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 Hilbert 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 non-commutative 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 K-functional 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 Hilbert space such that
tr(|x|p) < ∞. 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 (1∕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 Kaftal-Larson
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.
