Complex interpolation of families of Orlicz sequence spaces
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COMPLEX INTERPOLATION OF FAMILIES OF ORLICZ SEQUENCE SPACES BY
Willian Hans Goes Corrˆ ea∗ Departamento de Matem´ atica, Instituto de Matem´ atica e Estat´ıstica Universidade de S˜ ao Paulo, Rua do Mat˜ ao 1010, 05508-090 S˜ ao Paulo SP, Brazil e-mail: [email protected]
ABSTRACT
We study the complex interpolation and derivation process induced by a family of Orlicz sequence spaces. We present a concrete example of an interpolation family of three spaces inducing a centralizer that cannot be obtained from complex interpolation of two spaces.
1. Introduction Complex interpolation induces a homogeneous (usually nonlinear) map called the derivation map (see the next section for the definition). Given a K¨ othe function space X on a Polish space S with measure μ, a centralizer is a homogeneous map Ω : X → L0 (μ) for which there is a constant C > 0 such that for every x ∈ X, u ∈ L∞ (μ) we have Ω(ux) − uΩ(x)X ≤ Cu∞ xX . In [6, 7] Kalton showed that the derivations induced by complex interpolation of families of K¨othe function spaces are centralizers. Conversely, given a centralizer Ω on a superreflexive K¨ othe function space X, there is an interpolation family of K¨ othe function spaces such that the interpolation space at 0 is X and ∗ The present work was supported by S˜ ao Paulo Research Foundation (FAPESP),
processes 2016/25574-8 and 2018/03765-1. Received June 5, 2019 and in revised form September 12, 2019
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ˆ W. H. G. CORREA
Isr. J. Math.
the induced derivation is Ω. If the centralizer is real (i.e., Ω(f ) is a real function if f is real) then a family of two spaces is enough to induce Ω; otherwise, three spaces are enough. Kalton’s results leave unanswered the question of whether three spaces are necessary to obtain any centralizer. The answer is: sometimes yes; and a general argument is given in Section 3 while an explicit argument appears in Section 5. Indeed, the aim of this work is to give a concrete example of a family of three K¨othe sequence spaces inducing a centralizer that cannot be induced by interpolation of two spaces. In the search for such an example, one is tempted to use scales of p spaces or p,q spaces. However, the reiteration result [1, Theorem 4.15] shows that this approach is bound to fail. A natural candidate would then be a family formed by Orlicz spaces. In Section 4 we present a detailed treatment of complex interpolation for families of Orlicz sequence spaces and their associated derivations to finally present in Section 5 the desired example.
2. Complex interpolation and derivations We present our results in the context of Kalton’s interpolation method for K¨othe function spaces [7]. In what follows CN is endowed with the product topology. Kalton’s definition of K¨ othe function space is not the classic one. A K¨ othe function space X on N is a linear subspace of CN with a norm · X which makes it into a complete Banach space such that (for x ∈ CN \ X we make the convention that xX = ∞): (1) BX is closed in CN , where BX is the closed unit ball of X; (2) for every
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