Search: MSC category 46E30
( Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) )
1. CJM 2012 (vol 65 pp. 331)
 Kadets, Vladimir; Martín, Miguel; Merí, Javier; Werner, Dirk

Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces
We show that for spaces with 1unconditional bases
lushness, the alternative Daugavet property and numerical
index 1 are equivalent. In the class of rearrangement
invariant (r.i.) sequence spaces the only examples of spaces with
these properties are $c_0$, $\ell_1$ and $\ell_\infty$.
The only lush r.i. separable function space on $[0,1]$ is $L_1[0,1]$;
the same space is the only r.i. separable function space on $[0,1]$
with the Daugavet property over the reals.
Keywords:lush space, numerical index, Daugavet property, KÃ¶the space, rearrangement invariant space Categories:46B04, 46E30 

2. CJM 2007 (vol 59 pp. 276)
 Bernardis, A. L.; MartínReyes, F. J.; Salvador, P. Ortega

Weighted Inequalities for HardySteklov Operators
We characterize the pairs of weights $(v,w)$ for which the
operator $Tf(x)=g(x)\int_{s(x)}^{h(x)}f$ with $s$ and $h$
increasing and continuous functions is of strong type
$(p,q)$ or weak type $(p,q)$ with respect to the pair
$(v,w)$ in the case $0
Keywords:HardySteklov operator, weights, inequalities Categories:26D15, 46E30, 42B25 

3. CJM 2005 (vol 57 pp. 897)
 Berezhnoĭ, Evgenii I.; Maligranda, Lech

Representation of Banach Ideal Spaces and Factorization of Operators
Representation theorems are proved for Banach ideal spaces with the Fatou property
which are built by the Calder{\'o}nLozanovski\u\i\ construction.
Factorization theorems for operators in spaces more general than the Lebesgue
$L^{p}$ spaces are investigated. It is natural to extend the Gagliardo
theorem on the Schur test and the Rubio de~Francia theorem on factorization of the
Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for
the scales far from $L^{p}$spaces this is impossible. For the concrete integral operators
it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces
are not valid. Representation theorems for the Calder{\'o}nLozanovski\u\i\ construction
are involved in the proofs.
Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³nLozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f Categories:46E30, 46B42, 46B70 

4. CJM 2003 (vol 55 pp. 969)
 Glöckner, Helge

Lie Groups of Measurable Mappings
We describe new construction principles for infinitedimensional Lie
groups. In particular, given any measure space $(X,\Sigma,\mu)$ and
(possibly infinitedimensional) Lie group $G$, we construct a Lie
group $L^\infty (X,G)$, which is a Fr\'echetLie group if $G$ is so.
We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an
arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie
group, modelled on the locally convex direct sum $\bigoplus_{i\in I}
L(G_i)$.
Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25 

5. CJM 2003 (vol 55 pp. 204)
 Yan, Yaqiang

On the Nonsquare Constants of Orlicz Spaces with Orlicz Norm
Let $l^{\Phi}$ and $L^\Phi (\Omega)$ be the Orlicz sequence space and
function space generated by $N$function $\Phi(u)$ with Orlicz norm.
We give equivalent expressions for the nonsquare constants $C_J
(l^\Phi)$, $C_J \bigl( L^\Phi (\Omega) \bigr)$ in sense of James and
$C_S (l^\Phi)$, $C_S \bigl( L^\Phi(\Omega) \bigr)$ in sense of
Sch\"affer. We are devoted to get practical computational formulas
giving estimates of these constants and to obtain their exact value in
a class of spaces $l^{\Phi}$ and $L^\Phi (\Omega)$.
Keywords:James nonsquare constant, SchÃ¤ffer nonsquare constant, Orlicz sequence space, Orlicz function space Category:46E30 

6. CJM 2001 (vol 53 pp. 565)
 Hare, Kathryn E.; Sato, Enji

Spaces of Lorentz Multipliers
We study when the spaces of Lorentz multipliers from $L^{p,t}
\rightarrow L^{p,s}$ are distinct. Our main interest is the case when
$s
Keywords:multipliers, convolution operators, Lorentz spaces, Lorentzimproving multipliers Categories:43A22, 42A45, 46E30 

7. CJM 2000 (vol 52 pp. 920)
 Evans, W. D.; Opic, B.

Real Interpolation with Logarithmic Functors and Reiteration
We present ``reiteration theorems'' with limiting values
$\theta=0$ and $\theta = 1$ for a real interpolation method
involving brokenlogarithmic functors. The resulting spaces
lie outside of the original scale of spaces and to describe them
new interpolation functors are introduced. For an ordered couple
of (quasi) Banach spaces similar results were presented without
proofs by Doktorskii in [D].
Keywords:real interpolation, brokenlogarithmic functors, reiteration, weighted inequalities Categories:46B70, 26D10, 46E30 

8. CJM 2000 (vol 52 pp. 789)
 Kamińska, Anna; Mastyło, Mieczysław

The DunfordPettis Property for Symmetric Spaces
A complete description of symmetric spaces on a separable measure
space with the DunfordPettis property is given. It is shown that
$\ell^1$, $c_0$ and $\ell^{\infty}$ are the only symmetric sequence
spaces with the DunfordPettis property, and that in the class of
symmetric spaces on $(0, \alpha)$, $0 < \alpha \leq \infty$, the only
spaces with the DunfordPettis property are $L^1$, $L^{\infty}$, $L^1
\cap L^{\infty}$, $L^1 + L^{\infty}$, $(L^{\infty})^\circ$ and $(L^1 +
L^{\infty})^\circ$, where $X^\circ$ denotes the norm closure of $L^1
\cap L^{\infty}$ in $X$. It is also proved that all Banach dual
spaces of $L^1 \cap L^{\infty}$ and $L^1 + L^{\infty}$ have the
DunfordPettis property. New examples of Banach spaces showing that
the DunfordPettis property is not a threespace property are also
presented. As applications we obtain that the spaces $(L^1 +
L^{\infty})^\circ$ and $(L^{\infty})^\circ$ have a unique symmetric
structure, and we get a characterization of the DunfordPettis
property of some K\"otheBochner spaces.
Categories:46E30, 46B42 

9. CJM 2000 (vol 52 pp. 849)
 Sukochev, F. A.

Operator Estimates for Fredholm Modules
We study estimates of the type
$$
\Vert \phi(D)  \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D  D_0
\Vert^{\alpha}, \quad \alpha = \frac12, 1
$$
where $\phi(t) = t(1 + t^2)^{1/2}$, $D_0 = D_0^*$ is an unbounded
linear operator affiliated with a semifinite von Neumann algebra
$\calM$, $D  D_0$ is a bounded selfadjoint linear operator from
$\calM$ and $(1 + D_0^2)^{1/2} \in \emt$, where $\emt$ is a symmetric
operator space associated with $\calM$. In particular, we prove that
$\phi(D)  \phi(D_0)$ belongs to the noncommutative $L_p$space for
some $p \in (1,\infty)$, provided $(1 + D_0^2)^{1/2}$ belongs to the
noncommutative weak $L_r$space for some $r \in [1,p)$. In the case
$\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this
result continues to hold under the weaker assumption $(1 +
D_0^2)^{1/2} \in \calC_p$. This may be regarded as an odd
counterpart of A.~Connes' result for the case of even Fredholm
modules.
Categories:46L50, 46E30, 46L87, 47A55, 58B15 
