Skip to content
This repository was archived by the owner on Nov 19, 2017. It is now read-only.

Commit 112d642

Browse files
committed
Gaussian maximal function
1 parent 3a8a8a5 commit 112d642

15 files changed

+6278
-196
lines changed

#Gaussian-maximal-function.tex#

+766
Large diffs are not rendered by default.

.#Gaussian-maximal-function.tex

+1
Original file line numberDiff line numberDiff line change
@@ -0,0 +1 @@
1+

Gaussian-maximal-function.aux

+73
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,73 @@
1+
\relax
2+
\providecommand\hyper@newdestlabel[2]{}
3+
\providecommand{\usedref}[1]{\relax}
4+
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
5+
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
6+
\global\let\oldcontentsline\contentsline
7+
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
8+
\global\let\oldnewlabel\newlabel
9+
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
10+
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
11+
\AtEndDocument{\ifx\hyper@anchor\@undefined
12+
\let\contentsline\oldcontentsline
13+
\let\newlabel\oldnewlabel
14+
\fi}
15+
\fi}
16+
\global\let\hyper@last\relax
17+
\gdef\HyperFirstAtBeginDocument#1{#1}
18+
\providecommand\HyField@AuxAddToFields[1]{}
19+
\providecommand\HyField@AuxAddToCoFields[2]{}
20+
\abx@aux@sortscheme{nty}
21+
\abx@aux@cite{Sjogren1997}
22+
\abx@aux@cite{Portal2012}
23+
\abx@aux@cite{Maas2011b}
24+
\abx@aux@cite{Mauceri2007}
25+
\@writefile{toc}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
26+
\@writefile{lof}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
27+
\@writefile{lot}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
28+
\select@language{english}
29+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\select@language{english}}
30+
\@writefile{lof}{\defcounter {refsection}{0}\relax }\@writefile{lof}{\select@language{english}}
31+
\@writefile{lot}{\defcounter {refsection}{0}\relax }\@writefile{lot}{\select@language{english}}
32+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{The Mehler kernel and friends}}{1}{section.1}}
33+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{Notation}}{1}{subsection.1.1}}
34+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.2}{Setting}}{1}{subsection.1.2}}
35+
\newlabel{eq:Ornstein-Uhlenbeck-operator}{{1}{1}{Setting}{equation.1.1}{}}
36+
\newlabel{eq:Ornstein-Uhlenbeck-semigroup-integral}{{2}{1}{Setting}{equation.1.2}{}}
37+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.3}{The Mehler kernel}}{1}{subsection.1.3}}
38+
\newlabel{eq:Mehler-kernel}{{3}{1}{The Mehler kernel}{equation.1.3}{}}
39+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Some fine lemmata and definitions}}{1}{section.2}}
