Geodesic
Trudinger's inequality for double phase functionals with variable exponents
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 511-528
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Our aim in this paper is to establish Trudinger's inequality on Musielak-Orlicz-Morrey spaces $L^{\Phi ,\kappa }(G)$ under conditions on $\Phi $ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger's inequality for double phase functionals $\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}$, where $p(\cdot )$ and $q(\cdot )$ satisfy log-Hölder conditions and $a(\cdot )$ is nonnegative, bounded and Hölder continuous.
Our aim in this paper is to establish Trudinger's inequality on Musielak-Orlicz-Morrey spaces $L^{\Phi ,\kappa }(G)$ under conditions on $\Phi $ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger's inequality for double phase functionals $\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}$, where $p(\cdot )$ and $q(\cdot )$ satisfy log-Hölder conditions and $a(\cdot )$ is nonnegative, bounded and Hölder continuous.
DOI : 10.21136/CMJ.2021.0506-19
Classification : 31C15, 46E30
Keywords: Riesz potential; Trudinger's inequality; Musielak-Orlicz-Morrey space; double phase functional 
@article{10_21136_CMJ_2021_0506_19,
     author = {Maeda, Fumi-Yuki and Mizuta, Yoshihiro and Ohno, Takao and Shimomura, Tetsu},
     title = {Trudinger's inequality for double phase functionals with variable exponents},
     journal = {Czechoslovak Mathematical Journal},
     pages = {511--528},
     year = {2021},
     volume = {71},
     number = {2},
     doi = {10.21136/CMJ.2021.0506-19},
     mrnumber = {4263183},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2021.0506-19/}
}
TY  - JOUR
AU  - Maeda, Fumi-Yuki
AU  - Mizuta, Yoshihiro
AU  - Ohno, Takao
AU  - Shimomura, Tetsu
TI  - Trudinger's inequality for double phase functionals with variable exponents
JO  - Czechoslovak Mathematical Journal
PY  - 2021
SP  - 511
EP  - 528
VL  - 71
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2021.0506-19/
DO  - 10.21136/CMJ.2021.0506-19
LA  - en
ID  - 10_21136_CMJ_2021_0506_19
ER  - 
%0 Journal Article
%A Maeda, Fumi-Yuki
%A Mizuta, Yoshihiro
%A Ohno, Takao
%A Shimomura, Tetsu
%T Trudinger's inequality for double phase functionals with variable exponents
%J Czechoslovak Mathematical Journal
%D 2021
%P 511-528
%V 71
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2021.0506-19/
%R 10.21136/CMJ.2021.0506-19
%G en
%F 10_21136_CMJ_2021_0506_19
Maeda, Fumi-Yuki; Mizuta, Yoshihiro; Ohno, Takao; Shimomura, Tetsu. Trudinger's inequality for double phase functionals with variable exponents. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 511-528. doi: 10.21136/CMJ.2021.0506-19

[1] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften 314. Springer, Berlin (1996). | DOI | MR | JFM

[2] Ahmida, Y., Chlebicka, I., Gwiazda, P., Youssfi, A.: Gossez's approximation theorems in Musielak-Orlicz-Sobolev spaces. J. Funct. Anal. 275 (2018), 2538-2571. | DOI | MR | JFM

[3] Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St. Petersbg. Math. J. 27 (2016), 347-379. | DOI | MR | JFM

[4] 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 | MR | JFM

[5] Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218 (2015), 219-273. | DOI | MR | JFM

[6] Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215 (2015), 443-496. | DOI | MR | JFM

[7] Futamura, T., Mizuta, Y.: Continuity properties of Riesz potentials for functions in $L^{p(\cdot)}$ of variable exponent. Math. Inequal. Appl. 8 (2005), 619-631. | DOI | MR | JFM

[8] Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embedding for variable exponent Riesz potentials on metric spaces. Ann. Acad. Sci. Fenn., Math. 31 (2006), 495-522. | MR | JFM

[9] Futamura, T., Mizuta, Y., Shimomura, T.: Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent. J. Math. Anal. Appl. 366 (2010), 391-417. | DOI | MR | JFM

[10] Hästö, P.: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269 (2015), 4038-4048 corrigendum ibid. 271 240-243 2016. | DOI | MR | JFM

[11] 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 | MR | JFM

[12] Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Trudinger's inequality and continuity of potentials on Musielak-Orlicz-Morrey spaces. Potential Anal. 38 (2013), 515-535. | DOI | MR | JFM

[13] 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 | MR | JFM

[14] 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. | DOI | MR | JFM

[15] Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev embeddings for Riesz potential spaces of variable exponents near 1 and Sobolev's exponent. Bull. Sci. Math. 134 (2010), 12-36. | DOI | MR | JFM

[16] Mizuta, Y., Shimomura, T.: Differentiability and Hölder continuity of Riesz potentials of Orlicz functions. Analysis, München 20 (2000), 201-223. | DOI | MR | JFM

[17] 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. | DOI | MR | JFM

[18] Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034. Springer, Berlin (1983). | DOI | MR | JFM

[19] Nakai, E.: Generalized fractional integrals on Orlicz-Morrey spaces. Banach and Function Spaces Yokohama Publishers, Yokohama (2004), 323-333. | MR | JFM

[20] Ohno, T., Shimomura, T.: Trudinger's inequality for Riesz potentials of functions in Musielak-Orlicz spaces. Bull. Sci. Math. 138 (2014), 225-235. | DOI | MR | JFM

Cité par Sources :