A sublinear Sobolev inequality for $p$-superharmonic functions
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Abstract:
We establish a âsublinearâ Sobolev inequality of the form \[ \left (\int _{\mathbb {R}^n} u^{\frac {nq}{n-q}} dx\right )^{\frac {n-q}{nq}}\leq C \left (\int _{\mathbb {R}^n}|D u|^{q} dx\right )^{\frac {1}{q}}\] for all global $p$-superharmonic ($1<p<2$) functions $u$ in $\mathbb {R}^n$, $n\geq 2$, with $\inf _{\mathbb {R}^n} u=0$ and $p-1<q<1$. The same result also holds for the class of $\mathcal {A}$-superharmonic functions. More general sublinear trace inequalities, where Lebesgue measure is replaced by a general measure, are also considered.References
- D. R. Adams, Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25 (1971), 203â217. MR 287301
- David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441, DOI 10.1007/978-3-662-03282-4
- Stephen M. Buckley and Pekka Koskela, Sobolev-PoincarĂŠ inequalities for $p<1$, Indiana Univ. Math. J. 43 (1994), no. 1, 221â240. MR 1275460, DOI 10.1512/iumj.1994.43.43011
- Carme Cascante, JoaquĂn M. Ortega, and Igor E. Verbitsky, Trace inequalities of Sobolev type in the upper triangle case, Proc. London Math. Soc. (3) 80 (2000), no. 2, 391â414. MR 1734322, DOI 10.1112/S0024611500012260
- Carme Cascante, Joaquin M. Ortega, and Igor E. Verbitsky, On $L^p$-$L^q$ trace inequalities, J. London Math. Soc. (2) 74 (2006), no. 2, 497â511. MR 2269591, DOI 10.1112/S0024610706023064
- Marco Degiovanni, Alfredo Marzocchi, and Alessandro Musesti, Cauchy fluxes associated with tensor fields having divergence measure, Arch. Ration. Mech. Anal. 147 (1999), no. 3, 197â223. MR 1709215, DOI 10.1007/s002050050149
- Frank Duzaar and Giuseppe Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133 (2011), no. 4, 1093â1149. MR 2823872, DOI 10.1353/ajm.2011.0023
- Frank Duzaar and Giuseppe Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259 (2010), no. 11, 2961â2998. MR 2719282, DOI 10.1016/j.jfa.2010.08.006
- Emilio Gagliardo, ProprietĂ di alcune classi di funzioni in piĂš variabili, Ricerche Mat. 7 (1958), 102â137 (Italian). MR 102740
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
- Petri Juutinen, Peter Lindqvist, and Juan J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 (2001), no. 3, 699â717. MR 1871417, DOI 10.1137/S0036141000372179
- Tero Kilpeläinen and Jan MalĂ˝, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 4, 591â613. MR 1205885
- Tero Kilpeläinen and Jan MalĂ˝, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), no. 1, 137â161. MR 1264000, DOI 10.1007/BF02392793
- Tuomo Kuusi and Giuseppe Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (2013), no. 1, 215â246. MR 3004772, DOI 10.1007/s00205-012-0562-z
- L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115â162. MR 109940
- S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. 46 (1938), 471â497 (Russian); English transl.: Am. Math. Soc., Transl., II. Ser. 34 (1963), 39â68.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Igor E. Verbitsky, The Hessian Sobolev inequality and its extensions, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 6165â6179. MR 3393272, DOI 10.3934/dcds.2015.35.6165
Additional Information
- Nguyen Cong Phuc
- Affiliation: Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, Louisiana 70803
- Email: pcnguyen@math.lsu.edu
- Received by editor(s): March 28, 2016
- Published electronically: September 8, 2016
- Communicated by: Jeremy Tyson
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 327-334
- MSC (2010): Primary 31B35, 35J92; Secondary 31B15
- DOI: https://doi.org/10.1090/proc/13322
- MathSciNet review: 3565384