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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on sharp 1-dimensional Poincaré inequalities


Authors: Seng-Kee Chua and Richard L. Wheeden
Journal: Proc. Amer. Math. Soc. 134 (2006), 2309-2316
MSC (2000): Primary 26D10; Secondary 46E35
Published electronically: March 20, 2006
MathSciNet review: 2213704
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Abstract: Let $ 1<p<\infty$ and $ -\infty < a < b <\infty$. We show by using elementary methods that the best constant $ C$ (necessarily independent of $ a$ and $ b$) for which the 1-dimensional Poincaré inequality

$\displaystyle \Vert f-f_{av}\Vert _{{}_{\scriptstyle{L^1[a,b]}}} \le C (b-a)^{2-\frac{1}{p}} \Vert f'\Vert _{{}_{\scriptstyle{L^p[a,b]}}}$

holds for all Lipschitz continuous functions $ f$, with $ f_{av}=\int^b_a f/(b-a),$ is

$\displaystyle C=\frac{1}{2} (1+p')^{-1/p'}.$


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Additional Information

Seng-Kee Chua
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matcsk@nus.edu.sg

Richard L. Wheeden
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Email: wheeden@math.rutgers.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08545-5
PII: S 0002-9939(06)08545-5
Keywords: Poincar\'e inequalities, Sobolev inequalities, Hardy inequalities
Received by editor(s): March 3, 2005
Published electronically: March 20, 2006
Communicated by: Michael C. Lacey
Article copyright: © Copyright 2006 American Mathematical Society