Higher order estimates in complex interpolation theory (original) (raw)

A theory of complex interpolation for families of Banach spaces

Advances in Mathematics, 1982

A detailed development is given of a theory of complex interpolation for families of Banach spaces which extends the well-known theory for pairs of spaces. 203 000 I-8708/82/030203-27$05.00/O Copyri%t 0 1982 by Academic Rew. Inc. All &hts of reproduction i n my fm resawd.

Complex interpolation of some quasi-Banach spaces

Journal of Functional Analysis, 1986

We interpolate in the complex method some real-intermediate quasi-Banach spaces. This enables us for example to get in a unilied way complex interpolation of H,, spaces 0 <pO <pr 4 co, from the real interpolation results. The H, spaces could be the standard ones as well as weighted H, spaces, H,, spaces on product domains, etc. (' 1986 Academic Press. Inc An extension of the A. P. Calderon method of complex interpolation to quasi-Banach spaces was first considered by N. M. Riviere in [ 111. The main obstacle to a fully successful theory is the failure of the maximum principle for functions assuming values in a quasi-Banach space; see . We should point out that for the real interpolation method the situation is quite different; most central theorems from the Banach space setting hold in the quasi-Banach, and even in a more general situation.

Nigel Kalton and complex interpolation of compact operators

The fundamental theorem of complex interpolation is Boundedness Theorem: If, for j = 0, 1, a linear operator T is a bounded map from the Banach space X j to the Banach space Y j then, for each θ, 0 < θ < 1, T is a bounded map between the complex interpolation spaces [X 0 , X 1 ] θ and [Y 0 , Y 1 ] θ. Alberto Calderón, in his foundational presentation of this material fifty-one years ago [5], also proved the following companion result: Compactness Theorem/Question: Furthermore in some cases, if T is also a compact map from X 0 to Y 0 , then, for each θ, T is a compact map from [X 0 , X 1 ] θ to [Y 0 , Y 1 ] θ. The fundamental question of exactly which cases could be covered by such a result was not resolved then, and is still not resolved. The paper [20], which is the focus of this commentary, is a contribution to that question. We will not summarize in any detail here the contents of [20] or of related works. (Some of that may be done later in [27].) Rather we will take this opportunity to sketch the mathematical world, historical and current, in which that paper lives. We will see that there have been many very talented contributors and many fine contributions; however the core problem remains open. We will at least be able to announce some small new partial results.

Complex Interpolation of Operators and Optimal Domains

Integral Equations and Operator Theory, 2014

Let X0 and X1 be two order continuous Banach function spaces on a finite measure space, (E0, E1) a Banach space interpolation pair, and T : X0 + X1 → E0 + E1 an admissible operator between the pairs (X0, X1) and (E0, E1). If T θ : [X0, X1] [θ] → [E0, E1] [θ] is the interpolated operator by the first complex method of Calderón and m0, m1 and m θ are the vector measures coming from T | X 0 and T | X 1 and T θ , respectively, then we study the relationship between the optimal domain L 1 (m θ) of T θ and the complex interpolation space [L 1 (m0), L 1 (m1)] [θ] of the optimal domains of T | X 0 and T | X 1. Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p, q)power-concave operators.

Lecture notes on complex interpolation of compactness

We recall that the fundamental theorem of complex interpolation is the Boundedness Theorem: If, for j = 0, 1, a linear operator T is a bounded map from the Banach space X j to the Banach space Y j then, for each θ, 0 < θ < 1, T is a bounded map between the complex interpolation spaces [X 0 , X 1 ] θ and [Y 0 , Y 1 ] θ. Alberto Calderón, in his foundational presentation of this material fifty-one years ago [4], also proved the following companion result: Compactness Theorem/Question: Furthermore in some cases, if T is also a compact map from X 0 to Y 0 , then, for each θ, T is a compact map from [X 0 , X 1 ] θ to [Y 0 , Y 1 ] θ. The fundamental question of exactly which cases could be covered by such a result was not resolved then, and is still not resolved. In a previous paper [10] we surveyed several of the partial answers which have been obtained to this question, with particular emphasis on the work of Nigel Kalton in a joint paper [8] with one of us. This is a preliminary version of a set of lecture notes which will be a sequel to [10]. In them, for the most part, we will amplify upon various technical details of the contents of [8]. For example we plan to give a more explicit explanation of why the positive answer in [8] to the above question when (X 0 , X 1) is a couple of lattices holds without any restriction on those lattices, and we also plan to provide more detailed versions of some of the other proofs in that paper. The main purpose of this preliminary version is to present two apparently new small results, pointing out a previously unnoticed particular case where the answer to the above question is affirmative. As our title suggests, this and future versions of these notes are intended to be more accessible to graduate students than a usual research article.

Function theoretic results for complex interpolation families of Banach spaces

Transactions of the American Mathematical Society, 1984

The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville’s theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.

On certain Banach spaces in connection with interpolation theory

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.