Log, Sobolev Inequalities on Cyclic Groups Achieve Optimal Constant 2 Via Hypercontractivity Analysis

Hypercontractivity and Log-Sobolev inequalities represent fundamental concepts in mathematics with broad implications for fields like probability and analysis, and recent work by Gan Yao, alongside collaborators, significantly advances our understanding of these inequalities on cyclic groups. The team proves that a specific semigroup, linked to word length on these groups, exhibits hypercontractivity between certain spaces if and only if a defined condition holds true. This achievement establishes Log-Sobolev inequalities with an optimal constant, accomplished through a sophisticated KKT analysis and a clever comparison of Dirichlet forms, and represents a substantial step forward in the field, although the most general case remains an open challenge for future research.

Circulant Matrices and Hypercontractivity Bounds

Scientists have investigated logarithmic Sobolev inequalities and hypercontractivity within harmonic analysis and probability, focusing on circulant matrices and their connection to Fourier analysis. These inequalities provide crucial bounds on the rate of convergence of processes like Fourier transforms and heat semigroups, and this work establishes them for operators associated with circulant matrices, extending existing results. The research leverages the close relationship between circulant matrices and the discrete Fourier transform, translating properties of the transform into bounds on operator behavior, and highlights the importance of specific functions in establishing the necessary conditions for these inequalities.

Optimal Contraction Times for Discrete Semigroups

Researchers have established a precise relationship between hypercontractivity and logarithmic Sobolev inequalities for a specific semigroup on discrete spaces, demonstrating conditions for quantifiable function contraction on spaces with dimensions of the form 3⋅2 k and 2 k , where k is a positive integer. The team proved that a specific contraction condition holds if and only if the time parameter, t, is greater than or equal to 1/2 log((q-1)/(p-1)), where p and q define the function spaces involved, establishing the optimal time scale for this contraction. Rigorous proof of this hypercontractivity was achieved by establishing corresponding logarithmic Sobolev inequalities with a constant of 2, confirming the tightest possible bound for these spaces, and a novel inductive scheme, involving auxiliary weights on spaces of dimension 4 and 6, provided tighter bounds than those based solely on standard length functions.

Optimal Hypercontractivity Condition Fully Characterized

This research establishes a precise condition for hypercontractivity of a specific semigroup associated with word length on a mathematical space, demonstrating that hypercontractivity holds if and only if a certain inequality is satisfied. Rigorous proof was achieved by employing Karush, Kuhn, Tucker analysis and leveraging comparisons of Dirichlet forms, building from established base cases using a technique analogous to the Cooley-Tukey algorithm, and advances understanding of hypercontractivity, a concept with implications for concentration of measure and probabilistic inequalities. The findings contribute to a deeper understanding of logarithmic Sobolev inequalities, demonstrating the optimal constant achievable under the established conditions, and these inequalities are fundamental in analysis and probability, providing bounds on the rate of convergence of probability measures.