40+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{$m$inimal function}}{1}{subsection.2.1}}
41+
\newlabel{lem:m-xy-equivalence-no-cone}{{1}{1}{}{lemma.1}{}}
42+
\newlabel{eq:Gaussian-cone}{{4}{2}{$m$inimal function}{equation.2.4}{}}
43+
\newlabel{lem:m-xy-equivalence}{{2}{2}{}{lemma.2}{}}
44+
\newlabel{eq:Definition-local-region}{{5}{2}{}{equation.2.5}{}}
45+
\newlabel{eq:Definition-local-region-ball}{{6}{2}{}{equation.2.6}{}}
46+
\newlabel{eq:Definition-cut-off-set-D}{{7}{2}{$m$inimal function}{equation.2.7}{}}
47+
\newlabel{lem:Cone-Gaussians-comparable}{{3}{2}{}{lemma.3}{}}
48+
\newlabel{eq:Cone-Gaussians-comparable-1}{{8}{2}{$m$inimal function}{equation.2.8}{}}
49+
\newlabel{eq:Cone-Gaussians-comparable-2}{{9}{3}{$m$inimal function}{equation.2.9}{}}
50+
\newlabel{def:Global-region-cone-lemma}{{4}{3}{}{lemma.4}{}}
51+
\newlabel{eq:Global-region-cone-lemma-proof-1}{{10}{3}{$m$inimal function}{equation.2.10}{}}
52+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{On-diagonal estimates}}{3}{section.3}}
53+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Kernel estimates}}{3}{subsection.3.1}}
54+
\newlabel{lem:Time-part-Mehler-time-transform}{{5}{3}{}{lemma.5}{}}
55+
\newlabel{eq:Time-part-Mehler-time-transform}{{11}{3}{}{equation.3.11}{}}
56+
\newlabel{eq:Time-part-Mehler-time-transform-proof-1}{{12}{4}{Kernel estimates}{equation.3.12}{}}
57+
\newlabel{lem:Exponential-estimates}{{6}{4}{}{lemma.6}{}}
58+
\newlabel{eq:Exponential-estimates-1}{{13}{4}{}{equation.3.13}{}}
59+
\newlabel{lem:Kernel-estimates-1}{{7}{4}{}{lemma.7}{}}
60+
\newlabel{eq:Kernel-lemma-1-estimate}{{14}{4}{}{equation.3.14}{}}
61+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{On-diagonal kernel estimates on annuli}}{6}{subsection.3.2}}
62+
\newlabel{eq:C_k-annulus-decomposition}{{15}{6}{On-diagonal kernel estimates on annuli}{equation.3.15}{}}
63+
\newlabel{eq:C_k-annulus-decomposition-expand-nonzero}{{16}{6}{On-diagonal kernel estimates on annuli}{equation.3.16}{}}
64+
\newlabel{eq:C_k-annulus-decomposition-expand-zero}{{17}{6}{On-diagonal kernel estimates on annuli}{equation.3.17}{}}
65+
\newlabel{lem:On-diagonal-kernel-estimates-on-Ck}{{8}{6}{}{lemma.8}{}}
66+
\newlabel{lem:On-diagonal-kernel-estimates-on-Ck-proof-1}{{18}{6}{On-diagonal kernel estimates on annuli}{equation.3.18}{}}
67+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{The Ornstein-Uhlenbeck maximal function}}{7}{subsection.3.3}}
68+
\newlabel{tocindent-1}{0pt}
69+
\newlabel{tocindent0}{12.26888pt}
70+
\newlabel{tocindent1}{17.6065pt}
71+
\newlabel{tocindent2}{0pt}
72+
\newlabel{tocindent3}{0pt}
73+
\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{section*.1}}

