Interpolation between H p spaces and noncommutative generalizations. I
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We give an elementary proof that the Hp spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between H1 and H∞. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hubert transform and the classical factorisation of a function in Hp as a product of two functions in Hq and Hr with 1/q + 1/r = 1/p. This proof extends without any real extra difficulty to the noncommutative setting and to several Banach space valued extensions of Hp spaces. In particular, this proof easily extends to the couple Hp0(lq0), Hp1 (lq1), with 1 ≤ p0, p1, q0, q1 ≤ ∞. In that situation, we prove that the real interpolation spaces and the Kfunctional are induced (up to equivalence of norms) by the same objects for the couple LP0(lq0), Lp1(lq1). In another direction, let us denote by Cp the space of all compact operators x on Hubert space such that tr(xp) < ∞. Let Tp be the subspace of all upper triangular matrices relative to the canonical basis. If p = ∞, Cp is just the space of all compact operators. Our proof allows us to show for instance that the space HP(CP) (resp. Tp) is the interpolation space of parameter (l/p, p) between H1(C1) (resp. T1) and H∞(C∞) (resp. T∞). We also prove a similar result for the complex interpolation method. Moreover, extending a recent result of KaftalLarson and Weiss, we prove that the distance to the subspace of upper triangular matrices in C1 and C∞ can be essentially realized simultaneously by the same element. © 1992 by Pacific Journal of Mathematics.
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