Refining the Two-Dimensional Signed Small Ball Inequality
Abstract
The two-dimensional signed small ball inequality states that for all possible choices of signs, ∑|R|=2-nɛRhRL∞≳n,where the summation runs over all dyadic rectangles in the unit square and hR denotes the associated Haar function. This inequality first appeared in the work of Talagrand, and alternative proofs are due to Temlyakov and Bilyk & Feldheim (who showed that the supremum equals n+1 in all cases). We prove a stronger result: for all integers 0≤k≤n+1, all possible choices of signs, and all dyadic rectangles Q with |Q|≥2-n-1, x∈Q:∑|R|=2-nɛRhR=n+1-2k=|Q|2n+1n+1k.
- Publication:
-
Journal of Fourier Analysis and Applications
- Pub Date:
- August 2019
- DOI:
- 10.1007/s00041-018-9643-1
- arXiv:
- arXiv:1712.01206
- Bibcode:
- 2019JFAA...25.1921K
- Keywords:
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- Small ball inequality;
- Haar system;
- Geometric discrepancy;
- Primary 42C40;
- Secondary 11K36;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Combinatorics;
- Mathematics - Probability