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Lorentz Spaces of Weak-Type

Lorentz Spaces of Weak-Type By JAVIER SORIAt [Received 4 September 1996] 1. Introduction WE consider the problem of characterizing when the weak-type Lorentz space AP,oo(w), O < P < 00, is a Banach space. This space is defined as (see [5]) AP'''''(w) = {f; IIfIIM.OO(w) = ~~~ 1*(1)(l' W(S)dS) lip < oo}. where w is a weight in R+, f is a measurable function in R and f* denotes the non-increasing rearrangement of f (see [2] for standard notation). Our problem is thus to decide when there exists a norm on AP,oo(w), equivalent to the quasi-norm II f II AP,OO(w)' The strong version is defined as AP(w) = If; IIf11M(w) = (1"" U*(t»)PW(t)dt)'IP < oo}. Classical examples are obtained by choosing w(t) = 1. In this case AP,oo(w) = LP'oo and AP(w) = LP. With w(t) = t(p/q)-l, AP(w) = Lq,p. The analogous problem for the strong case was studied in [11]: for 0 < p < 00 we define w E B , if for all r > 0, 00 w(t) C l' --dt ~ - w(t)dt, r tP r 0 then, for 1 < p < 00, AP(w) is a Banach space, if and only if w E B . This http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

Lorentz Spaces of Weak-Type

Soria, Javier
The Quarterly Journal of Mathematics , Volume 49 (1) – Mar 1, 1998
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Publisher
Oxford University Press
Copyright
© Published by Oxford University Press.
Subject
Articles
ISSN
0033-5606
eISSN
1464-3847
DOI
10.1093/qmathj/49.1.93
Publisher site
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Abstract

By JAVIER SORIAt [Received 4 September 1996] 1. Introduction WE consider the problem of characterizing when the weak-type Lorentz space AP,oo(w), O < P < 00, is a Banach space. This space is defined as (see [5]) AP'''''(w) = {f; IIfIIM.OO(w) = ~~~ 1*(1)(l' W(S)dS) lip < oo}. where w is a weight in R+, f is a measurable function in R and f* denotes the non-increasing rearrangement of f (see [2] for standard notation). Our problem is thus to decide when there exists a norm on AP,oo(w), equivalent to the quasi-norm II f II AP,OO(w)' The strong version is defined as AP(w) = If; IIf11M(w) = (1"" U*(t»)PW(t)dt)'IP < oo}. Classical examples are obtained by choosing w(t) = 1. In this case AP,oo(w) = LP'oo and AP(w) = LP. With w(t) = t(p/q)-l, AP(w) = Lq,p. The analogous problem for the strong case was studied in [11]: for 0 < p < 00 we define w E B , if for all r > 0, 00 w(t) C l' --dt ~ - w(t)dt, r tP r 0 then, for 1 < p < 00, AP(w) is a Banach space, if and only if w E B . This

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The Quarterly Journal of Mathematics – Oxford University Press

Published: Mar 1, 1998

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