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Search: MSC category 46E30 ( Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) )

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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 1-unconditional 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, Köthe space, rearrangement invariant space
Categories:46B04, 46E30

2. CJM 2007 (vol 59 pp. 276)

Bernardis, A. L.; Martín-Reyes, F. J.; Salvador, P. Ortega
Weighted Inequalities for Hardy--Steklov 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:Hardy--Steklov 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}n--Lozanovski\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}n--Lozanovski\u\i\ construction are involved in the proofs.

Keywords:Banach ideal spaces, weighted spaces, weight functions,, Calderón--Lozanovski\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 infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie 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, Schäffer nonsquare constant, Orlicz sequence space, Orlicz function space

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, Lorentz-improving 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 broken-logarithmic 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, broken-logarithmic functors, reiteration, weighted inequalities
Categories:46B70, 26D10, 46E30

8. CJM 2000 (vol 52 pp. 789)

Kamińska, Anna; Mastyło, Mieczysław
The Dunford-Pettis Property for Symmetric Spaces
A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that $\ell^1$, $c_0$ and $\ell^{\infty}$ are the only symmetric sequence spaces with the Dunford-Pettis property, and that in the class of symmetric spaces on $(0, \alpha)$, $0 < \alpha \leq \infty$, the only spaces with the Dunford-Pettis 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 Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space 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 Dunford-Pettis property of some K\"othe-Bochner 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 self-adjoint 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 non-commutative $L_p$-space for some $p \in (1,\infty)$, provided $(1 + D_0^2)^{-1/2}$ belongs to the non-commutative 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

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