Generalized interpolation spaces

Williams, Vernon

doi:10.1090/S0002-9947-1971-0275149-3

Generalized interpolation spaces
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Trans. Amer. Math. Soc. 156 (1971), 309-334 Request permission

Abstract:

In this paper we introduce the notion of “generalized” interpolation space, and state and prove a “generalized” interpolation theorem. This apparently provides a foundation for an axiomatic treatment of interpolation space theory, for subsequently we show that the “mean” interpolation spaces of Lions-Peetre, the “complex” interpolation spaces of A. P. Calderón, and the “complex” interpolation spaces of M. Schechter are all generalized interpolation spaces. Furthermore, we prove that each of the interpolation theorems established respectively for the above-mentioned interpolation spaces is indeed a special case of our generalized interpolation theorem. In §III of this paper we use the generalized interpolation space concept to state and prove a “generalized” duality theorem. The very elegant duality theorems proved by Calderón, Lions-Peetre and Schechter, respectively, are shown to be special cases of our generalized duality theorem. Of special interest here is the isolation by the general theorem of the need in each of the separate theorems for certain “base” spaces to be duals of others. At the close of §II of this paper we employ our generalized interpolation theorem “structure” to construct new interpolation spaces which are neither complex nor mean spaces.

References

A.-P. Calderón, Intermediate spaces and interpolation, Studia Math. (Ser. Specjalna) Zeszyt 1 (1963), 31–34. MR 0147896
A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Peter D. Lax, Symmetrizable linear transformations, Comm. Pure Appl. Math. 7 (1954), 633–647. MR 68116, DOI 10.1002/cpa.3160070403
Jacques-Louis Lions, Une construction d’espaces d’interpolation, C. R. Acad. Sci. Paris 251 (1960), 1853–1855 (French). MR 119093
J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343, DOI 10.1007/BF02684796
Enrico Magenes, Spazi d’interpolazione ed equazioni a derivate parziali, Atti del Settimo Congresso dell’Unione Matematica Italiana (Genova, 1963) Edizioni Cremonese, Rome, 1965, pp. 134–197 (Italian). MR 0215082

Sur l’interpolation d’opérateurs

208

Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
Marcel Riesz, Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires, Acta Math. 49 (1927), no. 3-4, 465–497 (French). MR 1555250, DOI 10.1007/BF02564121
Martin Schechter, Complex interpolation, Compositio Math. 18 (1967), 117–147 (1967). MR 223880
Martin Schechter, General boundary value problems for elliptic partial differential equations, Comm. Pure Appl. Math. 12 (1959), 457–486. MR 125323, DOI 10.1002/cpa.3160120305
Martin Schechter, On $L^{p}$ estimates and regularity. I, Amer. J. Math. 85 (1963), 1–13. MR 188615, DOI 10.2307/2373179
Martin Schechter, On $L^{p}$ estimates and regularity. II, Math. Scand. 13 (1963), 47–69. MR 188616, DOI 10.7146/math.scand.a-10688
Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966

Theory of functions

A. H. Zemanian, Distribution theory and transform analysis. An introduction to generalized functions, with applications, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0177293