Abstract
Our aim in this note is to establish Hardy-Sobolev inequalities for double phase functionals \(\Phi (t,r)= r^p + (b(t) r)^q\) on the half space, as a continuation of our paper (Mizuta and Shimomura in Rocky Mount. J. Math.), where \(1 \le p < q\), b is non-negative and Hölder continuous in \([0,\infty )\) of order \(\theta \in (0,1]\). The Sobolev conjugate for \(\Phi \) is given by \(\Phi ^*(t,r)= r^{p^*} + (b(t) r)^{q^*}\), where \(p^*\) and \(q^*\) denote the Sobolev exponent of p and q, respectively, that is, \(1/p^* = 1/p - 1/n\) and \(1/q^* = 1/q - 1/n\). As applications, we study the boundary behavior of Sobolev functions on the half space.
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References
Alvino, A.: Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. A 14(5), 148–156 (1977)
Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. Partial Differ. Equ. 57(2), 1–48 (2018)
Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St Petersburg Math. J. 27, 347–379 (2016)
Byun, S.S., Cho, N., Youn, Y.: Global gradient estimates for a borderline case of double phase problems with measure data. J. Math. Anal. Appl. 501, 124072 (2021)
Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215(2), 443–496 (2015)
Colombo, M., Mingione, G.: Bounded minimizers of double phase variational integrals. Arch. Rat. Mech. Anal. 218, 219–273 (2015)
De Filippis, C., Palatucci, G.: Hölder regularity for nonlocal double phase equations. J. Differ. Equ. 267(1), 547–586 (2019)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2d edn. Cambridge University Press (1952)
Harjulehto, P., Hästö, P., Lee, M.: Hölder continuity of quasiminimizers and \(\omega \)-minimizers of functionals with generalized Orlicz growth. Ann. Sc. Norm. Super. Pisa Cl. Sci. XXII, 549–582 (2021)
Hästö, P.: The maximal operator on generalized Orlicz spaces [J. Funct. Anal. 269 4038–4048 (2015)]. J. Funct. Anal. 271(1), 240–243 (2016)
Hästö, P., Ok, J.: Maximal regularity for local minimizers of non-autonomous functionals. J. Eur. Math. Soc 24, 1285–1334 (2022)
Kufner, A., Opic, B.: Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219. Longman Group UK Limited, London (1990)
Kufner, A., Persson, L.-E., Samko, N.: Weighted inequalities of Hardy type, 2nd edn. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017)
Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s inequality for double phase functionals with variable exponents. Forum Math. 31, 517–527 (2019),
Maeda, F.-Y., Mizuta, Y., Shimomura, T.: Growth properties of Musielak-Orlicz integral means for Riesz potentials. Nonlinear Anal. 112, 69–83 (2015)
Maz’ya, V.G.: Sobolev spaces. Springer, Heidelberg (2011)
Mizuta, Y.: Potential theory in Euclidean spaces. Gakkōtosho, Tokyo (1996)
Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Campanato-Morrey spaces for the double phase functionals. Rev. Mat. Complut. 33, 817–834 (2020)
Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s theorem for double phase functionals. Math. Ineq. Appl. 23, 17–33 (2020)
Mizuta, Y., Shimomura, T.: Hardy-Sobolev inequalities in the half space for double phase functionals. Rocky Mount. J. Math 51, 2159–2169 (2021)
Mizuta, Y., Shimomura, T.: Boundary growth of Sobolev functions of monotone type for double phase functionals. Ann. Fenn. Math. 47, 23–37 (2022)
Mizuta, Y., Shimomura, T.: Hardy-Sobolev inequalities in the unit ball for double phase functionals. J. Math. Anal. Appl. 501, 124133 (2021)
Persson, L.-E., Samko, S.G.: A note on the best constants in some Hardy inequalities. J. Math. Inequal. 9(2), 437–447 (2015)
Ragusa, M.A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9(1), 710–728 (2020)
Shimomura, T.: \(L^q\)-mean limits for Taylor’s expansion of Riesz potentials of functions in Orlicz classes. Hiroshima Math. J. 27, 159–175 (1997)
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)
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Mizuta, Y., Shimomura, T. Hardy-Sobolev inequalities and boundary growth of Sobolev functions for double phase functionals on the half space. Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00686-5
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DOI: https://doi.org/10.1007/s11587-022-00686-5