Research proof index

Translation preserves the discrete normalized $L^p$ norm: for every group element $a$, $\|\tau_a f\|_{p}=\|f\|_{p}$.

Hölder's inequality, binary case.

Hölder's inequality, binary case.

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 $k\in\mathbb N$.

The Marcinkiewicz-Zygmund inequality for real-valued functions, with a slightly easier to bound constant than Real.marcinkiewicz_zygmund'. Note that RCLike.marcinkiewicz_zygmund is another version that works for both ℝ and ℂ at the expense of a slightly worse constant.

The Marcinkiewicz-Zygmund inequality for real-valued functions, with a slightly better constant than Real.marcinkiewicz_zygmund.