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Keywords:
maximal function; Riesz potential; Musielak-Orlicz space; Sobolev's inequality; metric measure space; non-doubling measure; multi-phase functional
maximal function; Riesz potential; Musielak-Orlicz space; Sobolev's inequality; metric measure space; non-doubling measure; multi-phase functional
Summary:
We are concerned with the boundedness of modified Hardy-Littlewood maximal operator $M_{\lambda }$ and Sobolev inequalities for the variable Riesz potentials $I_{\alpha (\cdot ),\tau }f$ on Musielak-Orlicz spaces $L^{\Phi }(X)$ over unbounded metric measure spaces, as an improvement of our recent paper, see T. Ohno, T. Shimomura (2025a). As an application, we give the boundedness of $M_{\lambda }$ and Sobolev inequalities for $I_{\alpha (\cdot ),\tau }f$ for the multi-phase functionals with variable exponents $$ \Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}+ b(x) t^{s(x)}, \quad x \in X, \ t \ge 0 , $$ where $p({\cdot })$, $q({\cdot })$, and $s({\cdot })$ are log-Hölder continuous, $p(x)<q(x) \le s(x)$ for $x\in X$, and $a({\cdot })$, and $b({\cdot })$ are nonnegative, bounded, and Hölder continuous.
References:
[1] Adamowicz, T., Harjulehto, P., Hästö, P.: Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. Math. Scand. 116 (2015), 5-22. DOI 10.7146/math.scand.a-20448 | MR 3322604 | Zbl 1316.42018
[2] 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. DOI 10.1007/s00526-018-1332-z | MR 3775180 | Zbl 1394.49034
[3] Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics 17. EMS, Zürich (2011). DOI 10.4171/099 | MR 2867756 | Zbl 1231.31001
[4] Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215 (2015), 443-496. DOI 10.1007/s00205-014-0785-2 | MR 3294408 | Zbl 1322.49065
[5] Cruz-Uribe, D. V., Shukla, P.: The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type. Stud. Math. 242 (2018), 109-139. DOI 10.4064/sm8556-6-2017 | MR 3778907 | Zbl 1397.42009
[6] Filippis, C. De, Oh, J.: Regularity for multi-phase variational problems. J. Differ. Equations 267 (2019), 1631-1670. DOI 10.1016/j.jde.2019.02.015 | MR 3945612 | Zbl 1422.49037
[7] Gogatishvili, A., Kokilashvili, V.: Criteria of strong type two-weighted inequalities for fractional maximal functions. Georgian Math. J. 3 (1996), 423-446. DOI 10.1007/BF02259772 | MR 1407384 | Zbl 0884.42015
[8] Hajłasz, P., Koskela, P.: Sobolev Met Poincaré. Memoirs of the American Mathematical Society 688. AMS, Providence (2000). DOI 10.1090/memo/0688 | MR 1683160 | Zbl 0954.46022
[9] Harjulehto, P., Hästö, P., Latvala, V.: Sobolev embeddings in metric measure spaces with variable dimension. Math. Z. 254 (2006), 591-609. DOI 10.1007/s00209-006-0960-8 | MR 2244368 | Zbl 1109.46037
[10] Hashimoto, D., Sawano, Y., Shimomura, T.: Gagliardo-Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces over quasi-metric measure spaces. Colloq. Math. 161 (2020), 51-66. DOI 10.4064/cm7535-4-2019 | MR 4085112 | Zbl 1464.46043
[11] Hedberg, L. I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36 (1972), 505-510. DOI 10.1090/S0002-9939-1972-0312232-4 | MR 0312232 | Zbl 0283.26003
[12] Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001). DOI 10.1007/978-1-4613-0131-8 | MR 1800917 | Zbl 0985.46008
[13] Kairema, A.: Sharp weighted bounds for fractional integral operators in a space of homogeneous type. Math. Scand. 114 (2014), 226-253. DOI 10.7146/math.scand.a-17109 | MR 3206387 | Zbl 1302.47075
[14] 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. DOI 10.1016/j.bulsci.2012.03.008 | MR 3007101 | Zbl 1267.46045
[15] 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. DOI 10.1515/forum-2018-0077 | MR 3918454 | Zbl 1423.46049
