# Ancient Hindu Geometry: The Science Of The Sulba by Bibhutibhushan Datta By Bibhutibhushan Datta

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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). As with the second estimate, suppose q ∈ [2, p + 2). Since F ∈ Hp , this function can be written as F = BG where G has no zeros with G Hp = F Hp and B is a Blaschke product. Accordingly, |F |p−q |F |q ≤ 2q−1 (|G|p |B|p−q |B |q + |B|p |G|p−q |G |q ).

2 When approximating the above integral by a Riemann sum, we are naturally led to sketching a decomposition result for functions in Qp . As with this aspect, we will still use B(z, r) = {w ∈ D : dD (z, w) < r} as the hyperbolic 0 < r-ball centered at z ∈ D. Moreover, for τ > 0 we say that a sequence of points {zj } in D is τ -separated, respectively τ -dense, provided ∞ inf dD (zj , zk ) ≥ τ, respectively D = j=k B(zj , τ ).

Given any Carleson box S(I). Under µf,p CMp < ∞ we make the following estimates: µTa,b f,p S(I) |Ta,b f (z)|2 (1 − |z|2 )p+2a−2 dm(z) = S(I) ≤ + S(I) S(I) S(2I) S(2I) |f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b + S(I) = D\S(2I) |f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b D\S(2I) 2 |f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b 2 dm(z) (1 − |z|2 )2−p−2a dm(z) (1 − |z|2 )2−p−2a 2 dm(z) (1 − |z|2 )2−p−2a Int1 + Int2 . For Int1 , we shall apply the classical Schur’s Lemma for bounded operators on L2 (D).