The Hardy operator averages a function over its past: (Hf)(x) = (1/x) ∫₀ˣ f(t) dt. It maps L^p to L^p for p > 1 — the average is smoother than the function. But for p = 1, the Hardy operator is unbounded: there exist integrable functions whose running average is not integrable.
The paper on a sharp logarithmic condition (arXiv: 2603.21252) identifies the precise correction. Subtract a natural term — the right compensator — and the modified Hardy operator maps into L¹ on the largest possible subspace. The condition characterizing this subspace is logarithmic: the function must satisfy a weighted integrability condition involving log factors that reflect obstructions at both x → 0 and x → ∞.
Two scales of difficulty exist simultaneously: the small-scale obstruction (near zero, where the average amplifies singularities) and the large-scale obstruction (near infinity, where the average decays too slowly). The logarithmic weight captures both. The discrete analogue — summable sequences and the discrete Hardy operator — has the same structure.
The through-claim: the Hardy operator's failure on L¹ is exactly logarithmic. The gap between “bounded on L^p for p > 1” and “unbounded on L¹” is filled by a logarithmic correction. The obstruction to L¹ boundedness is not a power-law divergence but a logarithmic one, and the sharp condition identifies the exact rate. The correction is the minimal surgery that restores boundedness.
2603.21252. Classical analysis / Hardy operator / Sobolev spaces / logarithmic conditions / sharp bounds.