Interpolation between weighted Lorentz spaces with respect to a vector measure
Maryam MOHSENİPOUR
^{
[1]
}
,
Ghadir SADEGHİ
^{
[2]
}
In this paper, we consider weighted Lorentz spaces with respect to a vector measure and derive some of their properties. We describe the interpolation with a parameter function of these spaces. As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\m\)$ and interpolation spaces for couples of LorentzZygmund spaces.
Lorentz spaces, interpolation, vector measures, Steffensen's inequality
 [1] R.G. Bartel, N. Dunford and J. Schwartz, Weak compactness and vector measures,
Canad. J. Math. 7, 289305, 1955.
 [2] C. Bennett and R. Sharply, Interpolation of Operators, Pure Appl. Math. 129, 469
pages, Academic Press, 1988.
 [3] J. Bergh, A generalization of Steffensen’s inequality, J. Math. Anal. Appl. 41, 187
191, 1973.
 [4] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math.
Wiss. 223, SpringerVerlag Berlin Heidelberg, 1976.
 [5] M.J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct.
Anal. 112, 480494, 1993.
 [6] G.P. Curbera, Operators into L1 of a vector measure and applications to Banach
lattices , Math. Ann. 293, 317330, 1992.
 [7] G.P. Curbera, When L1 of a vector measure is an ALspaces, Pacific. J. Math. 162,
287303, 1994.
 [8] G.P. Curbera, Banach space properties of L1 of a vector measure, Proc. Amer. Math.
Soc. 123, 37973806, 1995.
 [9] G.P. Curbera and W.J. Ricker, Vector measures, integration and application, in: Positivity,
127160, Birkhäuser Basel, 2007.
 [10] J. Diestel and J.J.Jr. Uhl, Vector Measures, Math. Surveys Monogr. 15, 1977.
 [11] R. del Campo, A. Fernandez and F. Mayoral, A note on real interpolation of Lp–spaces
of vector measures on δ–rings, J. Math. Anal. Appl. 405, 518529, 2013.
 [12] R. del Campo, A. Fernandez, A. Manzano, F. Mayoral and F. Naranjo, Complex
interpolation of Orlicz spaces with respect to a vector measure, Math. Nachr. 287,
2331, 2014.
 [13] D.E. Edmunds, P. Gurka and B. Opic, Sharpness of embeddings in logarithmic Bessel
Potential spaces, Proc. Roy. Soc. Edinburgh Sect. A. 126A, 9951009, 1996.
 [14] A. Fernandez, F. Mayoral and F. Naranjo, Real interpolation method on spaces of
scalar integrable functions with respect to vector measures, J. Math. Anal. Appl. 376,
203211, 2011.
 [15] A. Fernandez, F. Mayoral, F. Naranjo, C. Sáez and E.A. SánchezPérez, Spaces of
pintegrable functions with respect to a vector measure, Positivity 10, 116, 2006.
 [16] A. Fernandez, F. Mayoral, F. Naranjo and E.A. SánchezPérez, Complex interpolation
of spaces of integrable functions with respect to a vector measure, Collect. Math. 61,
241252, 2010.
 [17] A. Fernandez and F. Naranjo, Rybakov’s theorem for vector measures in Fréchet
spaces, Indag. Math. (N.S.) 8 (1), 3342, 1997.
 [18] I. Kluvanek and G. Knowles, Vector Measures and Control Systems, Note Mat. 58,
1975.
 [19] D.R. Lewis, Integration with respect to vector measures, Pacific. J. Math. 33, 157165,
1970.
 [20] D.R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16,
583599, 1973.
 [21] W.A.J. Luxemburg, Banach function spaces, Ph.D. Thesis, Delft Institute of Technology.
Assen, Netherlands, 1955.
 [22] L. Maligranda and L.E. Persson, Real interpolation between weighted $L^p$ and Lorentz
spaces, Bull. Polish Acad. Sci. Math. 35, 765778, 1987.
 [23] C. Merucci, Applications of interpolation with a function parameter to Lorentz Soblev
and Besov spaces, in: Interpolation Spaces and Allied Topics in Analysis, Lecture
Notes in Math. 1070, 183201, Springer, Berlin, Heidelberg, 1984.
 [24] S. Okada, The dual space of $L^1(\mu)$ for a vector measure $\mu$, J. Math. Anal. Appl. 177,
583599, 1993.
 [25] L.E. Persson, Interpolation with a parameter function, Math. Scand. 59, 199222,
1986.
 [26] E.A. Sánches Pérez, Compactness arguments for spaces of pintegrable functions with
respect to a vector measure and factorization of operators through LebesgueBochner
spaces, Illinois J. Math. 45, 907923, 2001.
 [27] G.F. Stefansson, $L^1$ of a vector measure $\mu$, Le Matematiche. 48, 219234, 1993.
Primary Language 
en