[16] Maeda, F.-Y., Ohno, T., Shimomura, T.: Boundedness of the maximal operator on Musielak-Orlicz-Morrey spaces. Tohoku Math. J. 69 (2017), 483-495. DOI 10.2748/tmj/1512183626 | MR 3732884 | Zbl 1387.42017
[17] Mizuta, Y., Shimomura, T., Sobukawa, T.: Sobolev's inequality for Riesz potentials of functions in non-doubling Morrey spaces. Osaka J. Math. 46 (2009), 255-271. MR 2531149 | Zbl 1186.31003
[18] Nazarov, F., Treil, S., Volberg, A.: Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 1997 (1997), 703-726. DOI 10.1155/S1073792897000469 | MR 1470373 | Zbl 0889.42013
[19] Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 1998 (1998), 463-487. DOI 10.1155/S1073792898000312 | MR 1626935 | Zbl 0918.42009
[20] Ohno, T., Shimomura, T.: Musielak-Orlicz-Sobolev spaces on metric measure spaces. Czech. Math. J. 65 (2015), 435-474. DOI 10.1007/s10587-015-0187-0 | MR 3360438 | Zbl 1363.46027
[21] Ohno, T., Shimomura, T.: Sobolev inequalities for Riesz potentials of functions in $L^{p(\cdot)}$ over nondoubling measure spaces. Bull. Aust. Math. Soc. 93 (2016), 128-136. DOI 10.1017/S0004972715001331 | MR 3436021 | Zbl 1354.46036
[22] Ohno, T., Shimomura, T.: Maximal and Riesz potential operators on Musielak-Orlicz spaces over metric measure spaces. Integral Equations Oper. Theory 90 (2018), Article ID 62, 18 pages. DOI 10.1007/s00020-018-2484-0 | MR 3856222 | Zbl 1401.42020
[23] Ohno, T., Shimomura, T.: Trudinger-type inequalities for variable Riesz potentials of functions in Musielak-Orlicz-Morrey spaces over metric measure spaces. Math. Nachr. 297 (2024), 1248-1274. DOI 10.1002/mana.202300265 | MR 4734972 | Zbl 1550.46032
[24] Ohno, T., Shimomura, T.: Boundedness of maximal operators and Sobolev inequalities on Musielak-Orlicz spaces over unbounded metric measure spaces. Bull. Sci. Math. 199 (2025), Article ID 103546, 47 pages. DOI 10.1016/j.bulsci.2024.103546 | MR 4837131 | Zbl 07991347
[25] Ohno, T., Shimomura, T.: Generalized Riesz potential operators on Musielak-Orlicz-Morrey spaces over unbounded metric measure spaces. Anal. Math. Phys. 15 (2025), Article ID 24, 47 pages. DOI 10.1007/s13324-025-01020-6 | MR 4859610 | Zbl 08013133
[26] Ragusa, M. A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9 (2020), 710-728. DOI 10.1515/anona-2020-0022 | MR 3985000 | Zbl 1420.35145
[27] Samko, N. G., Samko, S. G., Vakulov, B. G.: Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 335 (2007), 560-583. DOI 10.1016/j.jmaa.2007.01.091 | MR 2340340 | Zbl 1142.46018
[28] Sawano, Y.: Sharp estimates of the modified Hardy-Littlewood maximal operator on the nonhomogeneous space via covering lemmas. Hokkaido Math. J. 34 (2005), 435-458. DOI 10.14492/hokmj/1285766231 | MR 2159006 | Zbl 1088.42010
[29] Sawano, Y., Shigematsu, M., Shimomura, T.: Generalized Riesz potentials of functions in Morrey spaces $L^{(1,\varphi;\kappa)}(G)$ over non-doubling measure spaces. Forum Math. 32 (2020), 339-359. DOI 10.1515/forum-2019-0140 | MR 4069939 | Zbl 1436.42029
[30] Sawano, Y., Shimomura, T.: Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. Collect. Math. 64 (2013), 313-350. DOI 10.1007/s13348-013-0082-7 | MR 3084400 | Zbl 1280.31001
[31] Sawano, Y., Shimomura, T.: Maximal operator on Orlicz spaces of two variable exponents over unbounded quasi-metric measure spaces. Proc. Am. Math. Soc. 147 (2019), 2877-2885. DOI 10.1090/proc/14225 | MR 3973891 | Zbl 1416.42025
[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. DOI 10.1007/s00025-021-01490-7 | MR 4305494 | Zbl 1479.42055
[33] Stempak, K.: Examples of metric measure spaces related to modified Hardy-Littlewood maximal operators. Ann. Acad. Sci. Fenn., Math. 41 (2016), 313-314. DOI 10.5186/aasfm.2016.4119 | MR 3467713 | Zbl 1337.42023
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