Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A sublinear Sobolev inequality for $ p$-superharmonic functions

Author: Nguyen Cong Phuc
Journal: Proc. Amer. Math. Soc. 145 (2017), 327-334
MSC (2010): Primary 31B35, 35J92; Secondary 31B15
Published electronically: September 8, 2016
MathSciNet review: 3565384
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish a ``sublinear'' Sobolev inequality of the form

$\displaystyle \left (\int _{\mathbb{R}^n} u^{\frac {nq}{n-q}} dx\right )^{\frac... ...}\leq C \left (\int _{\mathbb{R}^n}\vert D u\vert^{q} dx\right )^{\frac {1}{q}}$

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 [Enhancements On Off] (What's this?)

  • [Ad] D. R. Adams, Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa (3) 25 (1971), 203-217. MR 0287301
  • [AH] 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
  • [BK] Stephen M. Buckley and Pekka Koskela, Sobolev-Poincaré inequalities for $ p<1$, Indiana Univ. Math. J. 43 (1994), no. 1, 221-240. MR 1275460,
  • [COV1] Carme Cascante, Joaquín 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, 391-414. MR 1734322,
  • [COV2] 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, 497-511. MR 2269591,
  • [DMM] Marco Degiovanni, Alfredo Marzocchi, and Alessandro Musesti, Cauchy fluxes associated with tensor fields having divergence measure, Arch. Ration. Mech. Anal. 147 (1999), no. 3, 197-223. MR 1709215,
  • [DM1] Frank Duzaar and Giuseppe Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133 (2011), no. 4, 1093-1149. MR 2823872,
  • [DM2] Frank Duzaar and Giuseppe Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259 (2010), no. 11, 2961-2998. MR 2719282,
  • [Gag] Emilio Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102-137 (Italian). MR 0102740
  • [HKM] 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
  • [JLM] 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, 699-717 (electronic). MR 1871417,
  • [KM1] Tero Kilpeläinen and Jan Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 4, 591-613. MR 1205885
  • [KM2] Tero Kilpeläinen and Jan Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), no. 1, 137-161. MR 1264000,
  • [KM] Tuomo Kuusi and Giuseppe Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (2013), no. 1, 215-246. MR 3004772,
  • [Nir] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. MR 0109940
  • [Sob] S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. 46 (1938), 471-497 (Russian); English transl.: Am. Math. Soc., Transl., II. Ser. 34 (1963), 39-68.
  • [Stei] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [Ver] Igor E. Verbitsky, The Hessian Sobolev inequality and its extensions, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 6165-6179. MR 3393272,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 31B35, 35J92, 31B15

Retrieve articles in all journals with MSC (2010): 31B35, 35J92, 31B15

Additional Information

Nguyen Cong Phuc
Affiliation: Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, Louisiana 70803

Received by editor(s): March 28, 2016
Published electronically: September 8, 2016
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society