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[[File:Rising sun lemma.svg|thumb|right|400px]]

In [[mathematical analysis]], the '''rising sun lemma''' is a [[lemma (mathematics)|lemma]] due to [[Frigyes Riesz]], used in the proof of the [[Hardy–Littlewood maximal theorem]].  The lemma was a precursor in one dimension of the [[Calderón–Zygmund lemma]]. <ref>{{harvnb|Stein|1998}}</ref>

The lemma is stated as follows:<ref>See: 
*{{harvnb|Riesz|1932}}
*{{harvnb|Zygmund|1977|p=31}}
*{{harvnb|Tao|2011|p=118-119}}
*{{harvnb|Duren|1970|loc=Appendix B}}</ref>

:Let ''g''(''x'') be a real-valued continuous function on the interval [''a'',''b''], and let ''E'' be the set of ''x'' &isin; (a,''b'') such that ''g''(''y'') > ''g''(''x'') for some ''y'' with ''x'' < ''y'' < ''b''.

:Then ''E'' is an open set, and can be written as a disjoint union of intervals
::<math>E=\bigcup_k (a_k,b_k)</math>
:such that ''g''(''a''<sub>''k''</sub>) = ''g''(''b''<sub>''k''</sub>), except possibly if ''a''<sub>''k''</sub> = ''a'' when ''g''(''b''<sub>''k''</sub>) ≥ ''g''(''a''<sub>''k''</sub>).

The colorful name of the lemma comes from imagining the graph of the function ''g'' as a mountainous landscape, with the sun shining horizontally from the right. The set ''E'' consist of points that are in the shadow. 

==Proof==
The set ''E'' is open, so a countable disjoint union of intervals (''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>).

The main step is to show that ''g''(''b''<sub>''n''</sub>) ≥ ''g''(''x'') for ''x'' in (''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>). If not take ''x'' with ''g''(''b''<sub>''n''</sub>) < ''g''(''x''). Let ''A'' be the closed subset of  [''x'',''b''<sub>''n''</sub>] consisting of points ''y'' such that ''g''(''y'') ≥ ''g''(''x''). It contains ''x'' but not ''b''<sub>''n''</sub>. It has a largest element, ''z'' say. Since ''z'' lies in ''E'', there is a ''y'' with ''z'' < ''y'' < ''b'' and ''g''(''y'') > ''g''(''z''). Since  ''b''<sub>''n''</sub> ∉ ''E'', ''g''(''t'') ≤ ''g''(''b''<sub>''n''</sub>) if 
''b''<sub>''n''</sub> ≤ ''t'' ≤ ''b''. Since ''g''(''y'') > ''g''(''z'') ≥ ''g''(''x'') > ''g''(''b''<sub>''n''</sub>), ''y'' must lie in (''z'', ''b''<sub>''n''</sub>). That contradicts the maximality of ''z''. Hence  ''g''(''b''<sub>''n''</sub>) ≥ ''g''(''a''<sub>''n''</sub>).  

If ''a''<sub>''n''</sub> ≠ ''a'', the reverse inequality holds.  In fact since ''a''<sub>''n''</sub> ∉ ''E'', ''g''(''y'') ≤ ''g''(''a''<sub>''n''</sub>) if ''a''<sub>''n''</sub> ≤ ''y'' ≤ ''b''.  So ''g''(''b''<sub>''n''</sub>) ≤ ''g''(''a''<sub>''n''</sub>). Hence  ''g''(''b''<sub>''n''</sub>) = ''g''(''a''<sub>''n''</sub>).  If ''g''(''x'') = ''g''(''a''<sub>''n''</sub>) at an interior point, then ''g''(''y'') ≤ ''g''(''x'') for ''x'' < ''y'' < ''b'', contradicting ''x'' ∈ ''E''.

==Notes==
{{reflist}}
==References==
*{{citation |last=Duren |first=Peter L. |title=Theory of H<sup>p</sup> Spaces |publisher=Dover Publications |location=New York |year=2000 |isbn=0-486-41184-2}}
*{{citation|last=Garling|first=D.J.H.|title=Inequalities: a journey into linear analysis|publisher=Cambridge University Press|year=2007|id=ISBN 978-0-521-69973-0}}
*{{citation |last=Korenovskyy |first=A. A. |coauthors=A. K. Lerner and A. M. Stokolos |year=2004 |month=November |title=On a multidimensional form of F. Riesz's "rising sun" lemma |journal=[[Proceedings of the American Mathematical Society]] |volume=133 |issue=5 |pages=1437–1440 |url=http://www.ams.org/proc/2005-133-05/S0002-9939-04-07653-1/home.html |language= |format= |accessdate=2008-07-21 |doi=10.1090/S0002-9939-04-07653-1}}
*{{citation |last=Riesz |first=Frédéric |authorlink=Frigyes Riesz |year=1932 |title=Sur un Théorème de Maximum de Mm. Hardy et Littlewood |journal=Journal of the [[London Mathematical Society]] |volume=7 |issue=1 |pages=10–13 |doi=10.1112/jlms/s1-7.1.10 |url=http://jlms.oxfordjournals.org/cgi/content/citation/s1-7/1/10 |accessdate=2008-07-21}}
*{{citation|title=Singular integrals: The Roles of Calderón and Zygmund|first=Elias|last=Stein|authorlink=Elias Stein|journal=Notices of the American Mathematical Society|volume=45|pages=1130–1140|url=http://www.ams.org/notices/199809/stein.pdf|year=1998|issue=9}}.
*{{citation|title=An Introduction to Measure Theory|volume=126|series=Graduate Studies in Mathematics|first=Terence|last= Tao|publisher=American Mathematical Society|year= 2011|id=ISBN 0821869191}}
*{{citation|last=Zygmund|first=Antoni|authorlink=Antoni Zygmund| title=Trigonometric series. Vol. I, II|edition=2nd|publisher= Cambridge University Press|year=1977|id= ISBN 0-521-07477-0}}

[[Category:Real analysis]]
[[Category:Lemmas]]