Subjects 
Mathematics

Journal Section 
Mathematics 
Authors 
Orcid: 0000000218900280 Author: Maryam MOHSENİPOUR Institution: Hakim Sabzevari University Country: Iran
Orcid: 0000000238153681 Author: Ghadir SADEGHİ (Primary Author) Institution: Hakim Sabzevari University Country: Iran

Dates 
Publication Date
: December 8, 2019

Bibtex 
@research article { hujms652608,
journal = {Hacettepe Journal of Mathematics and Statistics},
issn = {2651477X},
eissn = {2651477X},
address = {},
publisher = {Hacettepe University},
year = {2019},
volume = {48},
pages = {1590  1600},
doi = {10.15672/hujms.652608},
title = {Interpolation between weighted Lorentz spaces with respect to a vector measure},
key = {cite},
author = {MOHSENİPOUR, Maryam and SADEGHİ, Ghadir}
} 
APA

MOHSENİPOUR, M
, SADEGHİ, G
.
(2019).
Interpolation between weighted Lorentz spaces with respect to a vector measure.
Hacettepe Journal of Mathematics and Statistics
, 48 (6) ,
15901600 .
DOI: 10.15672/hujms.652608 
MLA

MOHSENİPOUR, M
, SADEGHİ, G
.
"Interpolation between weighted Lorentz spaces with respect to a vector measure".
Hacettepe Journal of Mathematics and Statistics 48 (2019
): 15901600 <https://dergipark.org.tr/en/pub/hujms/issue/50516/652608>

Chicago

MOHSENİPOUR, M
, SADEGHİ, G
.
"Interpolation between weighted Lorentz spaces with respect to a vector measure".
Hacettepe Journal of Mathematics and Statistics 48 (2019
): 15901600 
RIS 
TY  JOUR
T1  Interpolation between weighted Lorentz spaces with respect to a vector measure
AU  Maryam MOHSENİPOUR
, Ghadir SADEGHİ
Y1  2019
PY  2019
N1
 doi: 10.15672/hujms.652608 DO
 10.15672/hujms.652608 T2  Hacettepe Journal of Mathematics and Statistics
JF  Journal
JO  JOR
SP  1590
EP  1600
VL  48
IS  6
SN  2651477X2651477X
M3
 doi: 10.15672/hujms.652608 UR
 https://doi.org/10.15672/hujms.652608 Y2  2015
ER 

EndNote 
%0 Hacettepe Journal of Mathematics and Statistics Interpolation between weighted Lorentz spaces with respect to a vector measure
%A Maryam MOHSENİPOUR
, Ghadir SADEGHİ
%T Interpolation between weighted Lorentz spaces with respect to a vector measure
%D 2019
%J Hacettepe Journal of Mathematics and Statistics
%P 2651477X2651477X
%V 48
%N 6
%R doi: 10.15672/hujms.652608 %U 10.15672/hujms.652608 
ISNAD 
MOHSENİPOUR, Maryam
, SADEGHİ, Ghadir
.
"Interpolation between weighted Lorentz spaces with respect to a vector measure". Hacettepe Journal of Mathematics and Statistics
48
/
6
(December 2019):
15901600
. https://doi.org/10.15672/hujms.652608 
AMA 
MOHSENİPOUR M
, SADEGHİ G
.
Interpolation between weighted Lorentz spaces with respect to a vector measure.
Hacettepe Journal of Mathematics and Statistics.
2019;
48(6):
15901600.

Vancouver 
MOHSENİPOUR M
, SADEGHİ G
.
Interpolation between weighted Lorentz spaces with respect to a vector measure.
Hacettepe Journal of Mathematics and Statistics.
2019;
48(6):
16001590.
