By Michael Wilson

Littlewood-Paley idea is a necessary software 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 quite make feel. It does so by way of letting us keep watch over definite oscillatory endless sequence of capabilities by way of endless sequence of non-negative capabilities. starting within the Eighties, it was once chanced on that this keep watch over can be made a lot sharper than was once formerly suspected. the current booklet attempts to provide a gradual, well-motivated advent to these discoveries, the equipment at the back of them, their effects, and a few in their purposes.

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42) Md (w) ≤ Cw, then w ∈ Ad∞ . 7. 43) (La (f )(x))p MR (v) dx. 45) (La (f )(x))p Mr,d (v) dx. 46) and (S(f ))p v dx ≤ Cp,r The proofs of these inequalities are fast and easy. 43, we write, (f ∗ (x))p v dx ≤ (f ∗ (x))p MR (v) dx ≤ Cp,R (S(f ))p MR (v) dx, where the ﬁrst inequality comes from the fact that v ≤ MR (v) and the second is due to the Ad∞ property of MR (v). The proofs of the others are practically identical. 1. a) If v and w are two weights and MR (v) ≤ w almost everywhere for some R > 1, then (f ∗ (x))p v dx ≤ Cp,R (S(f ))p w dx and (S(f ))p v dx ≤ Cp,R (La (f )(x))p w dx hold for all 0 < p < ∞ and all ﬁnite linear sums f = I λI h(I) .

If T is of weak type (p1 , p1 ) and (p2 , p2 ), then it maps boundedly from Lp (X) into Lp (Y ). 2 An Elementary Introduction 33 Proof. Let f ∈ Lp , ﬁx λ > 0, and write f = f1 + f2 , where f1 (x) = if |f (x)| > λ; otherwise. ) that f1 ∈ Lp1 and f2 ∈ Lp2 . Let’s ﬁrst assume that p2 < ∞. Because T is sublinear, ν ({y ∈ Y : |T f (y)| > λ}) ≤ ν ({y ∈ Y : |T f1 (y)| > λ/2}) + ν ({y ∈ Y : |T f2 (y)| > λ/2}) . 47) p2 dµ(x). 47 by pλp−1 and integrate from 0 to ∞, we get a constant times ∞ 0 λp−1−p1 |f |>λ |f | dµ(x) |f | dλ = |f | p1 λp−1−p1 dλ dµ(x) 0 X = (p − p1 )−1 p |f | 1 |f | p−p1 dµ(x) X = (p − p1 )−1 p |f | dµ(x), X and that’s what we want.

Hint: Almost-everywhere convergence comes from the Lebesgue diﬀerentiation theorem. 11. Let F1 ⊂ F2 ⊂ F3 · · · be an increasing sequence of ﬁnite subsets of D such that ∪k Fk = D. 50) I∈Fk converges to f in the L2 norm. 50 converges to f in the Lp norm. ) We know that f p ∼ S(f ) p . Since f ∈ Lp , S(f − fFk ) ≤ S(f ) < ∞ and S(f − fFk ) → 0 almost everywhere. Therefore, by Dominated Convergence, S(f − fFk ) → 0 in Lp , implying the result. We could also go another way: the sequence of operators deﬁned by f → fFk is uniformly bounded on Lp (1 < p < ∞), and the sequence converges to the identity on ﬁnite linear sums of Haar functions, which are dense in Lp .