Research proof index

A positivity lemma for self-difference-convolutions. If $f=g\mathbin{\circleddash}g$ and the nonnegative weight $\nu$ has a factorization $\nu=h\mathbin{\circleddash}h$, then every natural power of $f$ has nonnegative weighted inner product with $\nu$: $\langle f^k,\nu\rangle\ge0$ for every $k\in\mathbb N$.

An unbalancing lemma in physical space. Suppose $\nu$ is a probability weight, $f$ is real-valued, and $f$ and $\nu$ admit self-difference-convolution factorizations $f=g\mathbin{\circleddash}g$ and $\nu=h\mathbin{\circleddash}h$. If $0<\varepsilon\le1$, $p\ne0$, and $\|f\|_{L^p(\nu)}\ge\varepsilon$, then some integer $p'$ satisfies $p'\le 2^{10}\varepsilon^{-2}p$ and $\|1+f\|_{L^{p'}(\nu)}\ge1+\varepsilon/2$.