Monotonicity of Beckner Inequalities

In this paper, we prove a monotonicity principle for Beckner inequalities of the form $$\frac{\mu (f^2)-\mu (f^p{)}^{2\mathrm{/}p}}{2-p}\le C\text{E}(f,f),f\in \text{D}(\text{E}),1\le p\le 2,$$ where $\text{E}$ 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 $${\mathrm{\Psi }}_p(f)\le \beta (r)\text{E}(f,f)+r\text{Osc}(f{)}^2,f\in \text{D}(\text{E}),r>0,$$ where $\beta$ 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.