Monotonicity of Beckner Inequalities

In this paper, we prove a monotonicity principle for Beckner inequalities of the form \[ \frac{μ(f^2)-μ(f^p)^{2/p}}{2-p}\leq C\E(f,f), \qquad f\in\D(\E),\quad 1\le p\le2, \] where $\E$ is a conservative symmetric Dirichlet form. If the inequality holds for every exponent $p$, it holds with the same constant $C$ for all $q\in[1,p]$. This resolves an open question posed in Chapter~6 of Wang's monograph~\cite{Wangbook}. We also study weak Beckner inequalities of the form \[ Ψ_p(f)\le β(r)\E(f,f)+r\,\Osc(f)^2,\qquad f\in\D(\E),\quad r>0, \] where $β$ is a rate function. We show that the same monotonicity principle preserves the entire rate function. We further derive consequences for polynomial rates, semigroup contractivity, and concentration.