Interpolation of operators with change of measures
1958; American Mathematical Society; Volume: 87; Issue: 1 Linguagem: Inglês
10.1090/s0002-9947-1958-0092943-6
ISSN1088-6850
Autores Tópico(s)Numerical methods in inverse problems
ResumoIntroduction.This paper has two purposes.First, it is to give general interpolation theorems which allow one to change measures simultaneously with changing exponents of the Lp classes concerned.These theorems are generalizations of theorems of M. Riesz [7] and Marcinkiewicz [4].The idea of interpolating between measure seems to have been implicit for some time.The general situation has, however, only recently been investigated.(See e.g.[2] and [8].)Our second purpose is to apply these techniques.In § §3 and 4, we shall prove theorems for unbounded orthonormal systems which generalize known results for bounded orthonormal systems.These theorems are extensions of theorems of Paley, Marcinkiewicz and Zygmund [5], and Pitt [6].§1 will be devoted to necessary definitions and statements of needed known results.§2 will contain the general interpolation theorems.1. Definitions and statement of general results.Let (M, 93J, p) and(N, yi, v) be two measure spaces.We will consider only measurable, real or complex-valued functions on these two spaces.In this paper we will establish a generalization of the well known convexity theorem of M. Riesz (Theorem (1.1) below) and give some applications of this new theorem to certain orthonormal series.In order to do this we will need to establish some notation and give a few definitions.Suppose T is a mapping of a class of functions on M into a class of functions on N. T is called a sublinear operator if it satisfies the following properties:(i) If/=/i+/2 and Tft (i=l, 2) are defined then Tf is defined; (ii) | T(fx+f2) | ^ | Tf\\ + | Tf2\ almost everywhere;(iii) For any scalar k, we have | T(kf)\ =\k\\ Tf\ almost everywhere.Let p, g2t 1 be two real numbers.We say that T is of type (p, q) in case T is defined for all functions/ in L"(M, W, ju) and there exists a positive number, K, independent of/such that llr/||3, ^ll/IU where
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