Research proof index

If g is continuous on (0, ∞), positive for t ≥ t₀, and has strictly negative derivative at any zero in (0, t₀), then g is positive on all of (0, ∞).

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 $k\in\mathbb N$.

Derivative bounds for q-expansion coefficients. Given ‖a n‖ ≤ n^k, produces bounds ‖a n * 2πin * exp(2πin z)‖ ≤ 2π * n^(k+1) * exp(-2πn * y_min) on compact K ⊆ {z : 0 < z.im}. This is a key hypothesis for D_qexp_tsum_pnat.

Serre derivative of E₆: serre_D 6 E₆ = - 2⁻¹ * E₄². Uses the dimension argument: 1. serre_D 6 E₆ is weight-8 slash-invariant (by serre_D_slash_invariant) 2. Weight-8 modular forms are 1-dimensional, spanned by E₄² 3. Constant term is -1/2 (from D E₆ → 0, E₂ → 1, E₆ → 1)

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. For measurable sets $F,G\subseteq\mathbb{R}$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$, the real-line Carleson operator satisfies

$\int_G^+ T f(x)\,dx \le C(q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

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.