Sobolev’s inequality for double phase functionals with variable exponents. (English) Zbl 1423.46049
Summary: Our aim in this paper is to establish generalizations of Sobolev’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 of order \(\theta\in (0,1]\).
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
Keywords:
Riesz potentials; maximal functions; Sobolev’s inequality; Musielak-Orlicz-Morrey spaces; double phase functionalsReferences:
| [1] | D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin, 1996.; Adams, D. R.; Hedberg, L. I., Function Spaces and Potential Theory (1996) · Zbl 0834.46021 |
| [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.; Ahmida, Y.; Chlebicka, I.; Gwiazda, P.; Youssfi, A., Gossez’s approximation theorems in Musielak-Orlicz-Sobolev spaces, J. Funct. Anal., 275, 9, 2538-2571 (2018) · Zbl 1405.42042 |
| [3] | A. Almeida, J. Hasanov and S. Samko, Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J. 15 (2008), 195-208.; Almeida, A.; Hasanov, J.; Samko, S., Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J., 15, 195-208 (2008) · Zbl 1263.42002 |
| [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.; Baroni, P.; Colombo, M.; Mingione, G., Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27, 347-379 (2016) · Zbl 1335.49057 |
| [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.; Baroni, P.; Colombo, M.; Mingione, G., Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57, 2 (2018) · Zbl 1394.49034 |
| [6] | C. Capone, D. Cruz-Uribe and A. Fiorenza, The fractional maximal operator and fractional integrals on variable \(L^p\) spaces, Rev. Mat. Iberoam. 23 (2007), no. 3, 743-770.; Capone, C.; Cruz-Uribe, D.; Fiorenza, A., The fractional maximal operator and fractional integrals on variable \(L^p\) spaces, Rev. Mat. Iberoam., 23, 3, 743-770 (2007) · Zbl 1213.42063 |
| [7] | M. Colombo and G. Mingione, Bounded minimizers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), 219-273.; Colombo, M.; Mingione, G., Bounded minimizers of double phase variational integrals, Arch. Ration. Mech. Anal., 218, 219-273 (2015) · Zbl 1325.49042 |
| [8] | M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), 443-496.; Colombo, M.; Mingione, G., Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215, 443-496 (2015) · Zbl 1322.49065 |
| [9] | L. Diening, Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \(L^{p(\,\cdot\,)}\) and \(W^{k,p(\,\cdot\,)}\), Math. Nachr. 263 (2004), no. 1, 31-43.; Diening, L., Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \(L^{p(\,\cdot\,)}\) and \(W^{k,p(\,\cdot\,)}\), Math. Nachr., 263, 1, 31-43 (2004) · Zbl 1065.46024 |
| [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.; Futamura, T.; Mizuta, Y.; Shimomura, T., Sobolev embeddings for Riesz potential space of variable exponent, Math. Nachr., 279, 13-14, 1463-1473 (2006) · Zbl 1109.31004 |
| [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.; Futamura, T.; Mizuta, Y.; Shimomura, T., Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent, J. Math. Anal. Appl., 366, 391-417 (2010) · Zbl 1193.46016 |
| [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.; Giannetti, F.; Passarelli di Napoli, A., Regularity results for a new class of functionals with nonstandard growth conditions, J. Differential Equations, 254, 1280-1305 (2013) · Zbl 1255.49064 |
| [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.; Guliyev, V. S.; Hasanov, J.; Samko, S., Boundedness of the maximal, potential and singular integral operators in the generalized variable exponent Morrey type spaces, J. Math. Sci., 170, 4, 423-443 (2010) · Zbl 1307.46018 |
| [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.; Guliyev, V. S.; Hasanov, J.; Samko, S., Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces, Math. Scand., 107, 285-304 (2010) · Zbl 1213.42077 |
| [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.; Hästö, P., The maximal operator on generalized Orlicz spaces, J. Funct. Anal., 269, 12, 4038-4048 (2015) · Zbl 1338.47032 |
| [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.; 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, 76-96 (2013) · Zbl 1267.46045 |
| [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.; Maeda, F.-Y.; Ohno, T.; Shimomura, T., Boundedness of maximal operator on Musielak-Orlicz-Morrey spaces, Tohoku Math. J., 69, 483-495 (2017) · Zbl 1387.42017 |
| [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.; Mizuta, Y.; Nakai, E.; Ohno, T.; Shimomura, T., Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent, Complex Var. Elliptic Equ., 56, 7-9, 671-695 (2011) · Zbl 1228.31004 |
| [19] | Y. Mizuta, T. Ohno and T. Shimomura, Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space \(L^{p(\,\cdot\,)}(\log L)^{q(\,\cdot\,)}\), J. Math. Anal. Appl. 345 (2008), 70-85.; Mizuta, Y.; Ohno, T.; Shimomura, T., Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space \(L^{p(\,\cdot\,)}(\log L)^{q(\,\cdot\,)}\), J. Math. Anal. Appl., 345, 70-85 (2008) · Zbl 1153.31002 |
| [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.; Mizuta, Y.; Shimomura, T., Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent, J. Math. Soc. Japan, 60, 583-602 (2008) · Zbl 1161.46305 |
| [21] | J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983.; Musielak, J., Orlicz Spaces and Modular Spaces (1983) · Zbl 0557.46020 |
| [22] | J. Ok, Gradient estimates for elliptic equations with \(L^{p(\,\cdot\,)}\log L\) growth, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Paper No. 26.; Ok, J., Gradient estimates for elliptic equations with \(L^{p(\,\cdot\,)}\log L\) growth, Calc. Var. Partial Differential Equations, 55, 2 (2016) · Zbl 1342.35090 |
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