Abstract. The class of p-spaces is defined to consist of those Banach spaces B such that linear transformations between spaces of numerical ¿"-functions naturally extend with the same bound to ß-valued ¿"-functions. Some properties of p-spaces are derived including norm inequalities which show that 2-spaces and Hubert spaces are the same and that ^-spaces are uniformly convex for 1 ë<7=i2 or p^q^2; this leads to the theorem that, for an amenable group, a convolution operator on Lp gives a convolution operator on Lq with the same or smaller bound. Group representations in p-spaces are examined. Logical elementarily of notions related to /»-spaces are discussed. 0. Introduction. Let R designate the field of real or complex numbers. We denote by 38 the category whose objects are complete normed linear spaces over R and whose morphisms are the bounded i?-linear transformations of norm f£ 1.Thus 38(B, C) is the unit ball of HOM (B, C), the latter being the Banach space of all bounded .R-linear transformations from B to C. The endofunctor C i-> HOM (B, C) has a left adjoint A h> A B. In more concrete terms, the tensor product may be viewed this way: each element t e A ® B has a representation r=2f an <8> bn where {an}<=A, {bA<^B and ||r|| ^2 IM ¡M <°o, indeed ||/|| is the infimum of 2 ||«n|| ||èn|| taken over all representations. The concrete viewpoint is given only as a heuristic crutch.Suppose (/x) is a measure space and 1 gp • -^ LAß.; ■).Suppose that iv) is also a measure space and y: Lpí¡jl; R) -> Lv'v; R) is a morphism.
Let t denote a point of Euclidean k-dimensional space Rk. If F(t) e LO(Rk), 1 < p < 2, F has a Fourier transform G(s) -fRh exp (is-t)F(t) dt. C shall denote any convex polyhedron in Rk containing the origin in its interior and nC, n > 0, its dilations. The following theorem was communicated to the author by S. Bochner.
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