Research proof index

Let $u_1\ne u_2$ be forest tops with $\mathcal I(u_1)\le\mathcal I(u_2)$. If a tile $p$ belongs to either tree and its spatial cube overlaps $\mathcal I(u_1)$, then the two top frequencies are separated at the scale of $p$ by

$2^{Zn/2}\le d_p(\mathcal Q(u_1),\mathcal Q(u_2)).$

Fix a cube $J$ in the forest’s remaining-cube family. Consider grid cubes $I$ at relative scale $s(J)-s'$, disjoint from the top cube $\mathcal I(u_1)$, whose enlarged balls meet $J$. The normalized average over $J$ of the square of the number of those enlarged balls containing each point is bounded by the scale-dependent constant $C(a,s')$.

For a tree in the forest, a point $x$ in a boundary cube $L$, and a bounded compactly supported function $f$, the oscillatory sum of the error $f-\operatorname{approxOnCube}(f)$ over the relevant scales is bounded pointwise by a constant times the tree’s boundary operator applied to the cube-wise approximation of $|f|$, evaluated at any other point $x'\in L$.

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. Assume $a\ge4$ and that the truncated Calderón-Zygmund operators $T_r$ satisfy the required uniform strong $L^2$ estimate for every $r>0$. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$, then the two-sided metric Carleson operator satisfies

$\int_G^+ \mathcal C_K f(x)\,dx\le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

Assume the phase space is countable, $a\ge4$, $1<q<2$, and every truncated Calderón-Zygmund operator $T_r$ has the required uniform strong $L^2$ bound. Then the two-sided metric Carleson operator is bounded from Lorentz $L^{q,1}$ to weak Lorentz $L^{q,\infty}$, with the explicit project constant $4C(a,q)/q$.

Let $\varphi$ be $K$-Lipschitz and bounded in norm by $B$ on $(a,b)$. For every integer frequency $n$,

$\left\lVert\int_a^b e^{inx}\varphi(x)\,dx\right\rVert \le 2\pi(b-a)\left(B+\frac{K(b-a)}2\right)\left(1+|n|(b-a)\right)^{-1}.$