An interpolation theorem for analytic families of operators acting on certain H^p spaces

Eugenio Hernandez

Vol. 110 (1984), No. 1, 113–118

DOI: 10.2140/pjm.1984.110.113

Abstract

The main objective of this paper is to obtain an interpolation theorem for families of operators acting on atomic H^p spaces, 0 < p ≤ 1. We prove that if 0 < p₀ < p₁ ≤ 1 and {T_z}, z ∈S = {z ∈ C∕0 ≤ Real z ≤ 1}, is an analytic and admissible family of linear transformations such that T_j+iy maps H^p_j into L^p_j, where −∞ < y < ∞, with norm not exceeding A_j, j = 0,1, then for all 𝜃, 0 ≤ 𝜃 ≤ 1, T_𝜃 maps H^r into L^r with norm not exceeding cA₀^1−𝜃A₁^𝜃, where 1∕r = (1 − 𝜃)∕p₀ + 𝜃∕p₁.

Mathematical Subject Classification 2000

Primary: 47B38

Secondary: 46E35, 46M35

Milestones

Received: 8 March 1982

Revised: 16 December 1982

Published: 1 January 1984