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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
DOI: https://doi.org/10.1090/proc/13322
Published electronically: September 8, 2016
MathSciNet review: 3565384
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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.

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Additional Information

Nguyen Cong Phuc
Affiliation: Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, Louisiana 70803
Email: pcnguyen@math.lsu.edu

DOI: https://doi.org/10.1090/proc/13322
Received by editor(s): March 28, 2016
Published electronically: September 8, 2016
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society