Research proof index

If the project’s inhomogeneous Lipschitz norm of $\varphi$ on the ball $B(z,R)$ is finite, then $\varphi$ is Lipschitz on that ball. A valid Lipschitz constant is the finite normalized norm divided by $R$.

If F z / exp(a·z) → C ≠ 0 at i∞, then D F / F → a/(2πi): the exponential contributes a/(2πi) and the bounded limit factor's log-derivative vanishes. Public so downstream files (e.g. #331's Θ₂ analysis) can reuse it.

At every point $x$, the magnitude of the Carleson sum over an antichain of tiles is bounded by a constant depending on $a$ times a maximal function of $f$. The maximal function uses, for each tile $p$, a ball centered at the tile center with radius $8D^{\mathfrak{s}(p)}$.

Division of MDifferentiable functions on ℍ is MDifferentiable, when the denominator is everywhere nonzero.

For measurable $f$ with $\lVert f(x)\rVert\le1$, the scale-$n$ quantity lcoConvergent, which records a supremum of truncated linearized Carleson integrals over rational radii, is a measurable function of $x$.

The compact normalized $L^p$ norm is unchanged by conjugating a function and reflecting its argument: $\|x\mapsto\overline{f(-x)}\|_p=\|f\|_p$.