blueprint: fixed small typos

blueprint/src/chapter/main.tex

@@ -969,12 +969,7 @@ \chapter{Proof of Metric Space Carleson}
 \left| {T}_{2,\sigma_1,\sigma_2, \tQ}f(x) \right|\, d\mu(x)
 \le \frac{2^{445a^3}}{(q-1)^6} \mu(G)^{1-\frac{1}{q}} \mu(F)^{\frac{1}{q}}.
 \end{equation}
-As the integrand is bounded by
-\begin{equation}\mathbf{1}_{G\setminus G_{n}}(x)
-\sum_{-S<s_1\le s_2<S}\sum_{\mfa\in\tilde{\Mf}}
-\left| {T}_{1,s_1, s_2, \mfa} f(x) \right|,
-\end{equation}
-which by interchange of summation and integration is seen to be integrable, we obtain by Lebesgue's dominated convergence theorem
+As the integrand is non-negative and non-decreasing in $n$, we obtain by the monotone convergence theorem
  \begin{equation} \label{Sqcut3}
     \int \mathbf{1}_{G}(x)
 \left| {T}_{2,\sigma_1,\sigma_2, \tQ}f(x) \right|\, d\mu(x)
@@ -4522,7 +4517,7 @@ \section{The quantitative \texorpdfstring{$L^2$}{L2} tree estimate}
     \label{local-dens2-tree-bound}
     Let $J \in \mathcal{J}(\fT(\fu))$ be such that there exist $\fq \in \fT(\fu)$ with $J \cap \scI(\fq) \ne \emptyset$. Then
     $$
-        \mu(F \cap J) \le 2^{200a^3 + 19} \dens_2(\fT(\fu))\,.
+        \mu(F \cap J) \le 2^{200a^3 + 19} \dens_2(\fT(\fu)) \mu(J)\,.
     $$
 \end{lemma}
 
@@ -4579,7 +4574,7 @@ \section{The quantitative \texorpdfstring{$L^2$}{L2} tree estimate}
     $$
         \le 2^{100a^3 + 10} \dens_2(\fT(\fu))^{1/2} \|f\|_2\,.
     $$
-    Combining this with \eqref{eq-both-factors-tree}, \eqref{eq-factor-L-tree} and $a \ge 4$ gives \eqref{eq-cor-tree-est-F}.
+    Combining this with \eqref{eq-both-factors-tree}, \eqref{eq-cor-tree-proof} and $a \ge 4$ gives \eqref{eq-cor-tree-est-F}.
 \end{proof}
 
 Now we prove the two auxiliary estimates.
@@ -4780,7 +4775,9 @@ \section{Almost orthogonality of separated trees}
          \mathfrak{S} := \{\fp \in \fT(\fu_1) \cup \fT(\fu_2) \ : \ d_{\fp}(\fcc(\fu_1), \fcc(\fu_2)) \ge 2^{Zn/2}\,\}.
     \end{equation}
     \Cref{correlation-separated-trees} follows by combining the definition \eqref{defineZ} of $Z$ with the following two lemmas.
-    \begin{lemma}[correlation distant tree parts]
+\end{proof}
+
+\begin{lemma}[correlation distant tree parts]
         \label{correlation-distant-tree-parts}
         \uses{Holder-van-der-Corput,Lipschitz-partition-unity,Holder-correlation-tree,lower-oscillation-bound}
         We have for all $\fu_1 \ne \fu_2 \in \fU$ with $\scI(\fu_1) \subset \scI(\fu_2)$ and all bounded $g_1, g_2$ with bounded support
@@ -4806,7 +4803,6 @@ \section{Almost orthogonality of separated trees}
             \le 2^{222a^3} 2^{-Zn 2^{-10a}} \prod_{j =1}^2 \| S_{2, \fu_j} g_j\|_{L^2(\scI(\fu_1)}\,.
         \end{equation}
     \end{lemma}
-\end{proof}
 
 In the proofs of both lemmas, we will need the following observation.
 
@@ -5129,7 +5125,7 @@ \subsection{H\"older estimates for adjoint tree operators}
         $$
         Then
         $$
-            s(J) \le \ps(\fp) \le s(J) + 10a^2 + 2\,.
+            s(J) \le \ps(\fp) \le s(J) +3\,.
         $$
     \end{lemma}
 
@@ -5145,24 +5141,24 @@ \subsection{H\"older estimates for adjoint tree operators}
         $$
         by our assumption. Thus $D^{\ps(\fp)} \ge 64 D^{s(J)}$, which contradicts \eqref{defineD} and $a \ge 4$.
 
