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Sobolev’s Inequality for Riesz Potentials of Functions in Musielak–Orlicz–Morrey Spaces Over Non-doubling Metric Measure Spaces

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  06 September 2019

Takao Ohno  and
Tetsu Shimomura
Takao Ohno
Affiliation:
Faculty of Education, Oita University, Dannoharu Oita-city 870-1192, Japan Email: [email protected]
Tetsu Shimomura
Affiliation:
Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan Email: [email protected]
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Abstract

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Keywords

maximal functionRiesz potentialMusielak–Orlicz–Morrey spaceSobolev’s inequalitymetric measure spacenon-doubling measuredouble phase functional

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 287 - 303
Copyright
© Canadian Mathematical Society 2019

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References

Adams, D. R., A note on Riesz potentials. Duke Math. J. 42(1975), 765–778.Google Scholar
Adams, D. R. and Hedberg, L. I., Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften, 314, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/978-3-662-03282-4Google Scholar
Ahmida, Y., Chlebicka, I., Gwiazda, P., and Youssfi, A., Gossez’s approximation theorems in Musielak–Orlicz–Sobolev spaces. J. Funct. Anal. 275(2018), no. 9, 2538–2571. https://doi.org/10.1016/j.jfa.2018.05.015Google Scholar
Almeida, A., Hasanov, J., and Samko, S., Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15(2008), 195–208.Google Scholar
Baroni, P., Colombo, M., and Mingione, G., Non-autonomous functionals, borderline cases and related function classes. St. Petersburg Math. J. 27(2016), 347–379. https://doi.org/10.1090/spmj/1392Google Scholar
Baroni, P., Colombo, M., and Mingione, G., Regularity for general functionals with double phase. Calc. Var. Partial Differential Equations 57(2018), no. 2, Art. 62. https://doi.org/10.1007/s00526-018-1332-zGoogle Scholar
Björn, A. and Björn, J., Nonlinear potential theory on metric spaces. EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zurich, 2011. https://doi.org/10.4171/099Google Scholar
Chiarenza, F. and Frasca, M., Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. (7) 7(1987), 273–279.Google Scholar
Colasuonno, F. and Squassina, M., Eigenvalues for double phase variational integrals. Ann. Mat. Pura Appl. (4) 195(2016), no. 6, 1917–1959. https://doi.org/10.1007/s10231-015-0542-7Google Scholar
Colombo, M. and Mingione, G., Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215(2015), no. 2, 443–496. https://doi.org/10.1007/s00205-014-0785-2Google Scholar
Cruz-Uribe, D. and Shukla, P., The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type. Studia Math. 242(2018), no. 2, 109–139. https://doi.org/10.4064/sm8556-6-2017Google Scholar
Guliyev, V. S., Hasanov, J., and Samko, S., Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107(2010), 285–304. https://doi.org/10.7146/math.scand.a-15156Google Scholar
Guliyev, V. S., Hasanov, J., and Samko, S., Boundedness of the maximal, potential and singular integral operators in the generalized variable exponent Morrey type spaces. J. Math. Sci. 170(2010), no. 4, 423–443. https://doi.org/10.1007/s10958-010-0095-7Google Scholar
Hajłasz, P. and Koskela, P., Sobolev met Poincaré. Mem. Amer. Math. Soc. 145(2000), no. 688. https://doi.org/10.1090/memo/0688Google Scholar
Harjulehto, P. and Hasto, P., Boundary regularity under generalized growth conditions. Z. Anal. Anwend. 38(2019), no. 1, 73–96. https://doi.org/10.4171/ZAA/1628Google Scholar
Hästö, P., Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2009), no. 2, 263–278. https://doi.org/10.4310/MRL.2009.v16.n2.a5Google Scholar
Hästö, P., The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269(2015), no. 12, 4038–4048. Corrigendum to: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 271(2016), no. 1, 240–243. https://doi.org/10.1016/j.jfa.2016.04.005Google Scholar
Hedberg, L. I., On certain convolution inequalities. Proc. Amer. Math. Soc. 36(1972), 505–510. https://doi.org/10.2307/2039187Google Scholar
Maeda, F.-Y., Mizuta, Y., Ohno, T., and Shimomura, T., Boundedness of maximal operators and Sobolev’s inequality on Musielak–Orlicz–Morrey spaces. Bull. Sci. Math. 137(2013), 76–96. https://doi.org/10.1016/j.bulsci.2012.03.008Google Scholar
