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We develop a theory of Lp spaces based on outer measures rather thanmeasures. This theory includes the classical Lp theory on measure spaces asspecial case. It also covers parts of potential theory and Carleson embeddingtheorems. The theory turns out to be an elegant language to describe aspects ofclassical singular integral theory such as paraproduct estimates and T(1)theorems, and it is particularly useful for generalizations of singularintegral theory in time-frequency analysis. We formulate and prove ageneralized Carleson embedding theorem and give a relatively short reduction ofbasic estimates for the bilinear Hilbert transform to this new Carlesonembedding theorem.

Lp theory for outer measures and two themes of Lennart Carleson united