“Rising sun lemma”的意思、由来-开放百科全书

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Rising sun lemma

释义

Proof
Notes
References

In mathematical analysis, the rising sun lemma is a 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.^[1]

The lemma is stated as follows:^[2]

Suppose g is a real-valued continuous function on the interval [a,b] and S is the set of x in [a,b] such that g(x) < g(y) for some y with x < y ≤ b. (Note that b is not in S, though a may be.) Define E = S ∩ (a,b).

Then E is an open set, and it may be written as a countable union of disjoint intervals

such that g(a_k) = g(b_k), unless a_k = a ∈ S for some k, in which case g(a) < g(b_k) for that one k. Furthermore, if x ∈ (a_k,b_k), then g(x) < g(b_k).

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

We need a lemma: Suppose [c,d) ⊂ S, but d ∉ S. Then g(c) < g(d).

To prove this, suppose g(c) ≥ g(d).

Then g achieves its maximum on [c,d] at some point z < d.

Since z ∈ S, there is a y in (z,b] with g(z) < g(y).

If y ≤ d, then g would not reach its maximum on [c,d] at z.

Thus, y ∈ (d,b], and g(d) ≤ g(z) < g(y).

This means that d ∈ S, which is a contradiction, thus establishing the lemma.

The set E is open, so it is composed of a countable union of disjoint intervals (a_k,b_k).

It follows immediately from the lemma that g(x) < g(b_k) for x in

(a_k,b_k).

Since g is continuous, we must also have g(a_k) ≤ g(b_k).

If a_k ≠ a or a ∉ S, then a_k ∉ S,

so g(a_k) ≥ g(b_k), for otherwise a_k ∈ S.

Thus, g(a_k) = g(b_k) in these cases.

Finally, if a_k = a ∈ S, the lemma tells us that g(a) < g(b_k).

Notes

1. ^{{harvnb|Stein|1998}}
2. ^See*{{harvnb|Riesz|1932}}*{{harvnb|Zygmund|1977|p=31}}*{{harvnb|Tao|2011|pp=118–119}}*{{harvnb|Duren|1970|loc=Appendix B}}

References

{{citation |last=Duren |first=Peter L. |title=Theory of H^p 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|isbn=978-0-521-69973-0}}
{{citation |last=Korenovskyy |first=A. A. |author2=A. K. Lerner |author3=A. M. Stokolos |date=November 2004 |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= |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|isbn=0821869191}}
{{citation|last=Zygmund|first=Antoni|authorlink=Antoni Zygmund| title=Trigonometric series. Vol. I, II|edition=2nd|publisher= Cambridge University Press|year=1977|isbn= 0-521-07477-0}}

2 : Real analysis|Lemmas

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