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A Note on Sharp 1-Dimensional Poincaré Inequalities

Seng-Kee Chua and Richard L. Wheeden
Proceedings of the American Mathematical Society
Vol. 134, No. 8 (Aug., 2006), pp. 2309-2316
Stable URL: http://www.jstor.org/stable/4098269
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
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 $\leftVert f - f_{av} \rightVert_{L^1[a, b]} \leq C(b - a)^{2-{\frac{1}{p}}} \leftVert f^\prime \rightVert_{L^p[a, b]}$ holds for all Lipschitz continuous functions f, with $f_{av} = \int_a^b f/(b - a)$, is $C = {\frac{1}{2}}(1 + p^\prime)^{-1/p^\prime}$.

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