Research proof index

For every bounded, finitely supported function $f$, almost every point $x$ admits a sequence of balls $B(c_i,r_i)$ that all contain $x$, whose radii tend to $0$ from above, and whose averages converge to $f(x)$:

$\dashint_{B(c_i,r_i)} f(y)\,dy\longrightarrow f(x).$

Convergence on a dense set of a collection of monotone function controls the limsup at a point if f is right continuous at a. We prove this under the assumption that α has both a bottom element and a top element. The bottom element is needed because otherwise limsup evaluated at the bottome element may give a junk value to break the inequality.

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. If every phase-linearized nontangential operator has the required uniform $L^2$ bound, then for measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, the linearized Carleson operator satisfies

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

Let $\mathfrak A$ be an antichain contained in the tile class $\mathfrak C(k,n)$, and let $f$ be measurable with $\lVert f(x)\rVert\le\mathbf 1_F(x)$. On $G\setminus G'$, the Carleson sum over the selected positive tiles in $\mathfrak A$ has an $L^1$ bound with exponential decay in the layer index $n$:

$\int_{G\setminus G'}^+\!\lVert\operatorname{CarlesonSum}f(x)\rVert\,dx\le C(a,q)\,\mu(G)^{1-1/q}\mu(F)^{1/q}\,2^{-((q-1)/(8a^4))n}.$

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 $R$.

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 $8D^{\mathfrak{s}(p)}$.