By Mario Milman

ISBN-10: 0387580816

ISBN-13: 9780387580814

This publication develops a idea of extrapolation areas with purposes to classical and smooth research. Extrapolation conception goals to supply a normal framework to review proscribing estimates in research. The e-book additionally considers the function that optimum decompositions play in proscribing inequalities incl. commutator estimates. lots of the effects provided are new or haven't seemed in ebook shape ahead of. a different characteristic of the e-book are the functions to different components of research. between them Sobolev imbedding theorems in several contexts together with logarithmic Sobolev inequalities are got, commutator estimates are hooked up to the speculation of comp. compactness, a reference to maximal regularity for summary parabolic equations is proven, sharp estimates for maximal operators in classical Fourier research are derived.

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**Extra info for Extrapolation and Optimal Decompositions: with Applications to Analysis**

**Example text**

Let Hi = {y E Qi: I n " f ( x ~ ) - n " f ( y ) l > A(t)N}, r where A(t) = ~l~l

To do so let r = Ixl - x 2 I , and let Qi , be cubes centered at xi, i = O, 1, with sides parallel to the coordinate axes with length equal to ~ r. Then, by simple geometry, there exists a constant c,, independent of the x~s such that [Q1 rl Q2[ > c . I Q i l , i -- 0, 1. Let Hi = {y E Qi: I n " f ( x ~ ) - n " f ( y ) l > A(t)N}, r where A(t) = ~l~l

Theorem 6 (cf. [721) ~_,{O-1L(LogL)I+~ Proof. 29) with norm equivalence independent of 6. Note t h a t the spaces L(LogL)~(T),fl E [0,2], have absolutely continuous norms. Then, for 0 e [0,1], we have isometrically (cf. 30) 24 CHAPTER 2. Calderdn [14]. First observe that, since we are dealing with spaces on a finite measure space, only large values of the Young's functions involved are important. In our case we shall take x > e ~. Using Calderdn's notation we write L(LogL) ~ = A51(L1), where An(x ) = x(log x)O,~ r [0,2], for x _> e ~.

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