdot )$ and $q(\cdot )$ satisfy log-Hölder conditions, $p(x)<q(x)$ for $x \in {\mathbb H} $, and $b(\cdot )$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$. (English) |
| Keyword: | variable exponent |
| Keyword: | fractional maximal function |
| Keyword: | Riesz potential |
| Keyword: | Sobolev's inequality |
| Keyword: | weighted Morrey space |
| Keyword: | double phase functional |
| MSC: | 31B15 |
| MSC: | 42B25 |
| MSC: | 46E30 |
| DOI: | 10.21136/CMJ.2023.0442-22 |
| . | |
| Date available: | 2023-11-23T12:26:04Z |
| Last updated: | 2023-11-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151955 |
| . | |
| Reference: | [1] Adams, D. R.: A note on Riesz potentials.Duke Math. J. 42 (1975), 765-778. Zbl 0336.46038, MR 0458158, 10.1215/S0012-7094-75-04265-9 |
| Reference: | [2] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory.Grundlehren der Mathematischen Wissenschaften 314. Springer, Berlin (1995). Zbl 0834.46021, MR 1411441, 10.1007/978-3-662-03282-4 |
| Reference: | [3] Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces.Georgian Math. J. 15 (2008), 195-208. Zbl 1263.42002, MR 2428465, 10.1515/GMJ.2008.195 |
| Reference: | [4] Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes.St. Petersbg. Math. J. 27 (2016), 347-379. Zbl 1335.49057, MR 3570955, 10.1090/spmj/1392 |
| Reference: | [5] Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase.Calc. Var. Partial Differ. Equ. 57 (2018), Article ID 62, 48 pages. Zbl 1394.49034, MR 3775180, 10.1007/s00526-018-1332-z |
| Reference: | [6] Byun, S.-S., Lee, H.-S.: Calderón-Zygmund estimates for elliptic double phase problems with variable exponents.J. Math. Anal. Appl. 501 (2021), Article ID 124015, 31 pages. Zbl 1467.35064, MR 4258791, 10.1016/j.jmaa.2020.124015 |
| Reference: | [7] Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator and fractional integrals on variable $L^p$ spaces.Rev. Mat. Iberoam. 23 (2007), 743-770. Zbl 1213.42063, MR 2414490, 10.4171/RMI/511 |
| Reference: | [8] Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals.Arch. Ration. Mech. Anal. 218 (2015), 219-273. Zbl 1325.49042, MR 3360738, 10.1007/s00205-015-0859-9 |
| Reference: | [9] Colombo, M., Mingione, G.: Regularity for double phase variational problems.Arch. Ration. Mech. Anal. 215 (2015), 443-496. Zbl 1322.49065, MR 3294408, 10.1007/s00205-014-0785-2 |
| Reference: | [10] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis.Applied and Numerical Harmonic Analysis. Birkhäuser, New York (2013). Zbl 1268.46002, MR 3026953, 10.1007/978-3-0348-0548-3 |
| Reference: | [11] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C. J.: Weighted norm inequalities for the maximal operator on variable Lebesgue spaces.J. Math. Anal. Appl. 394 (2012), 744-760. Zbl 1298.42021, MR 2927495, 10.1016/j.jmaa.2012.04.044 |
| Reference: | [12] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents.Lecture Notes in Mathematics 2017. Springer, Berlin (2011). Zbl 1222.46002, MR 2790542, 10.1007/978-3-642-18363-8 |
| Reference: | [13] Fazio, G. Di, Ragusa, M. A.: Commutators and Morrey spaces.Boll. Unione Mat. Ital., VII. Ser., A 5 (1991), 323-332. Zbl 0761.42009, MR 1138545 |
| Reference: | [14] Haj{ł}asz, P., Koskela, P.: Sobolev Met Poincaré.Memoirs of the American Mathematical Society 688. AMS, Providence (2000). Zbl 0954.46022, MR 1683160, 10.1090/memo/0688 |
| Reference: | [15] Hästö, P., Ok, J.: Calderón-Zygmund estimates in generalized Orlicz spaces.J. Differ. Equations 267 (2019), 2792-2823. Zbl 1420.35087, MR 3953020, 10.1016/j.jde.2019.03.026 |
| Reference: | [16] Kinnunen, J., Lindqvist, P.: The derivative of the maximal function.J. Reine Angew. Math. 503 (1998), 161-167. Zbl 0904.42015, MR 1650343, 10.1515/crll.1998.095 |
