Research proof index

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 $R=R_1$.

Let $1<q\le2$, let $q'$ be its Hölder conjugate, and suppose the linearized nontangential operators have the required uniform $L^2$ bound. For bounded measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf 1_F(x)$, the supremum over all truncated scale intervals $-B\le s_1\le s_2\le B$ satisfies

$\int_G^+\sup_{s_1\le s_2}\lVert T_{s_1,s_2}f(x)\rVert\,dx\le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

The stopped valued of the predictable part with respect to τₙ(c) is less than or equal to c.

The mesh step-extension of the discrete predictable part is strongly adapted.

We combine limsup_le_of_eventually_monotone_of_tendsto_on_dense and le_liminf_of_eventually_monotone_of_tendsto_on_dense to prove that F · a converges to f a if f is continuous at a.

For $x\in G\setminus G'$, summing the localized Carleson contribution over all tiles is exactly the same as summing the corresponding oscillatory kernel integral over the integer scales from $\sigma_1(x)$ to $\sigma_2(x)$. This is the identity that converts the discrete tile model back into the finitary operator.