Maeda, F.-Y., Mizuta, Y., Ohno, T., and Shimomura, T., Sobolev’s inequality for double phase functionals. Forum. Math. 31(2019), no. 2, 517–527. https://doi.org/10.1515/forum-2018-0077Google Scholar
Maeda, F.-Y., Ohno, T., and Shimomura, T., Boundedness of the maximal operator on Musielak–Orlicz–Morrey spaces. Tohoku Math. J. 69(2017), no. 4, 483–495. https://doi.org/10.2748/tmj/1512183626Google Scholar
Maeda, F.-Y., Sawano, Y., and Shimomura, T., Some norm inequalities in Musielak–Orlicz spaces. Ann. Acad. Sci. Fenn. Math. 41(2016), 721–744. https://doi.org/10.5186/aasfm.2016.4148Google Scholar
Mizuta, Y., Nakai, E., Ohno, T., and Shimomura, T., Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent. Complex Var. Elliptic Equ. 56(2011), no. 7–9, 671–695. https://doi.org/10.1080/17476933.2010.504837Google Scholar
Mizuta, Y., Ohno, T., and Shimomura, T., Sobolev inequalities for Musielak–Orlicz spaces. Manuscripta Math. 155(2018), no. 1–2, 209–227. https://doi.org/10.1007/s00229-017-0944-5Google Scholar
Mizuta, Y. and Shimomura, T., Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent. J. Math. Soc. Japan 60(2008), 583–602.Google Scholar
Mizuta, Y., Shimomura, T., and Sobukawa, T., Sobolev’s inequality for Riesz potentials of functions in non-doubling Morrey spaces. Osaka J. Math. 46(2009), 255–271.Google Scholar
Nakai, E., Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166(1994), 95–103. https://doi.org/10.1002/mana.19941660108Google Scholar
Nakai, E., Generalized fractional integrals on Orlicz–Morrey spaces. In: Banach and function spaces. Yokohama Publ., Yokohama, 2004, 323–333.Google Scholar
Nakai, E., Orlicz–Morrey spaces and the Hardy–Littlewood maximal function. Studia Math. 188(2008), 193–221. https://doi.org/10.4064/sm188-3-1Google Scholar
Ohno, T. and Shimomura, T., Trudinger’s inequality and continuity for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces on metric measure spaces. Nonlinear Anal. 106(2014), 1–17. https://doi.org/10.1016/j.na.2014.04.008Google Scholar
Ohno, T. and Shimomura, T., Musielak–Orlicz–Sobolev spaces on metric measure spaces. Czechoslovak Math. J. 65(2015), 140, 435–474. https://doi.org/10.1007/s10587-015-0187-0Google Scholar
Ohno, T. and Shimomura, T., Sobolev inequalities for Riesz potentials of functions in L p (⋅) over nondoubling measure spaces. Bull. Aust. Math. Soc. 93(2016), 128–136. https://doi.org/10.1017/S0004972715001331Google Scholar
Ohno, T. and Shimomura, T., Musielak–Orlicz–Sobolev spaces with zero boundary values on metric measure spaces. Czechoslovak Math. J. 66(2016), 141, 371–394. https://doi.org/10.1007/s10587-016-0262-1Google Scholar
Ohno, T. and Shimomura, T., Maximal and Riesz potential operators on Musielak–Orlicz spaces over metric measure spaces. Integral Equations Operator Theory 90(2018), no. 6, Art. 62. https://doi.org/10.1007/s00020-018-2484-0Google Scholar
Peetre, J., On the theory of 𝓛p, 𝜆 spaces. J. Funct. Anal. 4(1969), 71–87. https://doi.org/10.1016/0022-1236(69)90022-6Google Scholar
Samko, N., Samko, S., and Vakulov, B. G., Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 335(2007), no. 1, 560–583. https://doi.org/10.1016/j.jmaa.2007.01.091Google Scholar
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. https://doi.org/10.14492/hokmj/1285766231Google Scholar
Sawano, Y. and Shimomura, T., Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. Collect. Math. 64(2013), 313–350. https://doi.org/10.1007/s13348-013-0082-7Google Scholar
Sawano, Y. and Shimomura, T., Sobolev embeddings for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over non-doubling measure spaces. Integral Transforms Spec. Funct. 25(2014), 976–991. https://doi.org/10.1080/10652469.2014.955099Google Scholar
Sawano, Y. and Tanaka, H., Morrey spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21(2005), 1535–1544. https://doi.org/10.1007/s10114-005-0660-zGoogle Scholar
Stempak, K., Examples of metric measure spaces related to modified Hardy–Littlewood maximal operators. Ann. Acad. Sci. Fenn. Math. 41(2016), no. 1, 313–314. https://doi.org/10.5186/aasfm.2016.4119Google Scholar
Strömberg, J.-O., Weak type L 1 estimates for maximal functions on noncompact symmetric spaces. Ann. of Math. (2) 114(1981), no. 1, 115–126. https://doi.org/10.2307/1971380Google Scholar