| Reference: | [17] Kinnunen, J., Saksman, E.: Regularity of the fractional maximal function.Bull. Lond. Math. Soc. 35 (2003), 529-535. Zbl 1021.42009, MR 1979008, 10.1112/S0024609303002017 |
| Reference: | [18] 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 (2013), 76-96. Zbl 1267.46045, MR 3007101, 10.1016/j.bulsci.2012.03.008 |
| Reference: | [19] Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev's inequality for double phase functionals with variable exponents.Forum Math. 31 (2019), 517-527. Zbl 1423.46049, MR 3918454, 10.1515/forum-2018-0077 |
| Reference: | [20] Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponents.Complex Var. Elliptic Equ. 56 (2011), 671-695. Zbl 1228.31004, MR 2832209, 10.1080/17476933.2010.504837 |
| Reference: | [21] Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Maximal functions, Riesz potentials and Sobolev embeddings on Musielak-Orlicz-Morrey spaces of variable exponent in $\Bbb R^n$.Rev. Mat. Complut. 25 (2012), 413-434. Zbl 1273.31005, MR 2931419, 10.1007/s13163-011-0074-7 |
| Reference: | [22] Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Campanato-Morrey spaces for the double phase functionals with variable exponents.Nonlinear Anal., Theory Methods Appl., Ser. A 197 (2020), Article ID 111827, 18 pages. Zbl 1441.31004, MR 4073513, 10.1016/j.na.2020.111827 |
| Reference: | [23] Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev's inequalities for Herz-Morrey-Orlicz spaces on the half space.Math. Inequal. Appl. 21 (2018), 433-453. Zbl 1388.31009, MR 3776085, 10.7153/mia-2018-21-30 |
| Reference: | [24] Mizuta, Y., Ohno, T., Shimomura, T.: Boundedness of fractional maximal operators for double phase functionals with variable exponents.J. Math. Anal. Appl. 501 (2021), Article ID 124360, 16 pages. Zbl 1478.46028, MR 4258801, 10.1016/j.jmaa.2020.124360 |
| Reference: | [25] Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent.J. Math. Soc. Japan 60 (2008), 583-602. Zbl 1161.46305, MR 2421989, 10.2969/jmsj/06020583 |
| Reference: | [26] Mizuta, Y., Shimomura, T.: Hardy-Sobolev inequalities in the unit ball for double phase functionals.J. Math. Anal. Appl. 501 (2021), Article ID 124133, 17 pages. Zbl 1478.46037, MR 4258797, 10.1016/j.jmaa.2020.124133 |
| Reference: | [27] Mizuta, Y., Shimomura, T.: Sobolev type inequalities for fractional maximal functions and Green potentials in half spaces.Positivity 25 (2021), 1131-1146. Zbl 1481.46030, MR 4274309, 10.1007/s11117-021-00810-z |
| Reference: | [28] Mizuta, Y., Shimomura, T.: Boundedness of fractional integral operators in Herz spaces on the hyperplane.Math. Methods Appl. Sci. 45 (2022), 8631-8654. MR 4475228, 10.1002/mma.7425 |
| Reference: | [29] Mizuta, Y., Shimomura, T.: Sobolev type inequalities for fractional maximal functions and Riesz potentials in half spaces.Available at https://arxiv.org/abs/2305.13708 (2023), 22 pages. MR 4274309 |
| Reference: | [30] C. B. Morrey, Jr.: On the solutions of quasi-linear elliptic partial differential equations.Trans. Am. Math. Soc. 43 (1938), 126-166. Zbl 0018.40501, MR 1501936, 10.1090/S0002-9947-1938-1501936-8 |
| Reference: | [31] Ragusa, M. A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents.Adv. Nonlinear Anal. 9 (2020), 710-728. Zbl 1420.35145, MR 3985000, 10.1515/anona-2020-0022 |
| Reference: | [32] Sawano, Y., Shimomura, T.: Fractional maximal operator on Musielak-Orlicz spaces over unbounded quasi-metric measure spaces.Result. Math. 76 (2021), Article ID 188, 22 pages. Zbl 1479.42055, MR 4305494, 10.1007/s00025-021-01490-7 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
