Hostname: page-component-77c78cf97d-tlp4c Total loading time: 0 Render date: 2026-04-25T19:57:32.573Z Has data issue: false hasContentIssue false
  • English
  • Français

Trudinger-type inequalities in Musielak–Orlicz spaces and double phase functionals

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  27 March 2025

Takao Ohno Open the ORCID record for Takao Ohno [Opens in a new window]  and
Tetsu Shimomura Open the ORCID record for Tetsu Shimomura [Opens in a new window]
Takao Ohno*
Affiliation:
Faculty of Education, Oita University, Dannoharu Oita-city, 870-1192, Japan
Tetsu Shimomura
Affiliation:
Department of Mathematics, Graduate School of Humanities and Social Sciences, Hiroshima University, Higashi-Hiroshima, 739-8524, Japan e-mail: tshimo@hiroshima-u.ac.jp
*
e-mail: t-ohno@oita-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish Trudinger-type inequalities for variable Riesz potentials $J_{\alpha (\cdot ), \tau }f$ of functions in Musielak–Orlicz spaces $L^{\Phi }(X)$ over bounded metric measure spaces X equipped with lower Ahlfors $Q(x)$-regular measures under conditions on $\Phi $ which are weaker than conditions in the previous paper (Houston J. Math. 48 (2022), no. 3, 479–497). We also deal with the case $\Phi $ is the double phase functional with variable exponents. As an application, Trudinger-type inequalities are discussed for Sobolev functions.

Keywords

Riesz potentialsMusielak–Orlicz spacesTrudinger’s inequalitymetric measure spacedouble 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.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 4 , December 2025 , pp. 1192 - 1209
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Baroni, P., Colombo, M., and Mingione, G., Harnack inequalities for double phase functionals . Nonlinear Anal. 121(2015), 206–222.10.1016/j.na.2014.11.001CrossRefGoogle Scholar
Björn, A. and Björn, J., Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zurich, 2011.10.4171/099CrossRefGoogle Scholar
Colombo, M. and Mingione, G., Regularity for double phase variational problems . Arch. Ration. Mech. Anal. 215(2015), no. 2, 443–496.10.1007/s00205-014-0785-2CrossRefGoogle Scholar
Hajłasz, P. and Koskela, P., Sobolev met Poincaré . Mem. Amer. Math. Soc. 145(2000), no. 688, x+101 pp.Google Scholar
Harjulehto, P. and Hästö, P., Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, 2236, Springer, Cham, 2019.10.1007/978-3-030-15100-3CrossRefGoogle Scholar
Harjulehto, P., Hästö, P., and Latvala, V., Sobolev embeddings in metric measure spaces with variable dimension . Math. Z. 254(2006), no. 3, 591–609.10.1007/s00209-006-0960-8CrossRefGoogle Scholar
Hashimoto, D., Sawano, Y., and Shimomura, T., Gagliardo-Nirenberg inequality for generalized Riesz potentials of functions in Musielak–Orlicz spaces over quasi-metric measure spaces . Colloq. Math. 161(2020), 51–66.10.4064/cm7535-4-2019CrossRefGoogle Scholar
Hedberg, L. I., On certain convolution inequalities . Proc. Amer. Math. Soc. 36(1972), 505–510.CrossRefGoogle Scholar
Heinonen, J., Lectures on analysis on metric spaces. Springer-Verlag, New York, NY, 2001.CrossRefGoogle 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), 76–96.10.1016/j.bulsci.2012.03.008CrossRefGoogle Scholar
Maeda, F.-Y., Mizuta, Y., Ohno, T., and Shimomura, T., Sobolev’s inequality inequality for double phase functionals with variable exponents . Forum Math. 31(2019), 517–527.10.1515/forum-2018-0077CrossRefGoogle Scholar
Maeda, F.-Y., Mizuta, Y., Ohno, T., and Shimomura, T., Trudinger’s inequality for double phase functionals with variable exponents . Czechoslovak Math. J. 71(2021), no. 2, 511–528.10.21136/CMJ.2021.0506-19CrossRefGoogle Scholar
Nazarov, F., Treil, S., and Volberg, A., Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces . Internat. Math. Res. Notices (1998), no. 9, 463–487.10.1155/S1073792898000312CrossRefGoogle Scholar
Ohno, T. and Shimomura, T., Musielak-Orlicz-Sobolev spaces on metric measure spaces . Czechoslovak Math. J. 65(2015), no. 140, 435–474.10.1007/s10587-015-0187-0CrossRefGoogle Scholar
Ohno, T. and Shimomura, T., Sobolev inequalities for Riesz potentials of functions in ${L}^{p\left(\cdotp \right)}$ over nondoubling measure spaces. Bull. Aust. Math. Soc. 93(2016), 128–136.CrossRefGoogle Scholar
Ohno, T. and Shimomura, T., Maximal and Riesz potential operators on Musielak-Orlicz spaces over metric measure spaces . Integral Equ. Oper. Theory 90(2018), no. 6, Art. 62. 18 pp.CrossRefGoogle Scholar
Ohno, T. and Shimomura, T., Trudinger-type inequalities for variable Riesz potentials of functions in Musielak-Orlicz-Morrey spaces over metric measure spaces . Math. Nachr. 297(2024), no. 4, 1248–1274.10.1002/mana.202300265CrossRefGoogle Scholar
Ohno, T. and Shimomura, T., Trudinger-type inequalities in Musielak-Orlicz spaces . Houston J. Math. 48(2022), no. 3, 479–497.Google Scholar
Ragusa, M. A. and Tachikawa, A., Regularity for minimizers for functionals of double phase with variable exponents . Adv. Nonlinear Anal. 9(2020), no. 1, 710–728.10.1515/anona-2020-0022CrossRefGoogle Scholar
Sawano, Y., Sharp estimates of the modified Hardy-Littlewood maximal operator on the nonhomogeneous space via covering lemmas . Hokkaido Math. J. 34(2005), 435–458.10.14492/hokmj/1285766231CrossRefGoogle 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), 313–350.10.1007/s13348-013-0082-7CrossRefGoogle Scholar
Sawano, Y. and Shimomura, T., Maximal operator on Orlicz spaces of two variable exponents over unbounded quasi-metric measure spaces . Proc. Amer. Math. Soc. 147(2019), 2877–2885.10.1090/proc/14225CrossRefGoogle Scholar
Stein, E. M., Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ, 1970.Google Scholar
Strömberg, J.-O., Weak type ${L}^1$ estimates for maximal functions on noncompact symmetric spaces. Ann. Math. (2) 114(1981), no. 1, 115–126.10.2307/1971380CrossRefGoogle Scholar
Terasawa, Y., Outer measures and weak type $\left(1,1\right)$ estimates of Hardy-Littlewood maximal operators. J. Inequal. Appl. (2006), Article ID 15063, 13pp.Google Scholar
Trudinger, N. S., On imbeddings into Orlicz spaces and some applications . J. Math. Mech. 17(1967), 473–483.Google Scholar
Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory . Izv. Akad. Nauk SSSR Ser. Mat. 50(1986), 675–710.Google Scholar