Sobolev and Trudinger inequalities in weighted Morrey spaces for double phase functionals. (English) Zbl 1516.31016

Mizuta, Yoshihiro; Shimomura, Tetsu

Z. Anal. Anwend. 41, No. 3-4, 439-466 (2022).

Summary: Our aim in this paper is to establish Sobolev and Trudinger inequalities for Sobolev functions in weighted Morrey spaces. As an application, we extend these inequalities for double phase functionals \(\Phi (x, t) = t^p + (b (x) t)^q\), where \(1 < p < q\) and \(b (\cdot)\) is nonnegative, bounded and Hölder continuous of order \(\theta \in (0, 1]\). Our results are new even for the unweighted case.

MSC:

31B15	Potentials and capacities, extremal length and related notions in higher dimensions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

generalized Riesz potentials; weighted Morrey spaces; Sobolev inequality; Trudinger inequality

References:

[1]	A. Abdalmonem and A. Scapellato, Intrinsic square functions and commutators on Morrey-Herz spaces with variable exponents. Math. Methods Appl. Sci. 44 (2021), 12408-12425 · Zbl 1481.42013
[2]	A. Aberqi, J. Bennouna, O. Benslimane and M. A. Ragusa, Existence results for double phase problem in Sobolev-Orlicz spaces with variable exponents in complete manifold. Mediterr. J. Math. 19 (2022), 158 · Zbl 1491.35202
[3]	S. Baasandorj, S.-S. Byun and J. Oh, Calderón-Zygmund estimates for generalized double phase problems. J. Funct. Anal. 279 (2020), 108670 · Zbl 1448.35236
[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. · 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), 62 · Zbl 1394.49034
[6]	S.-S. Byun and J. Oh, Regularity results for generalized double phase functionals. Anal. PDE 13 (2020), 1269-1300 · Zbl 1477.49057
[7]	F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl. (7) 7 (1987), 273-279 · Zbl 0717.42023
[8]	F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals. Ann. Mat. Pura Appl. (4) 195 (2016), 1917-1959 · Zbl 1364.35226
[9]	M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218 (2015), 219-273 · Zbl 1325.49042
[10]	M. Colombo and G. Mingione, Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215 (2015), 443-496 · Zbl 1322.49065
[11]	C. De Filippis and G. Mingione, Lipschitz bounds and nonautonomous integrals. Arch. Ration. Mech. Anal. 242 (2021), 973-1057 · Zbl 1483.49050
[12]	P. Harjulehto and P. Hästö, Orlicz spaces and generalized Orlicz spaces. Lecture Notes in Math. 2236, Springer, Cham, 2019 · Zbl 1436.46002
[13]	P. Hästö and J. Ok, Calderón-Zygmund estimates in generalized Orlicz spaces. J. Differential Equations 267 (2019), 2792-2823 · Zbl 1420.35087
[14]	P. Hästö and J. Ok, Maximal regularity for local minimizers of non-autonomous functionals. J. Eur. Math. Soc. (JEMS) 24 (2022), 1285-1334 · Zbl 1485.49044
[15]	L. I. Hedberg, On certain convolution inequalities. Proc. Amer. Math. Soc. 36 (1972), 505-510 · Zbl 0283.26003
[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 · Zbl 1267.46045
[17]	F.-Y. Maeda, Y. Mizuta, T. Ohno and T. Shimomura, Sobolev’s inequality for double phase functionals with variable exponents. Forum Math. 31 (2019), 517-527 · Zbl 1423.46049
[18]	P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 391-409 · Zbl 0609.49009
[19]	P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstan-dard growth conditions. Arch. Ration. Mech. Anal. 105 (1989), 267-284 · Zbl 0667.49032
[20]	J. Meng, S. Wang and J. Zhang, Commutators of multilinear Calderón-Zygmund operator on weighted Herz-Morrey spaces with variable exponents. J. Funct. Spaces 2021 (2021), 9947489 · Zbl 1520.42010
[21]	Y. Mizuta, Potential theory in Euclidean spaces. GAKUTO Int. Ser. Math. Sci. Appl. 6, Gakkotosho, Tokyo, 1996
[22]	Y. Mizuta, Integral representations, differentiability properties and limits at infinity for Beppo Levi functions. Potential Anal. 6 (1997), 237-267 · Zbl 0885.31004
[23]	Y. Mizuta, E. Nakai, T. Ohno and T. Shimomura, Campanato-Morrey spaces for the double phase functionals. Rev. Mat. Complut. 33 (2020), 817-834 · Zbl 1452.31010
[24]	Y. Mizuta, E. Nakai, T. Ohno and T. Shimomura, Campanato-Morrey spaces for the double phase functionals with variable exponents. Nonlinear Anal. 197 (2020), 111827 · Zbl 1441.31004
[25]	Y. Mizuta, T. Ohno and T. Shimomura, Sobolev’s theorem for double phase functionals. Math. Inequal. Appl. 23 (2020), 17-33 · Zbl 1453.46021
[26]	Y. Mizuta and T. Shimomura, Weighted Morrey spaces of variable exponent and Riesz poten-tials. Math. Nachr. 288 (2015), 984-1002 · Zbl 1318.31009
[27]	Y. Mizuta and T. Shimomura, Boundary growth of Sobolev functions for double phase func-tionals. Ann. Acad. Sci. Fenn. Math. 45 (2020), 279-292 · Zbl 1439.31007
[28]	Y. Mizuta and T. Shimomura, Hardy-Sobolev inequalities in the unit ball for double phase functionals. J. Math. Anal. Appl. 501 (2021), 124133 · Zbl 1478.46037
[29]	Y. Mizuta and T. Shimomura, Sobolev type inequalities for fractional maximal functions and Riesz potentials in half spaces, Studia Math., to appear · Zbl 1575.46025
[30]	C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43 (1938), 126-166 · JFM 64.0460.02
[31]	J. Musielak, Orlicz spaces and modular spaces. Lecture Notes in Math. 1034, Springer, Berlin, 1983 · Zbl 0557.46020
[32]	E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz poten-tials on generalized Morrey spaces. Math. Nachr. 166 (1994), 95-103 · Zbl 0837.42008
[33]	J. Peetre, On the theory of L p ; spaces. J. Funct. Anal. 4 (1969), 71-87 · Zbl 0175.42602
[34]	M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9 (2020), 710-728 · Zbl 1420.35145
[35]	T. Shimomura and Y. Mizuta, Taylor expansion of Riesz potentials. Hiroshima Math. J. 25 (1995), 595-621 · Zbl 0863.31005
[36]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 675-710, 877 · Zbl 0599.49031