Read e-book online Applications of Orlicz Spaces (Pure and Applied Mathematics) PDF

By M.M. Rao

ISBN-10: 0585425140

ISBN-13: 9780585425146

ISBN-10: 0824707303

ISBN-13: 9780824707309

Offers formerly unpublished fabric at the fundumental pronciples and houses of Orlicz series and serve as areas. Examines the pattern direction habit of stochastic tactics. offers sensible functions in information and chance.

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Sample text

This is exemplified by the following result of considerable importance in applications. Theorem 11. , e I nil. Then there exists a strictly convex continuous Young function $1 such that L*(JJ,) = L*1^), on any measure space (fi, £, /Lt), and their norms are equivalent. Thus the spaces are isomorphic (but not isometric). Moreover, z/<& is ^-regular, so is $1. Proof. This result was proved in our earlier volume ([1], p. 153), but we include a simple construction for completeness. Thus we have r\u\ = \ V(t)dt.

4 Interpolation results on Orlicz spaces 37 Then by (4) we obtain: 1 i+ 1+ as asserted. D We now present an Orlicz space extension of the Riesz-Thorin theorem for a generalization and applications below. Theorem 2. Let 3>j,Qi,z = 1,2 be N-functions, (Q,E,/^), (Q', £', z/) be a-finite measure triples. If3>s,Qs are the intermediate N-functions, as in (3), consider the (complex) Orlicz spaces M^i C L $i (//), and Lfi^v^i — 1,2 and similarly M* S ,L^ S (^). ,/ € M*v (8) and [In (6)-(8), the operators are first defined on simple or bounded functions and then they have unique extensions to all of M$i or M®s without affecting the bounds.

If 1 < p < 2, let 1 < a < p < 2,$(u) = |w| a ,$ 0 (w) = u2 and s = * I ° . Then 0 < s < 1 and ^ w = w^ or $ a u = u p . 4,1 ^ from (17). Similarly let 2 < p < b < oo, $(u) = |w|6, $ 0 (w) = ^2 and s = ^Ef}Then 0 < s < 1 and 4> s (n) = w p, lim -s = p. lim ^^ = 9- and hence 2 ~ 6t°o 6t°o (25) follows from (17). D We also can obtain the modulus of convexity (and of smoothness) for a large class of uniformly convex (and smooth) Orlicz spaces from (17) and (18). 4 Interpolation results on Orlicz spaces 43 and (p(X, r) = sup {1(||/ + 0|| + ||/ - 0||) - 1 : / € S(X), \\g\\ = r}} .

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Applications of Orlicz Spaces (Pure and Applied Mathematics) by M.M. Rao

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