Abstract
Our aim in this paper is to establish generalizations of Sobolev’s inequality
for double phase functionals
Acknowledgements
We would like to express our thanks to the referees for their kind comments and helpful suggestions.
References
[1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin, 1996. 10.1007/978-3-662-03282-4Suche in Google Scholar
[2] Y. Ahmida, I. Chlebicka, P. Gwiazda and A. Youssfi, Gossez’s approximation theorems in Musielak–Orlicz–Sobolev spaces, J. Funct. Anal. 275 (2018), no. 9, 2538–2571. 10.1016/j.jfa.2018.05.015Suche in Google Scholar
[3] A. Almeida, J. Hasanov and S. Samko, Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J. 15 (2008), 195–208. 10.1515/GMJ.2008.195Suche in Google Scholar
[4] P. Baroni, M. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), 347–379. 10.1090/spmj/1392Suche in Google Scholar
[5] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 62. 10.1007/s00526-018-1332-zSuche in Google Scholar
[6]
C. Capone, D. Cruz-Uribe and A. Fiorenza,
The fractional maximal operator and fractional integrals on variable
[7] M. Colombo and G. Mingione, Bounded minimizers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), 219–273. 10.1007/s00205-015-0859-9Suche in Google Scholar
[8] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), 443–496. 10.1007/s00205-014-0785-2Suche in Google Scholar
[9]
L. Diening,
Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces
[10] T. Futamura, Y. Mizuta and T. Shimomura, Sobolev embeddings for Riesz potential space of variable exponent, Math. Nachr. 279 (2006), no. 13–14, 1463–1473. 10.1002/mana.200410432Suche in Google Scholar
[11] T. Futamura, Y. Mizuta and T. Shimomura, Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent, J. Math. Anal. Appl. 366 (2010), 391–417. 10.1016/j.jmaa.2010.01.053Suche in Google Scholar
[12] F. Giannetti and A. Passarelli di Napoli, Regularity results for a new class of functionals with nonstandard growth conditions, J. Differential Equations 254 (2013), 1280–1305. 10.1016/j.jde.2012.10.011Suche in Google Scholar
[13] V. S. Guliyev, J. Hasanov and S. Samko, 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. 10.1007/s10958-010-0095-7Suche in Google Scholar
[14] V. S. Guliyev, J. Hasanov and S. Samko, Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces, Math. Scand. 107 (2010), 285–304. 10.7146/math.scand.a-15156Suche in Google Scholar
[15] P. Hästö, 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. 10.1016/j.jfa.2015.10.002Suche in Google Scholar
[16] F.-Y. Maeda, Y. Mizuta, T. Ohno and T. Shimomura, Boundedness of maximal operators and Sobolev’s inequality on Musielak–Orlicz–Morrey spaces, Bull. Sci. Math. 137 (2013), 76–96. 10.1016/j.bulsci.2012.03.008Suche in Google Scholar
[17] F.-Y. Maeda, T. Ohno and T. Shimomura, Boundedness of maximal operator on Musielak–Orlicz–Morrey spaces, Tohoku Math. J. 69 (2017), 483–495. 10.2748/tmj/1512183626Suche in Google Scholar
[18] Y. Mizuta, E. Nakai, T. Ohno and T. Shimomura, Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent, Complex Var. Elliptic Equ. 56 (2011), no. 7–9, 671–695. 10.1080/17476933.2010.504837Suche in Google Scholar
[19]
Y. Mizuta, T. Ohno and T. Shimomura,
Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space
[20] Y. Mizuta and T. Shimomura, Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent, J. Math. Soc. Japan 60 (2008), 583–602. 10.2969/jmsj/06020583Suche in Google Scholar
[21] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983. 10.1007/BFb0072210Suche in Google Scholar
[22]
J. Ok,
Gradient estimates for elliptic equations with
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
- Infinite families of equivariantly formal toric orbifolds
- Bounds for GL3L-functions in depth aspect
- Nonlinear Dirichlet problems with unilateral growth on the reaction
- K-invariant cusp forms for reductive symmetric spaces of split rank one
- Cellularity in quotient spaces of topological groups
- Shifted convolution sums for higher rank groups
- Dimension quotients, Fox subgroups and limits of functors
- Large sums of Hecke eigenvalues of holomorphic cusp forms
- K-theory classification of graded ultramatricial algebras with involution
- Regularity of symbolic powers and arboricity of matroids
- On some problems concerning symmetrization operators
- An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
- Rankin–Selberg gamma factors of level zero representations of GLn
- Sobolev’s inequality for double phase functionals with variable exponents
- Independence of Artin L-functions
- Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
Artikel in diesem Heft
- Frontmatter
- Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
- Infinite families of equivariantly formal toric orbifolds
- Bounds for GL3L-functions in depth aspect
- Nonlinear Dirichlet problems with unilateral growth on the reaction
- K-invariant cusp forms for reductive symmetric spaces of split rank one
- Cellularity in quotient spaces of topological groups
- Shifted convolution sums for higher rank groups
- Dimension quotients, Fox subgroups and limits of functors
- Large sums of Hecke eigenvalues of holomorphic cusp forms
- K-theory classification of graded ultramatricial algebras with involution
- Regularity of symbolic powers and arboricity of matroids
- On some problems concerning symmetrization operators
- An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
- Rankin–Selberg gamma factors of level zero representations of GLn
- Sobolev’s inequality for double phase functionals with variable exponents
- Independence of Artin L-functions
- Oscillatory singular integral operators with Hölder class kernels on Hardy spaces