Hypercontractive Inequality Research Articles

Proof of a hypercontractive estimate via entropy

Consider the probability spaceW={−1, 1} n with the uniform (=product) measure. Letf: W →R be a function. Letf=Σf IXI be its unique expression as a multilinear polynomial whereX I=Π i∈I x i. For 1≤m≤n let\(f_{\hat m} \)=Σ|I|=m f IXI. LetT ɛ (f)=Σf Iɛ|I| X I where 0<ɛ<1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that $$\left| {T_\varepsilon \left( f \right)} \right|_2 \le \left| f \right|_{1 + \varepsilon ^2 } $$ This inequality has been used in several papers dealing with combinatorial and probabilistic problems. It is equivalent to the following inequality via duality: For anyq≥2 $$\left| {f_{\hat m} } \right|_q \le \left( {\sqrt {q - 1} } \right)^m \left| {f_{\hat m} } \right|_2 $$ In this paper we prove a special case with a slightly weaker constant, which is sufficient for most applications. We show $$\left| {f_{\hat m} } \right|_4 \le c^m \left| {f_{\hat m} } \right|_2 $$ where\(c = \sqrt[4]{{28}}\). Our proof uses probabilistic arguments, and a generalization of Shearer’s Entropy Lemma, which is of interest in its own right.

Comment on: “Hypercontractivity of Hamilton–Jacobi equations”, by S. Bobkov, I. Gentil and M. Ledoux

In their remarkable work [1], Bobkov, Gentil and Ledoux improve, generalize and simplify most previously known results about hypercontractivity, logarithmic Sobolev and transport inequalities. Among other results, they are able to recover and generalize the HWI inequalities introduced in [4], (1) DV ≥ KId ⇒ H(f |e−V ) ≤ W (f, e−V ) √ I(f |e−V )− K 2 W (f, e−V ). Here f and g are arbitrary probability distribution functions, H(f |g) ≡ ∫ f log(f/g) is the Kullback relative entropy of f w.r.t g, I(f |g) ≡ ∫ f |∇ log(f/g)|2 is the relative Fisher information of f w.r.t g, and

Trees, Not Cubes: Hypercontractivity, Cosiness, and Noise Stability

Noise sensitivity of functions on the leaves of a binary tree is studied, and a hypercontractive inequality is obtained. We deduce that the spider walk is not noise stable.

On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions

In the line of investigation of the works by D. Bakry and M. Emery (Lecture Notes in Math., Vol. 1123, pp. 175–206, Springer-Verlag, New York/Berlin 1985) and O. S. Rothaus ( J. Funct. Anal. 42 (1981) 102–109; J. Funct. Anal. 65 (1986) 358–367), we study an integral inequality behind the “ Γ 2 criterion” of D. Bakry and M. Emery (see previous reference) and its applications to hypercontractivity of diffusion semigroups. With, in particular, a short proof of the hypercontractivity property of the Ornstein-Uhlenbeck semigroup, our exposition unifies in a simple way several previous results, interpolating smoothly from the spectral gap inequalities to logarithmic Sobolev inequalities and even true Sobolev inequalities. We examine simultaneously the extremal functions for hypercontractivity and logarithmic Sobolev inequalities of the Ornstein-Uhlenbeck semigroup and heat semigroup on spheres.

Hypercontractive operators and Khinchin's inequality

Hypercontractive operators and Khinchin's inequality

Positivity improving operators and hypercontractivity

Here (1.1) is the famous Nelson hypercontractivity inequality [14], which has proved to be extremely useful in Euclidean quantum field theory (see e.g. [16]). Like (1.1) the estimate (1.2) follows very easily from the It6 calculus of Brownian motion [3]. Here, in order to make the paper self-contained and to introduce the reader to some general principles of interest, we submit a new proof of (1.2) by mimicing a standard proof of (1.1). We should like to add that (1.2) has at least a formal resemblance to the familiar correlation inequalities from statistical mechanics (see e.g. [11, 12, 16]). Using (1.1) and (1.2) it is simple to obtain similar estimates for the biquantization of a linear contraction on Hilbert space (Sect. 4). Now suppose A is a self-adjoint operator on L 2 of a probability space and set A(A)=maxspec(A)-sup(spec(A)\{maxspec(A)}), where spec(A) denotes the spectrum of A. Below we will state conditions that guarantee that the gap A (A) is proper. In fact, a simplified version of our main result (Theorem 5.3) gives a positive numerical lower bound of A (A) if (i) A is positivity preserving, (ii) sup { IlZf I1~; [I f [12 = 1 } = K 2, and (iii) inf{ [IZfllp/~v_~; I l f l l , = 1, f>0} = k > 0 some 0 < p < 1. Here (i)-(iii) imply that A has a unique ground state. It shall be empha-

Best constants in Young's inequality, its converse, and its generalization to more than three functions

The best possible constant D pqt in the inequality |∫∫ dx dy f(x)g(x − y) h(y)| ⩽ D pqt | f | p | g | q | h | t, p, q, t ⩾ 1, 1 p + 1 q + 1 t = 2 , is determined; the equality is reached if f, g, and h are appropriate Gaussians. The same is shown to be true for the converse inequality (0 < p, q < 1, t < 0), in which case the inequality is reversed. Furthermore, an analogous property is proved for an integral of k functions over n variables, each function depending on a linear combination of the n variables; some of the functions may be taken to be fixed Gaussians. Two applications are given, one of which is a proof of Nelson's hypercontractive inequality.

Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form

Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form