Measure, Integration & Real Analysis · Problem 4A.14

Product Measures — Sheldon Axler

Mathematics — Measure Theory · hardy-littlewood-maximal-inequality-strengthened

Problem

Suppose h∈L1(R)h \in \mathcal{L}^1(\mathbf{R}). Prove that ∣{b∈R:h∗(b)≥c}∣≤3c∥h∥1|\{b \in \mathbf{R} : h^*(b) \geq c\}| \leq \frac{3}{c}\|h\|_1 for every c>0c > 0. [This result slightly strengthens the Hardy–Littlewood maximal inequality (4.8) because the set on the left side above includes those b∈Rb \in \mathbf{R} such that h∗(b)=ch^*(b) = c. A much deeper strengthening comes from replacing the constant 3 in the Hardy–Littlewood maximal inequality with a smaller constant. In 2003, Antonios Melas answered what had been an open question about the best constant. He proved that the smallest constant that can replace 3 in the Hardy–Littlewood maximal inequality is (11+61)/12≈1.56752(11 + \sqrt{61})/12 \approx 1.56752; see Annals of Mathematics 157 (2003), 647–688.]

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