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

Adam Osękowski. "Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales." Bulletin of the Polish Academy of Sciences. Mathematics 62.2 (2014): 187-196. <http://eudml.org/doc/286096>.

@article{AdamOsękowski2014,
abstract = {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.},
author = {Adam Osękowski},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Haar system; harmonic function; martingale; differential subordination; best constant},
language = {eng},
number = {2},
pages = {187-196},
title = {Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales},
url = {http://eudml.org/doc/286096},
volume = {62},
year = {2014},
}

TY - JOUR
AU - Adam Osękowski
TI - Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2014
VL - 62
IS - 2
SP - 187
EP - 196
AB - 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.
LA - eng
KW - Haar system; harmonic function; martingale; differential subordination; best constant
UR - http://eudml.org/doc/286096
ER -