By M.M. Rao

Offers formerly unpublished fabric at the fundumental pronciples and homes of Orlicz series and serve as areas. Examines the pattern direction habit of stochastic techniques. offers sensible purposes in facts and likelihood.

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**Extra resources for Applications of Orlicz Spaces (Pure and Applied Mathematics)**

**Example text**

M*v (8) and [In (6)-(8), the operators are first defined on simple or bounded functions and then they have unique extensions to all of M$i or M®s without affecting the bounds. ] The proof based on the three-line theorem of complex analysis is an Orlicz space extension of that given by the classical procedure for the Lebesgue spaces, and the detailed argument is given in Rao and Ren [1], pp. 227-229, and will be omitted here. e. 2c]~sC2For the packing problem and other applications in the following chapters, a special multilinear formulation of the above result is needed.

26) or in Krasnoselskii and Rutickii ([1], pp. 15, 27, 33). [Recall that a function M(-) is called the principal part of an TVfunction TV(-), if it (is convex and) coincides with TV(-) for large values of the argument. This Mp above is a principal part of <&p. ] The following result will often be used in Chapters 2-5, and it is recently verified by Yan [1]. Theorem 15. Let <$, ^ be a pair of complementary N-functions. Then we have 2a$/3\& = 1 = 2a\&j3$, (20) and f) — Q ~t O^% /-? ^OO*\ Proof. We only show (23) = 1.

Then we can find an integer no > 1 such that |c| < 2n° and so $(cu) < $(2n°u) < K n °$(u),u > u0. Hence r cfrl it <"" iI •*• \ Jf )iff«/-*' ^^ r*~i *—^^ • •/[|/l>«o] By the convexity of p$ (or $), it follows that for any fi 6 L*,z — 1,2, p$(/i + /2) < |[p*(2/i) + p*(2/2)] < oo, with c = 2 in the above. Thus L* is linear. If //(f2) = +00, then UQ = 0 so that the same argument is valid showing the linearity of L*. 14 /. Introduction and Background Material For the converse implication when // is diffuse, let QQ € S,0 < /^(fJo) < oo, be a set on which fj, is diffuse.