The circle maximal function and nontangential approach regions

Assume countable choice. Identify the torus $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ with the Euclidean unit circle through $\phi ([t])=e^{2\pi it}$, and write $m$ for the normalized Haar measure of $\mathbb{T}$ (The one-dimensional torus and its normalized Haar integral), a probability measure on the Borel sets of $\mathbb{T}$. For $\zeta ,\eta \in \mathbb{T}$ put $d(\zeta ,\eta ):=\min \{\mathrm{\mid }s-t-k\mathrm{\mid }:k\in \mathbb{Z}\},\zeta =[t],\eta =[s],$ the circular distance; it is well defined because replacing $t$ or $s$ by an integer translate does not change the set of numbers $\mathrm{\mid }s-t-k\mathrm{\mid }$.

Centered arcs. For $\zeta \in \mathbb{T}$ and $0<h\le \frac{1}{2}$ put $I_h(\zeta ):=\{\eta \in \mathbb{T}:d(\zeta ,\eta )<h\}(0<h<\frac{1}{2}),I_{1\mathrm{/}2}(\zeta ):=\mathbb{T}\mathrm{.}$ Thus $I_h(\zeta )$ is the open circular arc of radius $h$ centered at $\zeta$, the point $\zeta$ itself included, and the half-circle case is deliberately the whole circle so that the antipode is not lost. The normalization is the one used throughout this pair: for $0<h\le \frac{1}{2}$, $m(I_h(\zeta ))=2h\mathrm{.}$ For $0<h<\frac{1}{2}$ one has $I_h([0])=q((-h,h))$ for the quotient map $q:\mathbb{R}\to \mathbb{T}$, and $q^{-1}q((-h,h))={\bigcup }_{k\in \mathbb{Z}}(-h+k,h+k)$ meets $[0,1)$ in $[0,h)\cup (1-h,1)$, a set of measure $2h$; translation invariance of $m$ (proved with the measure $m$ in The one-dimensional torus and its normalized Haar integral) moves this identity to every center. The case $h=\frac{1}{2}$ reads $m(\mathbb{T})=1=2\cdot \frac{1}{2}$ by the convention $I_{1\mathrm{/}2}(\zeta )=\mathbb{T}$.

Circle maximal function. Let $\mu$ be a finite regular complex Borel measure on $\mathbb{T}$ (Regular complex Borel measures), with total variation $\mathrm{\mid }\mu \mathrm{\mid }$ (The total variation |nu|(E) from countable measurable partitions, The total variation of a signed or complex measure is a positive measure). Define $M_{\mathbb{T}}\mu (\zeta ):={\sup }_{0<h\le 1\mathrm{/}2}\frac{\mathrm{\mid }\mu \mathrm{\mid }(I_h(\zeta ))}{m(I_h(\zeta ))},\zeta \in \mathbb{T}\mathrm{.}$ Each quotient is finite because $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})<+\mathrm{\infty }$, and each is nonnegative. Their supremum is therefore a well-defined extended nonnegative real: $M_{\mathbb{T}}\mu :\mathbb{T}\to [0,+\mathrm{\infty }]$. It may equal $+\mathrm{\infty }$, for example at an atom of $\mathrm{\mid }\mu \mathrm{\mid }$. For $f\in L^1(\mathbb{T},m)$ (The class $L^1(\mu )$ of integrable functions) define likewise $M_{\mathbb{T}}f(\zeta ):={\sup }_{0<h\le 1\mathrm{/}2}\frac{1}{m(I_h(\zeta ))}{\int }_{I_h(\zeta )}\mathrm{\mid }f\mathrm{\mid }dm\mathrm{.}$ The two definitions agree when $\mu =fm$ is the density measure of $f$, that is when $\mu (E)={\int }_{E}fdm$: then $\mathrm{\mid }\mu \mathrm{\mid }(E)={\int }_E\mathrm{\mid }f\mathrm{\mid }dm$ (A complex L^1 density defines a complex measure whose total variation is |h| dmu), so $\mathrm{\mid }\mu \mathrm{\mid }(I_h(\zeta ))={\int }_{I_h(\zeta )}\mathrm{\mid }f\mathrm{\mid }dm$ and $M_{\mathbb{T}}\mu =M_{\mathbb{T}}f$. The assignment $f\mapsto M_{\mathbb{T}}f$ is unchanged if $f$ is replaced by an almost-everywhere equal function, because the integrals over the arc agree.

Nontangential regions. For $A>1$ and $\zeta \in \mathbb{T}$ put ${\mathrm{\Gamma }}_A(\zeta ):=\{z\in \mathbb{D}:\mathrm{\mid }z-\zeta \mathrm{\mid }<A(1-\mathrm{\mid }z\mathrm{\mid })\}\subseteq \mathbb{D},$ and for $v:\mathbb{D}\to \mathbb{C}$ let $N_{A}v(\zeta ):={\sup }_{z\in {\mathrm{\Gamma }}_A(\zeta )}\mathrm{\mid }v(z)\mathrm{\mid }\in [0,+\mathrm{\infty }]\mathrm{.}$ Here $\mathrm{\mid }z-\zeta \mathrm{\mid }$ is the Euclidean modulus after identifying $\mathbb{T}$ with the unit circle. The sets are nested: ${\mathrm{\Gamma }}_A(\zeta )\subseteq {\mathrm{\Gamma }}_B(\zeta )$ for $1<A\le B$, and $0\in {\mathrm{\Gamma }}_A(\zeta )$ for every $A>1$ because $\mathrm{\mid }0-\zeta \mathrm{\mid }=1<A$. A point $z\in \mathbb{D}\setminus \{\zeta \}$ lies in ${\mathrm{\Gamma }}_A(\zeta )$ as soon as $A>\mathrm{\mid }z-\zeta \mathrm{\mid }\mathrm{/}(1-\mathrm{\mid }z\mathrm{\mid })$, and $\mathrm{\mid }z-\zeta \mathrm{\mid }\ge 1-\mathrm{\mid }z\mathrm{\mid }$ for every $z\in \mathbb{D}$, so every such $z$ lies in some ${\mathrm{\Gamma }}_A(\zeta )$ and ${\bigcup }_{A>1}{\mathrm{\Gamma }}_A(\zeta )=\mathbb{D}\setminus \{\zeta \}$. A complex-valued $v$ on $\mathbb{D}$ has nontangential limit $L$ at $\zeta$ if for every $A>1$ and every $\epsilon >0$ there is $\delta >0$ with $\mathrm{\mid }v(z)-L\mathrm{\mid }<\epsilon$ whenever $z\in {\mathrm{\Gamma }}_A(\zeta )$ and $\mathrm{\mid }z-\zeta \mathrm{\mid }<\delta$. Because the regions increase with $A$, it suffices to verify this for every integer $A\ge 2$: an arbitrary $A>1$ satisfies ${\mathrm{\Gamma }}_A(\zeta )\subseteq {\mathrm{\Gamma }}_m(\zeta )$ for every integer $m\ge A$.