A note on sharp 1-dimensional Poincaré inequalities
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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 \[ \|f-f_{av}\|_{{}_{\scriptstyle {L^1[a,b]}}} \le C (b-a)^{2-\frac {1}{p}} \|f’\|_{{}_{\scriptstyle {L^p[a,b]}}}\] holds for all Lipschitz continuous functions $f$, with $f_{av}=\int ^b_a f/(b-a),$ is \[ C=\frac {1}{2} (1+p’)^{-1/p’}.\]References
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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
- Received by editor(s): March 3, 2005
- Published electronically: March 20, 2006
- Communicated by: Michael C. Lacey
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2309-2316
- MSC (2000): Primary 26D10; Secondary 46E35
- DOI: https://doi.org/10.1090/S0002-9939-06-08545-5
- MathSciNet review: 2213704