From 055b3c3268b3bb7129bb708fe5616d6482d97d4d Mon Sep 17 00:00:00 2001 From: Alessio Rondelli Date: Tue, 19 Nov 2024 11:07:23 +0100 Subject: [PATCH] fixes --- nota5.log | 15 +++++---------- nota5.pdf | Bin 281177 -> 281244 bytes nota5.synctex.gz | Bin 81558 -> 81669 bytes nota5.tex | 5 +++-- 4 files changed, 8 insertions(+), 12 deletions(-) diff --git a/nota5.log b/nota5.log index 3d43c95..437d386 100644 --- a/nota5.log +++ b/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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -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 -[] - [] - -[2] +. 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> -Output written on nota5.pdf (6 pages, 281177 bytes). +Output written on nota5.pdf (6 pages, 281244 bytes). 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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 $$