Interpolation between Hp spaces and non-commutative generalizations (II).

Pisier, Gilles. "Interpolation between Hp spaces and non-commutative generalizations (II).." Revista Matemática Iberoamericana 9.2 (1993): 281-291. <http://eudml.org/doc/39438>.

@article{Pisier1993,
abstract = {We continue an investigation started in a preceding paper. We discuss tha classical result of Carleson connecting Carleson measures with the ∂-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution for the ∂-equation, which satisfies simultaneously a good L∞ estimate and a good L1 estimate. This appears as a special case of our main result which can be stated as follows:Let (Ω, A, μ) be any measure space. Consider a bounded operator u: H1 → L1(μ). Assume that on one hand u admits an extension u1: L1 → L1(μ) bounded with norm C1, and on the other hand that u admits an extension u∞: L∞ → L∞(μ) bounded with norm C∞. Then u admits an extension u' which is bounded simultaneously from L1 into L1(μ) and from L∞ into L∞(μ) and satisfies||u': L∞ → L∞(μ)|| ≤ C C∞||u': L1 → L1(μ)|| ≤ C C1where C is a numerical constant.},
author = {Pisier, Gilles},
journal = {Revista Matemática Iberoamericana},
keywords = {Espacios de Hardy; Espacios LP; Desigualdades; Interpolación; Funciones analíticas; Carleson measures; -equation},
language = {eng},
number = {2},
pages = {281-291},
title = {Interpolation between Hp spaces and non-commutative generalizations (II).},
url = {http://eudml.org/doc/39438},
volume = {9},
year = {1993},
}

TY - JOUR
AU - Pisier, Gilles
TI - Interpolation between Hp spaces and non-commutative generalizations (II).
JO - Revista Matemática Iberoamericana
PY - 1993
VL - 9
IS - 2
SP - 281
EP - 291
AB - We continue an investigation started in a preceding paper. We discuss tha classical result of Carleson connecting Carleson measures with the ∂-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution for the ∂-equation, which satisfies simultaneously a good L∞ estimate and a good L1 estimate. This appears as a special case of our main result which can be stated as follows:Let (Ω, A, μ) be any measure space. Consider a bounded operator u: H1 → L1(μ). Assume that on one hand u admits an extension u1: L1 → L1(μ) bounded with norm C1, and on the other hand that u admits an extension u∞: L∞ → L∞(μ) bounded with norm C∞. Then u admits an extension u' which is bounded simultaneously from L1 into L1(μ) and from L∞ into L∞(μ) and satisfies||u': L∞ → L∞(μ)|| ≤ C C∞||u': L1 → L1(μ)|| ≤ C C1where C is a numerical constant.
LA - eng
KW - Espacios de Hardy; Espacios LP; Desigualdades; Interpolación; Funciones analíticas; Carleson measures; -equation
UR - http://eudml.org/doc/39438
ER -