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Transactions of the American Mathematical Society

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Interpolation theorems for the pairs of spaces $ (L\sp{p},\,L\sp{\infty })$ and $ (L\sp{1},\,L\sp{q})$


Authors: George G. Lorentz and Tetsuya Shimogaki
Journal: Trans. Amer. Math. Soc. 159 (1971), 207-221
MSC: Primary 46M35; Secondary 46E30
DOI: https://doi.org/10.1090/S0002-9947-1971-0380447-9
MathSciNet review: 0380447
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Abstract: A Banach space $ Z$ has the interpolation property with respect to the pair $ (X,Y)$ if each $ T$, which is a bounded linear operator from $ X$ to $ X$ and from $ Y$ to $ Y$, can be extended to a bounded linear operator from $ Z$ to $ Z$. If $ X = {L^p},Y = {L^\infty }$ we give a necessary and sufficient condition for a Banach function space $ Z$ on $ (0,l),0 < l \leqq + \infty $, to have this property. The condition is that $ g \prec {}^pf$ and $ f \in Z$ should imply $ g \in Z$; here $ g \prec {}^pf$ means that $ {g^{ \ast p}} \prec {f^{ \ast p}}$ in the Hardy-Littlewood-Pólya sense, while $ {h^ \ast }$ denotes the decreasing rearrangement of the function $ \vert h\vert$.

If the norms $ \vert\vert T\vert{\vert _X},\vert\vert T\vert{\vert _Y}$ are given, we can estimate $ \vert\vert T\vert{\vert _Z}$. However, there is a gap between the necessary and the sufficient conditions, consisting of an unknown factor not exceeding $ {\lambda _p},{\lambda _p} \leqq {2^{1/q}},1/p + 1/q = 1$.

Similar results hold if $ X = {L^1},Y = {L^q}$. For all these theorems, the complete continuity of $ T$ on $ Z$ is assured if $ T$ has this property on $ X$ or on $ Y$, and if $ Z$ satisfies a certain additional necessary and sufficient condition, expressed in terms of $ \vert\vert{\sigma _a}\vert{\vert _Z},a > 0$, where $ {\sigma _a}$ is the compression operator $ {\sigma _a}f(t) = f(at),0 \leqq t < l$.


References [Enhancements On Off] (What's this?)

  • [1] D. W. Boyd, The Hilbert transform on rearrangement invariant spaces, Canad. J. Math. 19 (1967), 599-616. MR 35 #3383. MR 0212512 (35:3383)
  • [2] A. P. Calderón, Spaces between $ {L^1}$ and $ {L^\infty }$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273-299. MR 34 #3295. MR 0203444 (34:3295)
  • [3] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
  • [4] M. A. Krasnosel'skiĭ, On a theorem of M. Riesz, Dokl. Akad. Nauk SSSR 131 (1960), 246-248=Soviet Math. Dokl. 1 (1960), 229-231. MR 22 #9852. MR 0119086 (22:9852)
  • [5] M. A. Krasnosel'skiĭ et al., Integral operators in spaces of summable functions, ``Nauka", Moscow, 1966. (Russian) MR 34 #6568. MR 0206751 (34:6568)
  • [6] M. A. Krasnosel'skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961. MR 0126722 (23:A4016)
  • [7] S. G. Kreĭn and Ju. I. Petunin, Scales of Banach spaces, Uspehi Mat. Nauk 21 (1966), no. 2 (128), 89-168=Russian Math. Surveys 21 (1966), no. 2, 85-159. MR 33 #1719. MR 0193499 (33:1719)
  • [8] G. G. Lorentz, Bernstein polynomials, Math. Expositions, no. 8, Univ. of Toronto Press, Toronto, 1953. MR 15, 217. MR 0057370 (15:217a)
  • [9] -, On the theory of spaces $ \Lambda $, Pacific. J. Math. 1 (1951), 411-429. MR 13, 470. MR 0044740 (13:470c)
  • [10] G. G. Lorentz and T. Shimogaki, Interpolation theorems for operators in function spaces, J. Functional Analysis 2 (1968), 31-51. MR 41 #2423. MR 0257775 (41:2424)
  • [11] -, Majorants for interpolation theorems, Publ. Ramanujan Inst. No. 1 (1969), 115-122. MR 0275197 (43:954)
  • [12] W. A. J. Luxemburg, Rearrangement invariant Banach function spaces, Proc. Sympos. Analysis, Queen's Univ. 10 (1967), 83-144.
  • [13] W. Orlicz, On a class of operations over the space of integrable functions, Studia Math. 14 (1955), 302-309. MR 16, 834. MR 0068125 (16:834c)
  • [14] T. Shimogaki, On the complete continuity of operators in an interpolation theorem, J. Fac. Sci. Hokkaido Univ. Ser. I 20 (1968), 109-114. MR 39 2011. MR 0240665 (39:2011)
  • [15] -, An interpolation theorem on Banach function spaces, Studia Math. 31 (1968), 233-240. MR 38 #2618. MR 0234301 (38:2618)
  • [16] -, On an equivalence relation on semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I 18 (1964), 41-55. MR 30 #5149. MR 0174959 (30:5149)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0380447-9
Keywords: Interpolation property, interpolation theorem, quasi-order, rearrangement invariant Banach function space, space monotone with respect to a quasi-order, completely continuous operator, compression operator, Orlicz space
Article copyright: © Copyright 1971 American Mathematical Society

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