Semantics

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By Michael Wilson

Littlewood-Paley idea is an important instrument of Fourier research, with functions and connections to PDEs, sign processing, and chance. It extends a few of the advantages of orthogonality to occasions the place orthogonality doesn’t rather make feel. It does so by means of letting us regulate sure oscillatory limitless sequence of features by way of countless sequence of non-negative capabilities. starting within the Eighties, it used to be came upon that this keep an eye on might be made a lot sharper than was once formerly suspected. the current e-book attempts to provide a gradual, well-motivated advent to these discoveries, the tools at the back of them, their results, and a few in their purposes.

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Otherwise, after canceling, we get λ2 ≤ C log(e + |I|/|Eλ |), which, after some algebra, yields |Eλ | ≤ C1 exp(−C2 λ2 )|I|. 3? For each integer k, let F k denote the family of dyadic intervals I for which 2k < 1 |I| v dx ≤ 2k+1 , I 3 Exponential Square 41 and define Dk = {x ∈ R : Md (v)(x) > 2k }. We observe that ∪{I : I ∈ F k } ⊂ Dk , and that every dyadic I for which I v > 0 lies in one and only one of the sets F k . Now we write 2 (S(f ))2 v dx = I |λI | |I| v dx I 2 |λI | = k I∈F k ≤ 1 |I| v dx I 2 |λI | .

We claim that, if x ∈ / ∪k Jk , then La (floc )(x) ≤ 2γλ. Then x ∈ S(fs )(x) = S(floc )(x). 5 showing up again. 9λ}. ) by Chebyshev’s inequality. 1. 1, we can see that we have obtained something slightly stronger. 21 for all p, 0 < p < ∞, for finite linear sums of Haar functions. The only role this finiteness hypothesis played was to ensure that S(f ) and f ∗ belonged to Lp . 32) and, if S(f ) ∈ Lp , then S(f ) p p. 32) rather than having to assume them as hypotheses. 31: if f is identically 1, its square function is identically 0.

10 fails; we show this at the end of the chapter. 10 are available, in which Md (v) is replaced by bigger maximal functions of v—such as iterations of Md (·). However, the proofs of these results also make no use of exponential-square estimates. What they use is the theory of Orlicz spaces, which we will develop later in the book. 8. 10 for 1 < p < 2, show that it fails for p > 2, and show an extension to 0 < p ≤ 1. 8—will have to wait until we have looked at Orlicz space theory. 11) in which neither v nor w is assumed to belong to Ad∞ .

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