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 was a precursor in one dimension of the Calderón–Zygmund lemma.

The lemma is stated as follows:

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

$E=\bigcup _{k}(a_{k},b_{k})$

such that g(a_k) = g(b_k), except possibly if a_k = a when g(b_k) ≥ g(a_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

The set E is open, so a countable disjoint union of intervals (a_n, b_n).

The main step is to show that g(b_n) ≥ g(x) for x in (a_n, b_n). If not take x with g(b_n) < g(x). Let A be the closed subset of [x,b_n] consisting of points y such that g(y) ≥ g(x). It contains x but not b_n. 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_n ∉ E, g(t) ≤ g(b_n) if b_n ≤ t ≤ b. Since g(y) > g(z) ≥ g(x) > g(b_n), y must lie in (z, b_n). That contradicts the maximality of z. Hence g(b_n) ≥ g(a_n).

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

References

• Duren, Peter L. (2000), Theory of H^p Spaces, New York: Dover Publications, ISBN 0-486-41184-2
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• Riesz, Frédéric (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, retrieved 2008-07-21
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