Gaussian-maximal-function.bbl

+144
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,144 @@
1+
% $ biblatex auxiliary file $
2+
% $ biblatex bbl format version 2.1 $
3+
% Do not modify the above lines!
4+
%
5+
% This is an auxiliary file used by the 'biblatex' package.
6+
% This file may safely be deleted. It will be recreated by
7+
% biber as required.
8+
%
9+
\begingroup
10+
\makeatletter
11+
\@ifundefined{[email protected]}
12+
{\@latex@error
13+
{Missing 'biblatex' package}
14+
{The bibliography requires the 'biblatex' package.}
15+
\aftergroup\endinput}
16+
{}
17+
\endgroup
18+
19+
20+
\refsection{0}
21+
\sortlist{entry}{nty}
22+
\entry{Maas2011b}{article}{}
23+
\name{labelname}{3}{}{%
24+
{{hash=a2a6616efceb7a78b10661e00cd48076}{Maas}{M\bibinitperiod}{Jan}{J\bibinitperiod}{}{}{}{}}%
25+
{{hash=e4acd277d61cb5b06a5c302193d7f8d9}{Neerven}{N\bibinitperiod}{Jan}{J\bibinitperiod}{}{}{}{}}%
26+
{{hash=72181505e073b35a6d75a96812fb827c}{Portal}{P\bibinitperiod}{Pierre}{P\bibinitperiod}{}{}{}{}}%
27+
}
28+
\name{author}{3}{}{%
29+
{{hash=a2a6616efceb7a78b10661e00cd48076}{Maas}{M\bibinitperiod}{Jan}{J\bibinitperiod}{}{}{}{}}%
30+
{{hash=e4acd277d61cb5b06a5c302193d7f8d9}{Neerven}{N\bibinitperiod}{Jan}{J\bibinitperiod}{}{}{}{}}%
31+
{{hash=72181505e073b35a6d75a96812fb827c}{Portal}{P\bibinitperiod}{Pierre}{P\bibinitperiod}{}{}{}{}}%
32+
}
33+
\strng{namehash}{53b0c03d897cb816778fd91b36160d36}
34+
\strng{fullhash}{53b0c03d897cb816778fd91b36160d36}
35+
\field{sortinit}{M}
36+
\field{labeltitle}{{Whitney coverings and the tent spaces $T^{1,q}(\gamma)$ for the Gaussian measure}}
37+
\field{issn}{0004-2080}
38+
\field{journaltitle}{Arkiv för Matematik}
39+
\field{month}{04}
40+
\field{number}{2}
41+
\field{title}{{Whitney coverings and the tent spaces $T^{1,q}(\gamma)$ for the Gaussian measure}}
42+
\field{volume}{50}
43+
\field{year}{2011}
44+
\field{pages}{379\bibrangedash 395}
45+
\verb{doi}
46+
\verb 10.1007/s11512-010-0143-z
47+
\endverb
48+
\verb{url}
49+
\verb http://dx.doi.org/10.1007/s11512-010-0143-z%20http://link.springer.com/10.1007/s11512-010-0143-z
50+
\endverb
51+
\endentry
52+
\entry{Mauceri2007}{article}{}
53+
\name{labelname}{2}{}{%
54+
{{hash=2012d8f32e8f2f2096492918578921cd}{Mauceri}{M\bibinitperiod}{Giancarlo}{G\bibinitperiod}{}{}{}{}}%
55+
{{hash=27346ed0c5a61e40f4eb42518bc45e80}{Meda}{M\bibinitperiod}{Stefano}{S\bibinitperiod}{}{}{}{}}%
56+
}
57+
\name{author}{2}{}{%
58+
{{hash=2012d8f32e8f2f2096492918578921cd}{Mauceri}{M\bibinitperiod}{Giancarlo}{G\bibinitperiod}{}{}{}{}}%
59+
{{hash=27346ed0c5a61e40f4eb42518bc45e80}{Meda}{M\bibinitperiod}{Stefano}{S\bibinitperiod}{}{}{}{}}%
60+
}
61+
\strng{namehash}{6fe36e59fd97e1b48eff21835e8c1a77}
62+
\strng{fullhash}{6fe36e59fd97e1b48eff21835e8c1a77}
63+
\field{sortinit}{M}
64+
\field{labeltitle}{{$\textrm{BMO}$ and $H^1$ for the Ornstein-Uhlenbeck operator}}
65+
\field{issn}{0022-1236}
66+
\field{journaltitle}{Journal of Functional Analysis}
67+
\field{month}{11}
68+
\field{number}{1}
69+
\field{title}{{$\textrm{BMO}$ and $H^1$ for the Ornstein-Uhlenbeck operator}}
70+
\field{volume}{252}
71+
\field{year}{2007}
72+
\field{pages}{278\bibrangedash 313}
73+
\verb{doi}
74+
\verb 10.1016/j.jfa.2007.06.017
75+
\endverb
76+
\verb{file}
77+
\verb :home/aw/teuwen/Localdisk/Mendeley-Desktop/Mauceri, Meda - 2007 - $textrm{BMO}$ and $H1$ for the Ornstein-Uhlenbeck operator.pdf:pdf
78+
