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A family of complemented subspaces in VMO and its isomorphic classification

Authors:

Email author" target="_blank">Paul?F?X?Müller Email author

Institution:

1.Department of Mathematics,J. Kepler University Linz,Linz,Austria

Abstract:

In this paper we study families of spaces which are similar in spirit to the Rosenthal class. We letS ⁰ be the infinite dimensional sequence space where the norm of a given null-sequence (a _I) is given as follows,

$\left\| {(a_{\rm I} )} \right\|so = \left\| {\sum x_I a_I h_I } \right\|vmo + sup\left| {a_I } \right|$

. Here (x _I) is a fixed sequence of bounded scalars. We show that these spaces are isomorphic to complemented subspaces of VMO, and classify their isomorphic types as follows:S ⁰ is isomorphic either toc ₀, to (ΣBMO_n)₀, or to VMO. The spaceS ⁰ arises as endpoint of the scaleS ^p, 2≤p<∞, where the norm of a sequence (a _I) is given by

$\left\| {(a_{\rm I} )} \right\|sP = \left\| {\sum a_I x_{_I }^{1 - 2/p} \frac{{h_I }}{{|I|^{I/p} }}} \right\|_{Lp} + (\sum |a_I |^p )^{1/p}$

. The isomorphic types of this class are shown to beL ^p and ℓ^p.

Keywords:

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