Interpolation between HpH^p Spaces and non-commutative generalizations, II

  • Gilles Pisier

    Texas A&M University, College Station, United States


We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the ˉ\bar{\partial}-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution of the ˉ\bar{\partial}-equation, which satisfies simultaneously a good LL_\infty estimate and a good L1L_1 estimate. This appears as a special case of our main result which can be stated as follows: Let (Ω,A,μ)(\Omega, \mathcal A, \mu) be any measure space. Consider a bounded operator u:H1L1(μ)u : H^1 \rightarrow L_1(\mu). Assume that on one hand uu admits an extension u1:L1L1(μ)u_1 : L^1 \rightarrow L_1(\mu) bounded with norm C1C_1, and on the other hand that uu admits an extension u:LL(μ)u_\infty : L^\infty \rightarrow L_\infty(\mu) bounded with norm CC_\infty. Then uu admits an extension u~\tilde{u} which is bounded simultaneously from L1L^1 into L1(μ)L_1(\mu) and from LL^\infty into L(μ)L_\infty(\mu) and satisfies

u~:LL(μ)CC\| \tilde{u}: L_\infty \rightarrow L_\infty(\mu) \| ≤ C C_\infty
u~:L1L1(μ)CC1\| \tilde{u}: L_1 \rightarrow L_1 (\mu) \| ≤ C C_1

where CC is a numerical constant.

Cite this article

Gilles Pisier, Interpolation between HpH^p Spaces and non-commutative generalizations, II. Rev. Mat. Iberoam. 9 (1993), no. 2, pp. 281–291

DOI 10.4171/RMI/137