Research proof index

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 $R^t$.

If the set is inf-closed, elements of countablInfClosure can be written as countable intersections of antitone sequences of sets.

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.

The set of large left jump times has no accumulation points. TODO: maybe to_dual can be extended to simplify this proof as the proof of the second part is very similar to the first part.