Function theoretic results for complex interpolation families of Banach spaces (original) (raw)
Related papers
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.
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.
Interpolation of Vector-Valued Real Analytic Functions
Journal of the London Mathematical Society, 2002
Let ω ⊆ R d be an open domain. The sequentially complete DF-spaces E are characterized such that for each (some) discrete sequence (z n) ⊆ ω, a sequence of natural numbers (k n) and any family (x n,α) n∈N, |α|6kn ⊆ E the infinite system of equations ∂ |α| f ∂z α (z n) = x n,α for n ∈ N, α ∈ N d , |α| 6 k n , has an E-valued real analytic solution f.
Complex interpolation and convexity
Proceedings of the American Mathematical Society, 1987
The relations between the complex theory of interpolation for families of Banach spaces and the notion of uniform convexity are studied. It is proven that the moduli of uniform convexity vary smoothly with the interpolation spaces. A new notion of "distance" between Banach spaces is introduced.
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.
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.
Transactions of the American Mathematical Society, 1990
The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces {C n , II • liz} zEG ' where G is open and bounded in C k. The main result asserts that (under a mild assumption on the datum) the norm function (z, w)-+ IIwll; belongs to some anisotropic Sobolew class and is characterized by a nonlinear PDE of second order. The proof uses the duality theorem for the harmonic interpolation method (obtained earlier by the author). A new, simpler, proof of this duality relation is also presented in the paper. 8(Q(z)-18Q(z)) = O. Coifman has proposed the following generalization of this equation to higher dimensions: k
Holomorphic functions on Banach spaces
2006
This is a survey about some problems from the theory of holomorphic functions on Banach spaces which have attracted the attention of many researchers during the last thirty years.