-        For the second estimate, assume that $\ps(\fp) > s(J) +10a^2 + 2$. Since $J \in \mathcal{J}'$, we have $J \subsetneq \scI(\fu_1)$. Thus there exists $J' \in \mathcal{D}$ with $J \subset J'$ and $s(J') = s(J) + 1$, by \eqref{coverdyadic} and \eqref{dyadicproperty}. By definition of $\mathcal{J}'$, there exists some $\fp' \in \mathfrak{S}$ such that $\scI(\fp') \subset B(c(J'), 100 D^{s(J) + 2})$. On the other hand, since $B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset$, by the triangle inequality it holds that
+        For the second estimate, assume that $\ps(\fp) > s(J) + 3$. Since $J \in \mathcal{J}'$, we have $J \subsetneq \scI(\fu_1)$. Thus there exists $J' \in \mathcal{D}$ with $J \subset J'$ and $s(J') = s(J) + 1$, by \eqref{coverdyadic} and \eqref{dyadicproperty}. By definition of $\mathcal{J}'$, there exists some $\fp' \in \mathfrak{S}$ such that $\scI(\fp') \subset B(c(J'), 100 D^{s(J) + 2})$. On the other hand, since $B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset$, by the triangle inequality it holds that
         $$
-            B(c(J'), 100 D^{s(J) + 10a^2 + 2}) \subset B(\pc(\fp), 10 D^{\ps(\fp)})\,.
+            B(c(J'), 100 D^{s(J) + 3}) \subset B(\pc(\fp), 10 D^{\ps(\fp)})\,.
         $$
         Using the definition of $\mathfrak{S}$, we have
         $$
             2^{Zn/2} \le d_{\fp'}(\fcc(\fu_1), \fcc(\fu_2)) \le d_{B(c(J'), 100 D^{s(J) + 2})}(\fcc(\fu_1), \fcc(\fu_2))\,.
         $$
         By \eqref{seconddb}, this is
         $$
-            \le 2^{-10a} d_{B(c(J'), 100 D^{s(J) + 10a^2 + 2})}(\fcc(\fu_1), \fcc(\fu_2))
+            \le 2^{-100a} d_{B(c(J'), 100 D^{s(J) + 3})}(\fcc(\fu_1), \fcc(\fu_2))
         $$
         $$
-            \le 2^{-10a} d_{B(\pc(\fp), 10 D^{\ps(\fp)})}(\fcc(\fu_1), \fcc(\fu_2))\,,
+            \le 2^{-100a} d_{B(\pc(\fp), 10 D^{\ps(\fp)})}(\fcc(\fu_1), \fcc(\fu_2))\,,
         $$
         and by \eqref{firstdb} and the definition of $\mathfrak{S}$
         $$
-            \le 2^{-4a} d_{\fp}(\fcc(\fu_1), \fcc(\fu_2)) \le 2^{-4a} 2^{Zn/2}\,.
+            \le 2^{-94a} d_{\fp}(\fcc(\fu_1), \fcc(\fu_2)) \le 2^{-94a} 2^{Zn/2}\,.
         $$
         This is a contradiction, the second estimate follows.
     \end{proof}
@@ -5186,7 +5182,7 @@ \subsection{H\"older estimates for adjoint tree operators}
         By \Cref{limited-scale-impact}, this is at most
         \begin{equation}
             \label{eq-sep-tree-aux-3}
-            \sum_{s = s(J)}^{s(J) + 10a^2 + 2} \sum_{\substack{\fp \in \fP, \ps(\fp) = s\\ B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset}} \sup_{B^\circ{}(J)} |T_{\fp}^* g|\,.
+            \sum_{s = s(J)}^{s(J) + 3} \sum_{\substack{\fp \in \fP, \ps(\fp) = s\\ B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset}} \sup_{B^\circ{}(J)} |T_{\fp}^* g|\,.
         \end{equation}
         If $x \in E(\fp)$ and $B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset$, then
         $$
@@ -5198,16 +5194,16 @@ \subsection{H\"older estimates for adjoint tree operators}
         $$
         Using \eqref{definetp*}, \eqref{eq-Ks-size} and that $a \ge 4$, we bound \eqref{eq-sep-tree-aux-3} by
         $$
-            2^{103a^3}\sum_{s = s(J)}^{s(J) + 10a^2 + 2} \sum_{\substack{\fp \in \fP, \ps(\fp) = s\\B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset}} \frac{1}{\mu(B(c(J), 16 D^s)} \int_{E(\fp)} |g| \, \mathrm{d}\mu\,.
+            2^{103a^3}\sum_{s = s(J)}^{s(J) + 3} \sum_{\substack{\fp \in \fP, \ps(\fp) = s\\B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset}} \frac{1}{\mu(B(c(J), 16 D^s)} \int_{E(\fp)} |g| \, \mathrm{d}\mu\,.
         $$
         For each $I \in \mathcal{D}$, the sets $E(\fp)$ for $\fp \in \fP$ with $\scI(\fp) = I$ are pairwise disjoint by \eqref{defineep} and \eqref{eq-dis-freq-cover}. Further, if $B(\scI(\fp)) \cap B^\circ(J) \ne \emptyset$ and $\ps(\fp) \ge s(J)$, then $E(\fp) \subset B(c(J), 32 D^{\ps(\fp)})$. Thus the last display is bounded by
         $$
-            2^{103a^3}\sum_{s = s(J)}^{s(J) + 10a^2 + 2} \frac{1}{\mu(B(c(J), 32 D^s))} \int_{B(c(J), 16 D^s)} |g| \, \mathrm{d}\mu\,.
+            2^{103a^3}\sum_{s = s(J)}^{s(J) + 3} \frac{1}{\mu(B(c(J), 32 D^s))} \int_{B(c(J), 16 D^s)} |g| \, \mathrm{d}\mu\,.
         $$
         $$
-            \le \inf_{x' \in J} 2^{103a^3}(10a^2 + 3) M_{\mathcal{B}, 1} |g|\,.
+            \le \inf_{x' \in J} 2^{103a^3 +2} M_{\mathcal{B}, 1} |g|\,.
         $$
-        The lemma follows since for $a \ge 4$ it holds that $10a^2 + 3 \le 2^{a^3}$.
+        The lemma follows since $a \ge 4$.
     \end{proof}
 
     \begin{lemma}[scales impacting interval]