Rising sun lemma

From Wikipedia, the free encyclopedia

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 quickly gives a proof of the one-dimensional Calderón-Zygmund lemma, a fundamental result in harmonic analysis.

The lemma is stated as follows:

Let g(x) be a real-valued continuous function on the interval [0,a], and let E be the set of x ∈ (0,a) such that

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

such that g(a_k) ≤ g(b_k).

It can be shown in fact that the conclusion of the lemma can be ostensibly strengthened to g(a_k) = g(b_k). The colorful name of the lemma thus refers to the shape of the graph of the function g over each of the intervals (a_k, b_k).

[edit] References

• Duren, Peter (1970), Theory of H^p spaces, Academic Press .
• Korenovskyy, A.A.; Lerner, A.K. & Stokolos, A.M. (2004), “On a multidimensional form of F. Riesz's "Rising sun" lemma”, Proceedings of the American Mathematical Society 133 (5): 1437-1440, <http://www.ams.org/proc/2005-133-05/S0002-9939-04-07653-1/S0002-9939-04-07653-1.pdf> .
• Riesz, F (1932), “Sur un Théorème de Maximum de Mm. Hardy et Littlewood”, Journal of the London Mathematical Society 7 (1): 10-13, doi:10.1112/jlms/s1-7.1.10, <http://jlms.oxfordjournals.org/cgi/content/citation/s1-7/1/10> .