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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Degenerate Poincaré-Sobolev inequalities

Authors: Carlos Pérez and Ezequiel Rela
Journal: Trans. Amer. Math. Soc. 372 (2019), 6087-6133
MSC (2010): Primary 42B25; Secondary 42B20
Published electronically: August 5, 2019
MathSciNet review: 4024515
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Abstract: We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the $ A_p$ constants of the involved weights. We obtain inequalities of the form

$\displaystyle \left (\frac {1}{w(Q)}\int _Q\vert f-f_Q\vert^{q}w\right )^\frac ... ...l (Q)\left (\frac {1}{w(Q)}\int _Q \vert\nabla f\vert^p w\right )^\frac {1}{p},$    

with different quantitative estimates for both the exponent $ q$ and the constant $ C_w$. We derive those estimates together with a large variety of related results as a consequence of a general self-improving property shared by functions satisfying the inequality

$\displaystyle -\kern -10.5pt\int _Q \vert f-f_Q\vert d\mu \le a(Q) $

for all cubes $ Q\subset \mathbb{R}^n$ and where $ a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev type exponent $ p^*_w>p$ associated with the weight $ w$ and obtain further improvements involving $ L^{p^*_w}$ norms on the left-hand side of the inequality above. For the endpoint case of $ A_1$ weights, we reach the classical critical Sobolev exponent $ p^*=\frac {pn}{n-p}$, which is the largest possible, and provide different types of quantitative estimates for $ C_w$. We also show that this best possible estimate cannot hold with an exponent on the $ A_1$ constant smaller than $ 1/p$.

As a consequence of our results (and the method of proof), we obtain further extensions to two-weight Poincaré inequalities and to the case of higher order derivatives. Some other related results in the spirit of the work of Keith and Zhong on the open-ended condition of Poincaré inequality are obtained using extrapolation methods. We also apply our method to obtain similar estimates in the scale of Lorentz spaces.

We also provide an argument based on extrapolation ideas showing that there is no $ (p,p)$, $ p\geq 1$, Poincaré inequality valid for the whole class of $ RH_\infty $ weights by showing their intimate connection with the failure of Poincaré inequalities $ (p,p)$ in the range $ 0<p<1$.

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Additional Information

Carlos Pérez
Affiliation: Department of Mathematics, University of the Basque Country, IKERBASQUE (Basque Foundation for Science), Bilbao, Spain; and BCAM—Basque Center for Applied Mathematics, Bilbao, Spain

Ezequiel Rela
Affiliation: Department of Mathematics, Facultad de Ciencias Exactas y Naturales, University of Buenos Aires, Ciudad Universitaria Pabellón I, Buenos Aires 1428, Capital Federal, Argentina

Keywords: Poincar\'e--Sobolev inequalities, Muckenhoupt weights.
Received by editor(s): July 13, 2018
Published electronically: August 5, 2019
Additional Notes: The first author was supported by grant MTM2017-82160-C2-1-P of the Ministerio de Economía y Competitividad (Spain), grant IT-641-13 of the Basque Government, and IKERBASQUE
The second author was partially supported by grants UBACyT 20020130100403BA and PIP (CONICET) 11220110101018, by the Basque Government through the BERC 2014-2017 program, and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.
This work started during a visit of the second author to the Basque Center for Applied Mathematics (BCAM) in Bilbao, Spain. He is deeply grateful to all of the staff for their hospitality.
Article copyright: © Copyright 2019 American Mathematical Society