Research proof index

If the project’s inhomogeneous $t$-Hölder norm of $\varphi$ on the ball $B(z,R)$ is finite and $t\ge0$, then $\varphi$ is $t$-Hölder on that ball. A valid Hölder constant is the finite normalized norm divided by $R^t$.

In the function-distance space used for the real-line Carleson argument, every ball of radius $2R'$ can be covered by at most three balls of radius $R'$.

The function y ↦ Ks s x y is integrable.

If F has a Fourier expansion ∑_{m≥0} a_m exp(2πi(m+n₀)z) with n₀ > 0, and the coefficients are absolutely summable at height im z = c, then F = O(exp(-2π n₀ · im z)) at atImInfty. The key bound is: for im z ≥ c, ‖F(z)‖ ≤ (∑_m ‖a_m‖ · exp(-2π c m)) · exp(-2π n₀ · im z)

If F : ℍ → ℂ is O(exp(-c * im τ)) at atImInfty for some c > 0, then the restriction to the imaginary axis t ↦ F(it) is O(exp(-c * t)) at atTop.

Suppose the compatible phase system satisfies the following oscillatory cancellation estimate on every ball $B(x,r)$: for every Lipschitz amplitude $\varphi$ supported in the ball and every pair of phases $f,g$,

$\left\lVert\int_{B(x,r)}e^{i(f-g)}\varphi\right\rVert\le A\,\mu(B(x,r))\,\lVert\varphi\rVert_{\mathrm{Lip}}\,(1+d_{x,r}(f,g))^{-\tau}.$

Then the metric phase space satisfies the project’s IsCancellative property with exponent $\tau$.