E-viri
PDF
Recenzirano
Odprti dostop
-
Kravitz, Noah
The Journal of fourier analysis and applications, 15/8, Letnik: 25, Številka: 4Journal Article
The two-dimensional signed small ball inequality states that for all possible choices of signs, ∑ | R | = 2 - n ε R h R L ∞ ≳ n , where the summation runs over all dyadic rectangles in the unit square and h R 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 - n ε R h R = n + 1 - 2 k = | Q | 2 n + 1 n + 1 k .