\endverb
79+
\verb{url}
80+
\verb http://linkinghub.elsevier.com/retrieve/pii/S0022123607002613%20http://dx.doi.org/10.1016/j.jfa.2007.06.017
81+
\endverb
82+
\keyw{a,analisi armonica,analisi tempo-frequenza e,and the progetto cofinanziato,atomic hardy space,bmo,corresponding author,gauss measure,imaginary powers,laplaciani generalizzati,m,n,p,prin2005,project,riesz transform,singular integrals,teoria delle rappresentazioni,the italian g,work partially supported by}
83+
\endentry
84+
\entry{Portal2012}{unpublished}{}
85+
\name{labelname}{1}{}{%
86+
{{hash=72181505e073b35a6d75a96812fb827c}{Portal}{P\bibinitperiod}{Pierre}{P\bibinitperiod}{}{}{}{}}%
87+
}
88+
\name{author}{1}{}{%
89+
{{hash=72181505e073b35a6d75a96812fb827c}{Portal}{P\bibinitperiod}{Pierre}{P\bibinitperiod}{}{}{}{}}%
90+
}
91+
\strng{namehash}{72181505e073b35a6d75a96812fb827c}
92+
\strng{fullhash}{72181505e073b35a6d75a96812fb827c}
93+
\field{sortinit}{P}
94+
\field{labeltitle}{{Maximal and quadratic Gaussian Hardy spaces}}
95+
\field{abstract}{Building on the author's recent work with Jan Maas and Jan van Neerven, this paper establishes the equivalence of two norms (one using a maximal function, the other a square function) used to define a Hardy space on $\R^{n}$ with the gaussian measure, that is adapted to the Ornstein-Uhlenbeck semigroup. In contrast to the atomic Gaussian Hardy space introduced earlier by Mauceri and Meda, the $h^{1}(\R^{n};d\gamma)$ space studied here is such that the Riesz transforms are bounded from $h^{1}(\R^{n};d\gamma)$ to $L^{1}(\R^{n};d\gamma)$. This gives a gaussian analogue of the seminal work of Fefferman and Stein in the case of the Lebesgue measure and the usual Laplacian.}
96+
\field{annotation}{Preprint}
97+
\field{eprinttype}{arXiv}
98+
\field{month}{03}
99+
\field{title}{{Maximal and quadratic Gaussian Hardy spaces}}
100+
\field{year}{2012}
101+
\verb{eprint}
102+
\verb 1203.1998
103+
\endverb
104+
\verb{url}
105+
\verb http://arxiv.org/abs/1203.1998
106+
\endverb
107+
\endentry
108+
\entry{Sjogren1997}{article}{}
109+
\name{labelname}{1}{}{%
110+
{{hash=58d187f94da7560ff4a5d473d1cd37ba}{Sjögren}{S\bibinitperiod}{Peter}{P\bibinitperiod}{}{}{}{}}%
111+
}
112+
\name{author}{1}{}{%
113+
{{hash=58d187f94da7560ff4a5d473d1cd37ba}{Sjögren}{S\bibinitperiod}{Peter}{P\bibinitperiod}{}{}{}{}}%
114+
}
115+
\list{publisher}{1}{%
116+
{Birkhäuser Boston}%
117+
}
118+
\strng{namehash}{58d187f94da7560ff4a5d473d1cd37ba}
119+
\strng{fullhash}{58d187f94da7560ff4a5d473d1cd37ba}
120+
\field{sortinit}{S}
121+
\field{labeltitle}{{Operators associated with the hermite semigroup — A survey}}
122+
\field{issn}{1069-5869}
123+
\field{journaltitle}{The Journal of Fourier Analysis and Applications}
124+
\field{month}{01}
125+
\field{number}{S1}
126+
\field{title}{{Operators associated with the hermite semigroup — A survey}}
127+
\field{volume}{3}
128+
\field{year}{1997}
129+
\field{pages}{813\bibrangedash 823}
130+
\verb{doi}
131+
\verb 10.1007/BF02656487
132+
\endverb
133+
\verb{file}
134+
\verb :home/aw/teuwen/Localdisk/Mendeley-Desktop/Sjögren - 1997 - Operators associated with the hermite semigroup — A survey.pdf:pdf
135+
\endverb
136+
\verb{url}
137+
\verb http://link.springer.com/10.1007/BF02656487
138+
\endverb
139+
\keyw{hermite,ornstein-uhlenbeck}
140+
\endentry
141+
\endsortlist
142+
\endrefsection
143+
\endinput
144+

0 commit comments

Comments
 (0)