Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014), p. 187-196 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $(h_{k})_{k≥0}$ be the Haar system on [0,1]. We show that for any vectors $a_{k}$ from a separable Hilbert space and any $ε_{k} ∈ [-1,1]$, k = 0,1,2,..., we have the sharp inequality $||∑_{k=0}^{n} ε_{k}a_{k}h_{k}||_{W([0,1])} ≤ 2||∑_{k=0}^{n} a_{k}h_{k}||_{L^{∞}([0,1])}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{∞}$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound $||Y||_{W(Ω)} ≤ 2||X||_{L^{∞}(Ω)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

Détail

Publié le : 2014-01-01

Zbl 1303.60034

EUDML-ID : urn:eudml:doc:286096