Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis

Let X be a rearrangement invariant function space on [0,1]. We consider the subspace Radi X of X which consists of all functions of the form , where xk are arbitrary independent functions from X and rk are usual Rademacher functions independent of {xk}. We prove that Radi X is complemented in X if and only if both X and its Köthe dual space X^′ possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [0,∞). This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)].     

Lacunary Walsh series in rearrangement invariant spaces

We prove that the classical results by Rodin and Semenov and by Lindenstrauss and Tzafriri on the subspace generated by the Rademacher system in rearrangement invariant spaces also hold for lacunary Walsh series.     

Rademacher series in symmetric spaces

НАИДЕНО НЕОБхОДИМОЕ И ДОстАтОЧНОЕ УслОВИ Е НА РьД пО сИстЕМЕ РАДЕМАхЕРА Д ль тОгО, ЧтОБы ЕгО кОЁФФИцИЕН ты пРИНАДлЕжАлИ НЕкО тОРОМУ сИММЕтРИЧНОМУ кООРД ИНАтНОМУ пРОстРАНст ВУ, НАпРИМЕР,l _Р (1<p<2). ИжУЧАЕ тсь ВОпРОс О тОМ, кОгДА РАжлИЧНыЕ ФУНкцИОНА льНыЕ пРОстРАНстВА ИНДУцИ РУУт НА МНОжЕстВЕ кОЁ ФФИцИЕНтОВ РАжлИЧНыЕ НОРМы.     

Bases of rearrangement invariant spaces

We prove that if E is a rearrangement-invariant space, then a boundedly complete basis exists in E, if and only if one of the following conditions holds: 1) E is maximal and E ≠ L ₁[0, 1]; 2) a certain (any) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a boundedly complete basis in E. As a corollary, we state the following assertion: Any (certain) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a spanning basis in a separable rearrangement-invariant space E, if and only if the adjoint space E* is separable. We prove that in any separable rearrangement-invariant space E the Haar system either forms an unconditional basis, or a strongly conditional one. The Haar system represents a strongly conditional basis in a separable rearrangement-invariant space, if and only if at least one of the Boyd indices of this space is trivial.     

Embedding property on rearrangement invariant spaces

Let X be a rearrangement invariant space in R ⁿ and $W_X^{r_1 ,...,r_n } $ be an anisotropic Sobolev space which is a generalization of $W_p^{r_1 ,...,r_n } $ . The main subject of this paper is to prove the embedding theorem for $W_X^{r_1 ,...,r_n } $ .     

Mixed norm spaces and rearrangement invariant estimates

Our main goal in this work is to further improve the mixed norm estimates due to Fournier [13], and also Algervik and Kolyada [1], to more general rearrangement invariant (r.i.) spaces. In particular we find the optimal domains and the optimal ranges for these embeddings between mixed norm spaces and r.i. spaces.     

On Rosenthal's inequality and rearrangement invariant spaces

Khabarovsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 1, pp. 32–37, January–February, 1993.     

Sobolev type inequalities for rearrangement invariant spaces

In the setting of rearrangement invariant spaces, optimal Sobolev inequalities (via the gradient) are well understood. By means of an alternative functional, we obtain new Sobolev inequalities which are finer than (and not necessarily equivalent to) the ones mentioned above.     

Disjointly homogeneous rearrangement invariant spaces via interpolation

A Banach lattice E is called p-disjointly homogeneous  , 1≤p≤∞$1\le p\le \infty$, when every sequence of pairwise disjoint normalized elements in E   has a subsequence equivalent to the unit vector basis of _?p${?}_p$. Employing methods from interpolation theory, we clarify which r.i. spaces on [0,1]$[0,1]$ are p  -disjointly homogeneous. In particular, for every 1<p<∞$1<p<\infty$ and any increasing concave function φ   on [0,1]$[0,1]$, which is not equivalent to neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function φ  . Moreover, it is shown that given 1<p<∞$1<p<\infty$ and an increasing concave function φ with non-trivial dilation indices, there is a unique p-disjointly homogeneous space among all interpolation spaces between the Lorentz and Marcinkiewicz spaces associated with φ.     

Wavelet bases in rearrangement invariant function spaces

We point out that the well known characterization of spaces () in terms of orthogonal wavelet bases extends to any separable rearrangement invariant Banach function space on (equipped with Lebesgue measure) with nontrivial Boyd's indices. Moreover we show that such bases are unconditional bases of .

     

Discrete hilbert transforms and rearrangement invariant sequence spaces

A pair of rearrangement inequalities are obtained for a discrete analogue of the Hilbert transform which lead to necessary and sufficient conditions for certain discrete analogues of the Hilbert transform to be bouonded as linear operators between rearrangement invariant sequence spaces. In particular, if X is a rearrangement invariant space with indices α and β, then 0<β≤α<1 is both necessary and sufficient for these transforms to be bounded from X into itself, which generalizes a well known result of M. Riesz. Applications are made to discerete Hilbert transforms in higher dimensions, in particular, the discrete Riesz transforms are bounded from X into itself if and only if 0<β≤α<1.     

Martingale inequalities in rearrangement invariant function spaces

The Burkholder-Davis-Gundy equivalence of the square function and maximal function of a martingale is extended to the setting of rearrangement invariant function spaces. Supported in part by NSF DMS-8703815 and U.S.-Isreal Binational Science Foundation. Supported in part by U.S.-Israel Binational Science Foundation.     

Operator-valued martingale transforms in rearrangement invariant spaces and applications

In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved. Some well-known results are generalized and unified. Applications are given to classical operators such as the maximal operator and the p-variation operator of vector-valued martingales, then we can very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces. These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and the Boyd indices of the rearrangement invariant function spaces. Finally we give an equivalent characterization of UMD Banach lattices, and also prove the Fefferman-Stein theorem in the rearrangement invariant function spaces setting.