On the UMD constant of the space $\ell {₁}^N$

Adam Osękowski. "On the UMD constant of the space $ℓ₁^{N}$." Colloquium Mathematicae 142.1 (2016): 135-147. <http://eudml.org/doc/283650>.

@article{AdamOsękowski2016,
abstract = {Let N ≥ 2 be a given integer. Suppose that $df = (dfₙ)_\{n≥0\}$ is a martingale difference sequence with values in $ℓ₁^\{N\}$ and let $(εₙ)_\{n≥0\}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $ℙ(sup_\{n≥0\} ||∑_\{k=0\}^\{n\} ε_\{k\} df_\{k\}||_\{ℓ₁^\{N\}\} ≥ 1) ≤ (ln N + ln(3ln N))/(1 - (2ln N)^\{-1\}) sup_\{n≥0\} ||∑_\{k=0\}^\{n\} df_\{k\}||_\{ℓ₁^\{N\}\}$. It is shown that this result is asymptotically sharp in the sense that the least constant $C_\{N\}$ in the above estimate satisfies $lim_\{N→ ∞\} C_\{N\}/ln N = 1$. The novelty in the proof is the explicit verification of the ζ-convexity of the space $ℓ₁^\{N\}$.},
author = {Adam Osękowski},
journal = {Colloquium Mathematicae},
language = {eng},
number = {1},
pages = {135-147},
title = {On the UMD constant of the space $ℓ₁^\{N\}$},
url = {http://eudml.org/doc/283650},
volume = {142},
year = {2016},
}

TY - JOUR
AU - Adam Osękowski
TI - On the UMD constant of the space $ℓ₁^{N}$
JO - Colloquium Mathematicae
PY - 2016
VL - 142
IS - 1
SP - 135
EP - 147
AB - Let N ≥ 2 be a given integer. Suppose that $df = (dfₙ)_{n≥0}$ is a martingale difference sequence with values in $ℓ₁^{N}$ and let $(εₙ)_{n≥0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $ℙ(sup_{n≥0} ||∑_{k=0}^{n} ε_{k} df_{k}||_{ℓ₁^{N}} ≥ 1) ≤ (ln N + ln(3ln N))/(1 - (2ln N)^{-1}) sup_{n≥0} ||∑_{k=0}^{n} df_{k}||_{ℓ₁^{N}}$. It is shown that this result is asymptotically sharp in the sense that the least constant $C_{N}$ in the above estimate satisfies $lim_{N→ ∞} C_{N}/ln N = 1$. The novelty in the proof is the explicit verification of the ζ-convexity of the space $ℓ₁^{N}$.
LA - eng
UR - http://eudml.org/doc/283650
ER -