Research proof index

For $0<y\le1$ and $y/2\le y'\le1$, with both $y$ and $y'$ nonnegative, the negative-side Hilbert kernel satisfies the quantitative regularity estimate

$\lVert k(-y)-k(-y')\rVert \le 2^6\,\frac{1}{|y|}\,\frac{|y-y'|}{|y|}.$

If $\varphi$ is supported in the ball $B(z,R)$, then the oscillatory integral with phase difference $f-g$ satisfies

$\left\lVert\int e^{i(f(x)-g(x))}\varphi(x)\,dx\right\rVert \le C(a)\,\mu(B(z,R))\,\lVert\varphi\rVert_{\mathrm{Hol},\tau;B(z,2R)}\,(1+d_{z,R}(f,g))^{-1/(2a^2+a^3)}.$

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.

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 $\tau$.