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Next, we check (i) and (ii). (i) For the first inequality, let q ∈ (0, min{2, p}]. Clearly, the inequality is valid for p = q = 2. So we just consider following two cases. Case 1: 0 < q < 2 < p. 4. Mixture of Derivative and Quotient 37 thereupon producing F pHp I(F ; p, q). Case 2: 0 < q < p < 2. By the first Littlewood–Paley inequality above, H¨older’s inequality and the Hardy–Stein identity for Fr , we find p Hp Fr I(Fr ; p, p) I(Fr ; p, q) I(F ; p, q) 2−p 2−q 2−p 2−q I(Fr ; p, 2) Fr p(p−q) 2−q Hp p−q 2−q , thereupon implying F pHp I(F ; p, q).

22(p+q)n (1 − |w|2 )p Thus ∞ µ CMp ,q ≤ sup w∈D n=1 ∞ sup w∈D n=1 En \En−1 µ(En ) 22n(p+q) (1 − |w|2 )p ∞ µ (1 − |w|2 )q dµ(z) |1 − wz| ¯ p+q CMp 2−nq , n=1 as desired. For a, b > 0 we define formally a Ta,b operator; that is, the following integral operator: (1 − |w|2 )b−1 Ta,b f (z) = f (w)dm(w), z ∈ D. ¯ a+b D |1 − wz| In the next result we show that CMp is invariant under Ta,b . 2. Let p ∈ (0, 2), a > 2−p 2 , b > 2 and f be Lebesgue measurable on 2 2 p D. If |f (z)| (1−|z| ) dm(z) belongs to CMp then |Ta,b f (z)|2 (1−|z|2 )p+2a−2 dm(z) also belongs to CMp .

Isomorphism, Decomposition and Discreteness Accordingly, the operator T is bounded from L2 (D) to L2 (D). Once the function f in Tf is replaced by g(w) = (1 − |w|2 )p/2 |f (w)|1S(2I) (w), where 1E stands for the characteristic function of E, we find 2 Int1 g(w)k(z, w) dm(w) D D dm(z) D |g(z)|2 dm(z) |I|p µf,p CMp . To handle Int2 , we note that both 2n |I| |1 − wz|, ¯ z ∈ S(I), w ∈ S(2n+1 I) \ S(2n I), and S(2n I) (1 − |z|2 )c−2 dm(z) (2n |I|)c , c > 1, hold for all n ∈ N ∪ {0}. When writing ∞ S(2n+1 I) \ S(2n I), D \ S(2I) = n=1 and using a > (2 − p)/2 as well as p ∈ (0, 2), we then get by the H¨ older inequality Int2 ∞ S(I) n=1 S(2n+1 I)\S(2n I) ∞ −(a+b) (2 |I|) n S(I) S(2n+1 I) n=1 2a+p ∞ |I| |I| ∞ S(2n+1 I) −a− p 2 (2 |I|) n 2 |f (w)| dm(w) (1 − |w|2 )1−b −(a+b) (2 |I|) n n=1 2a+p |f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b µf,p S(2 2 |f (w)| dm(w) (1 − |w|2 )1−b n+1 I) 1 2 dm(z) (1 − |z|2 )2−p−2a dm(z) (1 − |z|2 )2−p−2a 2 2 n=1 ∞ |I|p 2−na 2 µf,p CMp .

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