1992/12/04 by Gilles Pisier, Pisier, Gilles
Mathematics · #47D #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:47D
paper · pdf · doi:10.48550/arxiv.math/9212204
arxiv created 1992/12/04 · arxiv updated 2016/09/06
We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the \d-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 \d-equation, which satisfies simultaneously a good L_ı estimate and a good L1 estimate. This appears as a special case of our main result which can be stated as follows: Let (Ω,\calA,μ) be any measure space. Consider a bounded operator u:H1\ra L1(μ). Assume that on one hand u admits an extension u1:L1\ra L1(μ) bounded with norm C1, and on the other hand that u admits an extension u_ı:L^ı\ra L_ı(μ) bounded with norm C_ı. Then u admits an extension \wu which is bounded simultaneously from L1 into L1(μ) and from L^ı into L_ı(μ) and satisfies \eqalign‖ u\colon L_∞ → L_∞(μ)‖≤ CC_∞\cr ‖ u\colon L1→ L1(μ)‖≤ CC1 where C is a numerical constant.