Intermediate spaces and interpolation, the complex method

Intermediate spaces and interpolation, the complex method

T. XXIV. (1964) | A. P. Calderón
This paper discusses the complex method of interpolation, introduced by A. P. Calderón and independently by J. L. Lions. The paper is divided into two main parts. The first part covers the properties of intermediate spaces derived from the properties of interpolating spaces, including a detailed study of duality and a second complex method of interpolation yielding intermediate spaces $[B_0, B_1]^*$. The second part focuses on determining intermediate spaces between given spaces, particularly between Banach lattices of functions and various classes of Hölder continuous functions. The paper also introduces a general class $A(B, X)$ and provides a representation for functions in this class. The presentation is structured with paragraphs 1 to 14 containing definitions and statements of results, followed by proofs in paragraphs 15 to 34. The paper systematically uses functions with values in a Banach space and references other works on interpolation methods. Key results include the construction of Banach spaces $B_s$ and $B^s$ from interpolation pairs $(B^0, B^1)$, the extension of linear mappings to multilinear mappings, and the study of interpolation between function spaces related to Lipschitz functions.This paper discusses the complex method of interpolation, introduced by A. P. Calderón and independently by J. L. Lions. The paper is divided into two main parts. The first part covers the properties of intermediate spaces derived from the properties of interpolating spaces, including a detailed study of duality and a second complex method of interpolation yielding intermediate spaces $[B_0, B_1]^*$. The second part focuses on determining intermediate spaces between given spaces, particularly between Banach lattices of functions and various classes of Hölder continuous functions. The paper also introduces a general class $A(B, X)$ and provides a representation for functions in this class. The presentation is structured with paragraphs 1 to 14 containing definitions and statements of results, followed by proofs in paragraphs 15 to 34. The paper systematically uses functions with values in a Banach space and references other works on interpolation methods. Key results include the construction of Banach spaces $B_s$ and $B^s$ from interpolation pairs $(B^0, B^1)$, the extension of linear mappings to multilinear mappings, and the study of interpolation between function spaces related to Lipschitz functions.
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