By M.M. Rao

Provides formerly unpublished fabric at the fundumental pronciples and houses of Orlicz series and serve as areas. Examines the pattern course habit of stochastic tactics. presents functional purposes in records and chance.

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

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Thus we have r\u\ = \ V(t)dt. /o (4) 32 /. Introduction and Background Material Define a new function (f>i(t) = 0 and set /• / Joo 0 and strictly increasing, $1 is strictly convex. Moreover, it is clear that $(u) < $i(u) < 2$(w),u € R Hence / e L*(jz) iff / € L*i(yi) and ||/||w < ||/||(4>l) < 2||/||w. Thus, the spaces are isomorphic. The A2-regularity is immediate from construction. D In the above reference an uncountable collection of distinct Young functions generating equivalent norms is given.

Moreover, ||T/|| = | \Tf \\sp in this case. Hence, we have II7/H = sup{||:Z>G/)||* : ||^||« < 1} < K\\f\\*, (8) which implies (7), since In the reverse direction, suppose (7) holds for simple functions /, g. Then f*g G L*(R) and 0 < K < oo is an absolute constant. It suffices to consider 0 < /, g since for the general case / = fi — /2, g = g\ — gi implies / * g = fi * gi - /2 * g\ - f\ * #2 + h * 92 and the result will follow, by linearity, for all /, g in L*(R) (and then similarly for complex functions).

Consider the statements: (i) $ is AZ-regular, (ii) L* = L®, (Hi) lim ||/||$ — 0, p*(/)-+o (iv) linip^^oo ||/||$ = oo. Then (i) <=> (ii), (Hi) 4= (i) => (iv). If p, is diffuse on a set of positive measure, then all are equivalent conditions. The proof of this result is found in Rao and Ren ([1], p. 83), and the last statement uses the same argument as in the above book (p. 46). However, we include the proof that (iii) => (i) when p, is diffuse on a set r^ 0 ,0 < /i(^o) < CXD, to indicate the argument.

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