fixes

nota5.log

@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/nixos.org) (preloaded format=pdflatex 1980.1.1)  18 NOV 2024 15:45
+This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/nixos.org) (preloaded format=pdflatex 1980.1.1)  19 NOV 2024 10:57
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@@ -606,20 +606,15 @@ File: ulasy.fd 1998/08/17 v2.2e LaTeX symbol font definitions
 
 {/nix/store/mhdx3hkmpns8i6czk9ygf2yc7wxlkpy0-texlive-combined-full-2023-final/s
 hare/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] Excluding 'comment' comment
-. Excluding 'comment' comment.
-Overfull \hbox (82.38083pt too wide) detected at line 184
-[]
- []
-
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+. Excluding 'comment' comment. [2]
 LaTeX Font Info:    Trying to load font information for U+dsrom on input line 2
-24.
+25.
 
 (/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
 exmfdist/tex/latex/doublestroke/Udsrom.fd
 File: Udsrom.fd 1995/08/01 v0.1 Double stroke roman font definitions
 )
-Overfull \hbox (29.63692pt too wide) detected at line 235
+Overfull \hbox (29.63692pt too wide) detected at line 236
 []
  []
 
@@ -677,7 +672,7 @@ ublic/doublestroke/dsrom10.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-tex
 live-combined-full-2023-final-texmfdist/fonts/type1/public/amsfonts/symbols/msb
 m10.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023
 -final-texmfdist/fonts/type1/public/amsfonts/symbols/msbm7.pfb>
-Output written on nota5.pdf (6 pages, 281177 bytes).
+Output written on nota5.pdf (6 pages, 281244 bytes).
 PDF statistics:
  216 PDF objects out of 1000 (max. 8388607)
  145 compressed objects within 2 object streams

nota5.tex

@@ -178,7 +178,8 @@ Let's first tackle the case with $f$ Lipschitz. Let $x\in\R^N$. We have
 The second inequality for the Lipschitz case is done in a completely analogous manner. Let's consider the Holder case
 \end{comment}
 \begin{align*}
-\left| \partial_{x_ix_j}V^{s,t}_\nu f (x) \right|&=\left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)f(y)dy \right|\leq \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)(f(y)-f(e^{(t-s)B}x))dy \right|+ \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)dyf(e^{(t-s)B}x) \right|\\
+\left| \partial_{x_ix_j}V^{s,t}_\nu f (x) \right|&=\left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)f(y)dy \right| \\
+&\leq \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)(f(y)-f(e^{(t-s)B}x))dy \right|+ \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)dyf(e^{(t-s)B}x) \right|\\
 &\leq \int |\partial_{x_ix_j}p^\nu(s,x;t,y)||e^{(t-s)B}x-y|^\alpha_B dy + \left| \partial_{x_ix_j}\underbrace{\int p^\nu(s,x;t,y) dy}_{=1} \right||f(e^{(t-s)B}x)|\\
 &\leq \frac{C_{B,\alpha}}{|t-s|}\int \Gamma^+(t-s,x-y)|x-e^{-(t-s)B}y|^\alpha_B dy + 0\leq \frac{C_{B,\alpha}}{|t-s|^{1-\frac{\alpha}{2}}}.
 \end{align*}
@@ -219,7 +220,7 @@ Now if we consider $d_B(x,y)=|x-y|_B$ and $K^\epsilon=\left\lbrace x\in\R^N\ |\
 $$
 \sup_{x\in K^\epsilon}|f(x)-f_j(x)|\leq \sup_{x\in K^\epsilon}\left( |f(x)-f(y_x)| + |f(y_x)-f_j(y_x)| + |f_j(y_x)-f_j(x)|\right)\leq \sup_{x\in K^\epsilon}\left( 2\epsilon^\alpha + \epsilon \right)\leq C_\alpha\epsilon^\alpha.
 $$
-where $y_x$ is a point in $K$ such that $|x-y|_B<\epsilon$. $C_\alpha$ may be taken uniformly of $\epsilon$ as long as $\epsilon\leq 1$.
+where $y_x$ is a point in $K$ such that $|x-y_x|_B<\epsilon$. $C_\alpha$ may be taken uniformly of $\epsilon$ as long as $\epsilon\leq 1$.
 
 Let $g(x)=\max\left( 0, 1-\frac{d_B(x,K)}{\epsilon} \right)$, evidently $g\in bLip\subseteq C^{\alpha}_B$ and $\mathds{1}_K\leq g\leq \mathds{1}_{K^\epsilon}$. Thus by taking $n$ big enough we have by convergence against $C^{\alpha}_B$ functions that
 $$