Strong Contraction and Influences in Tail Spaces
Strong Contraction and Influences in Tail Spaces
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尾部空间的强烈收缩和影响
DOI：
10.1090/tran/6916
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
K. Oleszkiewicz
中科院分区：
文献类型：
--
作者：
Steven M. Heilman;Elchanan Mossel;K. Oleszkiewicz
We study contraction under a Markov semi-group and influence bounds for functions in $L^2$ tail spaces, i.e. functions all of whose low level Fourier coefficients vanish. It is natural to expect that certain analytic inequalities are stronger for such functions than for general functions in $L^2$. In the positive direction we prove an $L^{p}$ Poincar\'{e} inequality and moment decay estimates for mean $0$ functions and for all $1<p<\infty$, proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. That is, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$, with all influences bounded by $C \log n/n$ for some constants $0<c,C< \infty$. We also construct a function $f\colon\{-1,1\}^{n}\to\{0,1\}$ with nonzero mean whose remaining Fourier coefficients vanish up to level $c' \log n$, with the sum of the influences bounded by $C'(\mathbb{E}f)\log(1/\mathbb{E}f)$ for some constants $0<c',C'<\infty$.
We study contraction under a Markov semi-group and influence bounds for functions in $L^2$ tail spaces, i.e. functions all of whose low level Fourier coefficients vanish. It is natural to expect that certain analytic inequalities are stronger for such functions than for general functions in $L^2$. In the positive direction we prove an $L^{p}$ Poincar\'{e} inequality and moment decay estimates for mean $0$ functions and for all $1<p<\infty$, proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. That is, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$, with all influences bounded by $C \log n/n$ for some constants $0<c,C< \infty$. We also construct a function $f\colon\{-1,1\}^{n}\to\{0,1\}$ with nonzero mean whose remaining Fourier coefficients vanish up to level $c' \log n$, with the sum of the influences bounded by $C'(\mathbb{E}f)\log(1/\mathbb{E}f)$ for some constants $0<c',C'<\infty$.