Interpolation of Linear Operators

The aim of this paper is to prove a generalization of a well-known convexity theorem of M. Riesz [8]. The Riesz theorem was originally deduced by "real-variable" techniques. Later, Thorin [10], Tamarkin and Zygmund [9], and Thorin [ll] introduced convexity properties of analytic functions in their study of Riesz's theorem. These ideas were put in especially suggestive form by A. P. Calderon and A. Zygmund [3]. It is the last mentioned approach to the Riesz theorem which is our starting point. In the interpolation theorem we shall prove, we vary not only over the Lebesgue spaces in question, but we also vary the linear operators in question. An exact statement will be found in Theorem 1. Part II contains the first application of the interpolation theorem. We shall consider "Bochner-Riesz" summability of multiple Fourier series and Fourier integrals; we prove that we have Lp norm convergence (for Kp< 00) for the Bochner-Riesz means below the critical index. These results are contained in Theorems 3 and 4. A second application will be found in Part III. We shall show that a theorem of Pitt for Fourier Series may be proved for all uniformly bounded orthonormal systems. The fact that Pitt's theorem may hold in general circumstances was suggested by Professor A. Zygmund to the author. The last result is interesting when reapplied to the case of Fourier series via the familiar device of rearrangements. The result contains well-known inequalities of F. Riesz and R. E. A. C. Paley, as well as an inequality recently proved by I. I. Hirschman(2).