[Paper Review] On multidimensional F. Riesz's "Rising Sun" Lemma

A. A. Korenovskyy, Andrei K. Lerner+1 more|ArXiv.org|Aug 22, 2003

advanced mathematical theories3 references3 citations

TL;DR

This paper establishes a multidimensional analogue of F. Riesz's 'Rising Sun' Lemma using a generalized dyadic process on rectangles in ℝⁿ. It proves that for any absolutely continuous measure and integrable function with average ≤ A on a rectangle I₀, there exists a countable family of disjoint rectangles where the average equals A, and f(x) ≤ A μ-a.e. outside their union—extending the one-dimensional result to higher dimensions via a novel dyadic-like construction on rectangles, not cubes.

ABSTRACT

A multidimensional version of the Riesz rising sun lemma is proved by means of a generalized dyadic process.

Motivation & Objective

• To extend F. Riesz’s one-dimensional 'Rising Sun' Lemma to higher dimensions, where cubes fail to yield the same sharp result.
• To overcome the geometric limitations of cubes in ℝⁿ (n ≥ 2) by using rectangles instead.
• To develop a generalized dyadic process that mimics the covering and differentiation properties of dyadic cubes in higher dimensions.
• To provide a new proof technique for the multidimensional lemma that avoids reliance on the strong maximal function.

Proposed method

• Divide a rectangle I₀ into two subrectangles by bisecting its longest side.
• Select subrectangles based on whether their average value of f is less than or greater than A, using the absolute continuity of μ to adjust the dividing hyperplane.
• When one subrectangle has average > A and the other < A, shift the hyperplane continuously until one subrectangle achieves average exactly A.
• Include the subrectangle with average A in the family {Iⱼ}, and recursively subdivide the one with average < A.
• The resulting family {Jⱼ} of subdivided rectangles exhibits a 'dyadic' inclusion property: if two intersect, one contains the other.
• Use the Vitaly covering property of {Jⱼ} to apply differentiation of integrals, concluding f(x) ≤ A μ-a.e. on E = I₀ \ ∪Iⱼ.

Experimental results

Research questions

• RQ1 Can F. Riesz’s one-dimensional 'Rising Sun' Lemma be generalized to ℝⁿ for n ≥ 2 using a similar equality condition on averages?
• RQ2 Why do cubes fail to support a sharp multidimensional analogue of the Riesz lemma, unlike rectangles?
• RQ3 Is there a generalized dyadic process on rectangles that replicates the covering and differentiation properties of dyadic cubes in higher dimensions?
• RQ4 Can the proof avoid dependence on the strong maximal function while achieving the same sharp result as in the one-dimensional case?
• RQ5 Does the generalized dyadic structure on rectangles allow for differentiation of integrals on the exceptional set E, ensuring f ≤ A a.e. on E?

Key findings

• A multidimensional version of the Riesz 'Rising Sun' Lemma holds for rectangles in ℝⁿ, not cubes, due to geometric constraints in higher dimensions.
• The lemma guarantees the existence of a countable family of pairwise disjoint rectangles {Iⱼ} ⊂ I₀ such that the average of f over each Iⱼ equals A.
• On the set E = I₀ \ ∪Iⱼ, f(x) ≤ A holds for μ-almost every x, ensuring the exceptional set is controlled.
• The proof relies on a generalized dyadic process on rectangles that induces a differential basis with the Vitaly covering property.
• The method avoids the strong maximal function and instead uses a continuous adjustment of hyperplanes to achieve exact average A on selected subrectangles.
• The result provides a sharp alternative to the Calderón-Zygmund lemma, which only guarantees averages in [A, 2ⁿA], by achieving exact equality A.