Interpolation of abstract Cesàro, Copson and Tandori spaces

Abstract

We study real and complex interpolation of abstract Cesàro, Copson and Tandori spaces, including the description of Calderón-Lozanovskiǐ construction for those spaces. The results may be regarded as generalizations of interpolation for Cesàro spaces Ces p (I) in the case of real method, but they are new even for Ces p (I) in the case of complex method. Some results for more general interpolation functors are also presented. The investigations show an interesting phenomenon that there is a big difference between interpolation of Cesàro function spaces in the cases of finite and infinite interval.

Symmetrization, factorization and arithmetic of quasi-Banach function spaces

“…In fact, applying Theorem 1(iv) from [36], Theorem 4 from [40] (since F ′ and G are symmetric), the equality E (p) = ( E) (p) and again Theorem 1(iv) from [36] we obtain…”

“…where H is a Hardy operator and f (x) = ess sup t∈I, t≥x |f (t)| (cf. [1], [39], [40]). For example, if E = L p (I) the respective space CL p (I) is the classical Cesàro function space denoted usually by Ces p (I) .…”

“…Suppose we try to prove this result similarly as for I = (0, ∞) . Unfortunately, we are not able to apply Theorem 4 from [40] because the respective space F ′ (1/w) is not symmetric. Thus, for Theorem 4 from [40], we need to assume that H, H * are bounded on F ′ (1/w) and M (E, F ) which do not seem to be reasonable.…”

Local approach to order continuity in Cesàro function spaces

“…for 0 < x ∈ I. For a Banach ideal space X on I we define an abstract Cesàro space CX = CX(I) by CX := {f ∈ L 0 (I) : C|f | ∈ X}, with the norm f CX = C|f | X (see [9], [10], [22], [23], [24]). Let us note that for nonsymmetric space X the space CX need not have a weak unit even if X has it (see [22,Example 2]), so in general supp(CX) ⊂ supp(X).…”

Rotundity and monotonicity properties of selected Cesàro function spaces

“…Furthermore, sometimes the cases Ces p [0,1] and Ces p [0, ∞) are essentially different (see an isomorphic description of the Köthe dual of Ces àro spaces in [1,34], see also [23] for the respective isometric description). The spaces generated by the Cesàro operator (including abstract Cesàro spaces) have been considered by Curbera, Delgado, Soria, Ricker, Leśnik and Maligranda in several papers (see [13][14][15][16][34][35][36]). …”