Research proof index

For measurable $f$ with $\lVert f(x)\rVert\le1$, the scale-$n$ quantity lcoConvergent, which records a supremum of truncated linearized Carleson integrals over rational radii, is a measurable function of $x$.

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. In the project’s cancellative metric-space setting, assume the associated nontangential operators satisfy the required uniform $L^2$ bound. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, then the Carleson operator obeys the restricted estimate

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

For $0\le t\le1$, the reciprocal of $1-2^{-t}$ is controlled in the extended nonnegative reals by

$(1-2^{-t})^{-1}\le 2t^{-1}.$

For an $L^2$ function on a circle of period $T$, the squared $L^2$ norm of its $N$th partial Fourier sum equals the sum of the squared magnitudes of its Fourier coefficients from $-N$ through $N$:

$\lVert S_Nf\rVert_2^2=\sum_{n=-N}^{N}|\widehat f(n)|^2.$

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. For measurable sets $F,G\subseteq\mathbb{R}$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$, the real-line Carleson operator satisfies

$\int_G^+ T f(x)\,dx \le C(q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

If $f$ is integrable on the open annulus $\{y:R_1<d(x,y)<R_2\}$, then varying the inner radius from the right changes the annular integral continuously at $R_1$:

$R\longmapsto\int_{R<d(x,y)<R_2}f(y)\,dy$

is right-continuous at $R=R_1$.