Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The theory of $ p$-spaces with an application to convolution operators.


Author: Carl Herz
Journal: Trans. Amer. Math. Soc. 154 (1971), 69-82
MSC: Primary 22.65
DOI: https://doi.org/10.1090/S0002-9947-1971-0272952-0
MathSciNet review: 0272952
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The class of p-spaces is defined to consist of those Banach spaces B such that linear transformations between spaces of numerical $ {L_p}$-functions naturally extend with the same bound to B-valued $ {L_p}$-functions. Some properties of p-spaces are derived including norm inequalities which show that 2-spaces and Hilbert spaces are the same and that p-spaces are uniformly convex for $ 1 < p < \infty $. An $ {L_q}$-space is a p-space iff $ p \leqq q \leqq 2$ or $ p \geqq q \geqq 2$; this leads to the theorem that, for an amenable group, a convolution operator on $ {L_p}$ gives a convolution operator on $ {L_q}$ with the same or smaller bound. Group representations in p-spaces are examined. Logical elementarity of notions related to p-spaces are discussed.


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

  • [1] S. Bochner, Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind, Fund. Math. 20 (1933), 262-276.
  • [2] S. Bochner, Stable laws of probability and completely monotone functions, Duke Math. J. 3 (1937), 726-728. MR 1546026
  • [3] J. Bretagnolle, D. Dacunha-Castelle, and J.-L. Krivine, Lois stables et espaces $ {L^p}$, Ann. Inst. H. Poincaré Sect. B 2 (1965/66), 231-259. MR 34 #3605. MR 0203757 (34:3605)
  • [4] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414. MR 1501880
  • [5] C. S. Herz, A class of negative-definite functions, Proc. Amer. Math. Soc. 14 (1963), 670-676. MR 28 #1477. MR 0158251 (28:1477)
  • [6] P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris, 1937.
  • [7] J. Marcinkiewicz and A. Zygmund, Quelques inégalités pour les opèrations linèaires, Fund. Math. 32 (1939), 115-121.
  • [8] H. Nakano, Über normierte teilweisegeordnete Moduln, Proc. Imp. Acad. Tokyo 17 (1941), 311-317. MR 7, 249. MR 0014174 (7:249g)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22.65

Retrieve articles in all journals with MSC: 22.65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0272952-0
Keywords: Category of Banach spaces, tensor product, $ {L_p}$-space, p-space, locally compact group, representation, representative function, regular representation on $ {L_p}$, convolution operator, measure space, Bochner integral, negative-definite function, ultraproduct
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society