Rising sun lemma

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 harv|Stein|1998.

: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:: $g(x) > inf g|_{ [0,x] }$ :Then "E" is an open set, and can be written as a disjoint union of intervals:: $E=igcup_k (a_k,b_k)$ :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").

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